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3.1015 \(\int \frac {A+B x}{x^{7/2} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=307 \[ -\frac {2 \left (-a A c-a b B+A b^2\right )}{a^3 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (-\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a^3 \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 A}{5 a x^{5/2}} \]

[Out]

-2/5*A/a/x^(5/2)+2/3*(A*b-B*a)/a^2/x^(3/2)-2*(-A*a*c+A*b^2-B*a*b)/a^3/x^(1/2)-arctan(2^(1/2)*c^(1/2)*x^(1/2)/(
b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)*(A*b^2-a*b*B-a*A*c+(-a*B*(-2*a*c+b^2)+A*(-3*a*b*c+b^3))/(-4*a*c+b
^2)^(1/2))/a^3/(b-(-4*a*c+b^2)^(1/2))^(1/2)-arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/
2)*c^(1/2)*(A*b^2-a*b*B-a*A*c+(a*B*(-2*a*c+b^2)-A*(-3*a*b*c+b^3))/(-4*a*c+b^2)^(1/2))/a^3/(b+(-4*a*c+b^2)^(1/2
))^(1/2)

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Rubi [A]  time = 1.96, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {828, 826, 1166, 205} \[ -\frac {2 \left (-a A c-a b B+A b^2\right )}{a^3 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (-\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a^3 \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 A}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(7/2)*(a + b*x + c*x^2)),x]

[Out]

(-2*A)/(5*a*x^(5/2)) + (2*(A*b - a*B))/(3*a^2*x^(3/2)) - (2*(A*b^2 - a*b*B - a*A*c))/(a^3*Sqrt[x]) - (Sqrt[2]*
Sqrt[c]*(A*b^2 - a*b*B - a*A*c - (a*B*(b^2 - 2*a*c) - A*(b^3 - 3*a*b*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(A*b^2 - a*b
*B - a*A*c + (a*B*(b^2 - 2*a*c) - A*(b^3 - 3*a*b*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[
b + Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
 q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx &=-\frac {2 A}{5 a x^{5/2}}+\frac {\int \frac {-A b+a B-A c x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx}{a}\\ &=-\frac {2 A}{5 a x^{5/2}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}+\frac {\int \frac {-a b B+A \left (b^2-a c\right )+(A b-a B) c x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{a^2}\\ &=-\frac {2 A}{5 a x^{5/2}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{a^3 \sqrt {x}}+\frac {\int \frac {a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )-c \left (A b^2-a b B-a A c\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{a^3}\\ &=-\frac {2 A}{5 a x^{5/2}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{a^3 \sqrt {x}}+\frac {2 \operatorname {Subst}\left (\int \frac {a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )-c \left (A b^2-a b B-a A c\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=-\frac {2 A}{5 a x^{5/2}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{a^3 \sqrt {x}}-\frac {\left (c \left (A b^2-a b B-a A c-\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{a^3}-\frac {\left (c \left (A b^2-a b B-a A c+\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=-\frac {2 A}{5 a x^{5/2}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{a^3 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (A b^2-a b B-a A c-\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (A b^2-a b B-a A c+\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]  time = 0.79, size = 337, normalized size = 1.10 \[ \frac {-\frac {6 a^2 A}{x^{5/2}}+\frac {30 \left (a A c+a b B-A b^2\right )}{\sqrt {x}}-\frac {15 \sqrt {2} \sqrt {c} \left (-\frac {\left (a B \left (b \sqrt {b^2-4 a c}-2 a c+b^2\right )-A \left (b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}-3 a b c+b^3\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (A \left (-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}-3 a b c+b^3\right )+a B \left (b \sqrt {b^2-4 a c}+2 a c-b^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c}}+\frac {10 a (A b-a B)}{x^{3/2}}}{15 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(7/2)*(a + b*x + c*x^2)),x]

[Out]

((-6*a^2*A)/x^(5/2) + (10*a*(A*b - a*B))/x^(3/2) + (30*(-(A*b^2) + a*b*B + a*A*c))/Sqrt[x] - (15*Sqrt[2]*Sqrt[
c]*(-(((a*B*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c]) - A*(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*
a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((a*B*(-b
^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c]) + A*(b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[
(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c])/(15*a
^3)

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fricas [B]  time = 5.46, size = 7971, normalized size = 25.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/30*(15*sqrt(2)*a^3*x^3*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a
^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (a^7*b^2 - 4*
a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*
(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*
a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 +
31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^
2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a
^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(sqrt(2)*(B^3*a^3*b^8 - 3*A*B^2*a^2*b^9 + 3*A^2*B*a*b^10 - A^3*b
^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77*A^2*B*a^5*b^2 - 54*A^3*a^4*b^3)*c^4 -
 (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*B*a^4*b^4 - 77*A^3*a^3*b^5)*c^3 + (20*B^3*a^5*b^4 - 81*A*B^2*a^4
*b^5 + 105*A^2*B*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*A*B^2*a^3*b^7 + 30*A^2*B*a^2*b^8 - 11*A^3
*a*b^9)*c - (B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b - 3*A*a^9*b^2)*c^2 - (7*B*a^9*b^3 - 8*A*a^8*
b^4)*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2
*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B
*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 +
 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a
^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(
a^14*b^2 - 4*a^15*c)))*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4
*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (a^7*b^2 - 4*a^
8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A
^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^
5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31
*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*
b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^1
4*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) + 4*(A^4*a^3*c^7 + (5*A^3*B*a^3*b - 6*A^4*a^2*b^2)*c^6 - (B^4*a^5 - 7
*A*B^3*a^4*b + 9*A^2*B^2*a^3*b^2 + 2*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^5 + (3*B^4*a^4*b^2 - 11*A*B^3*a^3*b^3 + 12
*A^2*B^2*a^2*b^4 - 3*A^3*B*a*b^5 - A^4*b^6)*c^4 - (B^4*a^3*b^4 - 3*A*B^3*a^2*b^5 + 3*A^2*B^2*a*b^6 - A^3*B*b^7
)*c^3)*sqrt(x)) - 15*sqrt(2)*a^3*x^3*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^
3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (
a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*
a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2
 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*
B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7
 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a
*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(-sqrt(2)*(B^3*a^3*b^8 - 3*A*B^2*a^2*b^9 + 3*A^2*B*a
*b^10 - A^3*b^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77*A^2*B*a^5*b^2 - 54*A^3*a
^4*b^3)*c^4 - (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*B*a^4*b^4 - 77*A^3*a^3*b^5)*c^3 + (20*B^3*a^5*b^4 -
 81*A*B^2*a^4*b^5 + 105*A^2*B*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*A*B^2*a^3*b^7 + 30*A^2*B*a^2
*b^8 - 11*A^3*a*b^9)*c - (B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b - 3*A*a^9*b^2)*c^2 - (7*B*a^9*b
^3 - 8*A*a^8*b^4)*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^
4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b
^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^
3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b
^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4
*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3
 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (a
^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a
^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2
- 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B
*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7
+ 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*
b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) + 4*(A^4*a^3*c^7 + (5*A^3*B*a^3*b - 6*A^4*a^2*b^2)*c^6 -
 (B^4*a^5 - 7*A*B^3*a^4*b + 9*A^2*B^2*a^3*b^2 + 2*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^5 + (3*B^4*a^4*b^2 - 11*A*B^3
*a^3*b^3 + 12*A^2*B^2*a^2*b^4 - 3*A^3*B*a*b^5 - A^4*b^6)*c^4 - (B^4*a^3*b^4 - 3*A*B^3*a^2*b^5 + 3*A^2*B^2*a*b^
6 - A^3*B*b^7)*c^3)*sqrt(x)) + 15*sqrt(2)*a^3*x^3*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*
A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2
*a*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^
4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^
2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*
b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*
A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*
b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(sqrt(2)*(B^3*a^3*b^8 - 3*A*B^2*a^2*b^9
 + 3*A^2*B*a*b^10 - A^3*b^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77*A^2*B*a^5*b^
2 - 54*A^3*a^4*b^3)*c^4 - (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*B*a^4*b^4 - 77*A^3*a^3*b^5)*c^3 + (20*B
^3*a^5*b^4 - 81*A*B^2*a^4*b^5 + 105*A^2*B*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*A*B^2*a^3*b^7 +
30*A^2*B*a^2*b^8 - 11*A^3*a*b^9)*c + (B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b - 3*A*a^9*b^2)*c^2
- (7*B*a^9*b^3 - 8*A*a^8*b^4)*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A
^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A
^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5
*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116
*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2
*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^
2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a
*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*
b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*
B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^
4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^
3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^
9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) + 4*(A^4*a^3*c^7 + (5*A^3*B*a^3*b - 6*A^4*a^
2*b^2)*c^6 - (B^4*a^5 - 7*A*B^3*a^4*b + 9*A^2*B^2*a^3*b^2 + 2*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^5 + (3*B^4*a^4*b^
2 - 11*A*B^3*a^3*b^3 + 12*A^2*B^2*a^2*b^4 - 3*A^3*B*a*b^5 - A^4*b^6)*c^4 - (B^4*a^3*b^4 - 3*A*B^3*a^2*b^5 + 3*
A^2*B^2*a*b^6 - A^3*B*b^7)*c^3)*sqrt(x)) - 15*sqrt(2)*a^3*x^3*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*
A*B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2
*b^4 + 7*A^2*a*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B
*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a
^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*
A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^
4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 1
8*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(-sqrt(2)*(B^3*a^3*b^8 - 3*
A*B^2*a^2*b^9 + 3*A^2*B*a*b^10 - A^3*b^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77
*A^2*B*a^5*b^2 - 54*A^3*a^4*b^3)*c^4 - (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*B*a^4*b^4 - 77*A^3*a^3*b^5
)*c^3 + (20*B^3*a^5*b^4 - 81*A*B^2*a^4*b^5 + 105*A^2*B*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*A*B
^2*a^3*b^7 + 30*A^2*B*a^2*b^8 - 11*A^3*a*b^9)*c + (B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b - 3*A*
a^9*b^2)*c^2 - (7*B*a^9*b^3 - 8*A*a^8*b^4)*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3
*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3
*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 7
2*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*
a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 -
 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A
*B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*
b^4 + 7*A^2*a*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*
a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^
7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A
^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4
*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18
*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) + 4*(A^4*a^3*c^7 + (5*A^3*B*a^3
*b - 6*A^4*a^2*b^2)*c^6 - (B^4*a^5 - 7*A*B^3*a^4*b + 9*A^2*B^2*a^3*b^2 + 2*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^5 +
(3*B^4*a^4*b^2 - 11*A*B^3*a^3*b^3 + 12*A^2*B^2*a^2*b^4 - 3*A^3*B*a*b^5 - A^4*b^6)*c^4 - (B^4*a^3*b^4 - 3*A*B^3
*a^2*b^5 + 3*A^2*B^2*a*b^6 - A^3*B*b^7)*c^3)*sqrt(x)) - 4*(3*A*a^2 - 15*(B*a*b - A*b^2 + A*a*c)*x^2 + 5*(B*a^2
 - A*a*b)*x)*sqrt(x))/(a^3*x^3)

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giac [B]  time = 2.15, size = 5013, normalized size = 16.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-1/4*((2*b^6*c^2 - 18*a*b^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2
*b^2*c^2 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3
 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 2*(
b^2 - 4*a*c)*b^4*c^2 + 10*(b^2 - 4*a*c)*a*b^2*c^3 - 8*(b^2 - 4*a*c)*a^2*c^4)*A*a^2 - (2*a*b^5*c^2 - 16*a^2*b^3
*c^3 + 32*a^3*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5 + 8*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3
*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c^2 + 8*(
b^2 - 4*a*c)*a^2*b*c^3)*B*a^2 + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^7 - 10*sqrt(2)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a^2*b^5*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*a*b^7*c + 32*sqrt(2)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 12*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + sqrt(2)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + 20*a^2*b^5*c^2 - 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 -
16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3
 - 64*a^3*b^3*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 64*a^4*b*c^4 + 2*(b^2 - 4*a*c)*a*b^5
*c - 12*(b^2 - 4*a*c)*a^2*b^3*c^2 + 16*(b^2 - 4*a*c)*a^3*b*c^3)*A*abs(a) - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^2*b^6 - 9*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^2*b^5*c - 2*a^2*b^6*c + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 + 10*sqrt(2)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 18*a^3*b^4*c^2 - 16*sq
rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 5*sqrt(2
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 48*a^4*b^2*c^3 + 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4
*c^4 + 32*a^5*c^4 + 2*(b^2 - 4*a*c)*a^2*b^4*c - 10*(b^2 - 4*a*c)*a^3*b^2*c^2 + 8*(b^2 - 4*a*c)*a^4*c^3)*B*abs(
a) + (2*a^2*b^6*c^2 - 14*a^3*b^4*c^3 + 24*a^4*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^2*b^6 + 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^4*b^2*c^2 - 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^3*b^2*c^3 - 2*(b^2 - 4*a*c)*a^2*b^4*c^2 + 6*(b^2 - 4*a*c)*a^3*b^2*c^3)*A - (2*a^3*b^5*c^2 - 12*a^4*b^3*c^3 +
16*a^5*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b
^4*c - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^2 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c
^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 2*(b^2 - 4*a*c)*a^3*b^3*c^2 + 4*(
b^2 - 4*a*c)*a^4*b*c^3)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^3*b + sqrt(a^6*b^2 - 4*a^7*c))/(a^3*c)))/((a^5*b
^4 - 8*a^6*b^2*c - 2*a^5*b^3*c + 16*a^7*c^2 + 8*a^6*b*c^2 + a^5*b^2*c^2 - 4*a^6*c^3)*abs(a)*abs(c)) + 1/4*((2*
b^6*c^2 - 18*a*b^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*
c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2
- 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 8*sqrt
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a*b^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 2*(b^2 - 4*a
*c)*b^4*c^2 + 10*(b^2 - 4*a*c)*a*b^2*c^3 - 8*(b^2 - 4*a*c)*a^2*c^4)*A*a^2 - (2*a*b^5*c^2 - 16*a^2*b^3*c^3 + 32
*a^3*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c -
 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
 - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 4*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c^2 + 8*(b^2 - 4*a
*c)*a^2*b*c^3)*B*a^2 - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^7 - 10*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^2*b^5*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c + 2*a*b^7*c + 32*sqrt(2)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 12*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + sqrt(2)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a*b^5*c^2 - 20*a^2*b^5*c^2 - 32*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 16*sqrt(2
)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 6*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 + 64*a^3
*b^3*c^3 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 - 64*a^4*b*c^4 - 2*(b^2 - 4*a*c)*a*b^5*c + 12*(
b^2 - 4*a*c)*a^2*b^3*c^2 - 16*(b^2 - 4*a*c)*a^3*b*c^3)*A*abs(a) + 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^2*b^6 - 9*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b
^5*c + 2*a^2*b^6*c + 24*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 + 10*sqrt(2)*sqrt(b*c - sqrt(b^2 -
 4*a*c)*c)*a^3*b^3*c^2 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 - 18*a^3*b^4*c^2 - 16*sqrt(2)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*c^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 5*sqrt(2)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 + 48*a^4*b^2*c^3 + 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^4 - 32
*a^5*c^4 - 2*(b^2 - 4*a*c)*a^2*b^4*c + 10*(b^2 - 4*a*c)*a^3*b^2*c^2 - 8*(b^2 - 4*a*c)*a^4*c^3)*B*abs(a) + (2*a
^2*b^6*c^2 - 14*a^3*b^4*c^3 + 24*a^4*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b
^6 + 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^
2 - 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^
3 - 2*(b^2 - 4*a*c)*a^2*b^4*c^2 + 6*(b^2 - 4*a*c)*a^3*b^2*c^3)*A - (2*a^3*b^5*c^2 - 12*a^4*b^3*c^3 + 16*a^5*b*
c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
 - sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c - 8*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b*c^2 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 2*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 2*(b^2 - 4*a*c)*a^3*b^3*c^2 + 4*(b^2 - 4*a
*c)*a^4*b*c^3)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^3*b - sqrt(a^6*b^2 - 4*a^7*c))/(a^3*c)))/((a^5*b^4 - 8*a^
6*b^2*c - 2*a^5*b^3*c + 16*a^7*c^2 + 8*a^6*b*c^2 + a^5*b^2*c^2 - 4*a^6*c^3)*abs(a)*abs(c)) + 2/15*(15*B*a*b*x^
2 - 15*A*b^2*x^2 + 15*A*a*c*x^2 - 5*B*a^2*x + 5*A*a*b*x - 3*A*a^2)/(a^3*x^(5/2))

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maple [B]  time = 0.10, size = 913, normalized size = 2.97 \[ -\frac {3 \sqrt {2}\, A b \,c^{2} \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}-\frac {3 \sqrt {2}\, A b \,c^{2} \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {\sqrt {2}\, A \,b^{3} c \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}+\frac {\sqrt {2}\, A \,b^{3} c \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}+\frac {2 \sqrt {2}\, B \,c^{2} \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a}+\frac {2 \sqrt {2}\, B \,c^{2} \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a}-\frac {\sqrt {2}\, B \,b^{2} c \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}-\frac {\sqrt {2}\, B \,b^{2} c \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}-\frac {\sqrt {2}\, A \,c^{2} \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {\sqrt {2}\, A \,c^{2} \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {\sqrt {2}\, A \,b^{2} c \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}-\frac {\sqrt {2}\, A \,b^{2} c \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}-\frac {\sqrt {2}\, B b c \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {\sqrt {2}\, B b c \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {2 A c}{a^{2} \sqrt {x}}-\frac {2 A \,b^{2}}{a^{3} \sqrt {x}}+\frac {2 B b}{a^{2} \sqrt {x}}+\frac {2 A b}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 B}{3 a \,x^{\frac {3}{2}}}-\frac {2 A}{5 a \,x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(7/2)/(c*x^2+b*x+a),x)

[Out]

c^2/a^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A-
c/a^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^
2-3*c^2/a^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))
*c)^(1/2)*c*x^(1/2))*A*b+c/a^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+
(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^3+c/a^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((
b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b+2*c^2/a/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1
/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B-c/a^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+
b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^2-c^2/a^2*2^(1/2)/((-b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A+c/a^3*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^2-3*c^2/a^2/(-4*a*
c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(
1/2))*A*b+c/a^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)
^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^3-c/a^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c
+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b+2*c^2/a/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
anh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B-c/a^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^
(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^2-2/5*A/a/x^(5/2)+2/3/a^2/x^(
3/2)*A*b-2/3*B/a/x^(3/2)+2*A/a^2*c/x^(1/2)-2/a^3/x^(1/2)*A*b^2+2/a^2/x^(1/2)*B*b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (\frac {3 \, A a^{3}}{x^{\frac {5}{2}}} + 15 \, {\left ({\left (b^{3} - 2 \, a b c\right )} A - {\left (a b^{2} - a^{2} c\right )} B\right )} \sqrt {x} - \frac {15 \, {\left (B a^{2} b - {\left (a b^{2} - a^{2} c\right )} A\right )}}{\sqrt {x}} + \frac {5 \, {\left (B a^{3} - A a^{2} b\right )}}{x^{\frac {3}{2}}}\right )}}{15 \, a^{4}} - \int -\frac {{\left ({\left (b^{3} c - 2 \, a b c^{2}\right )} A - {\left (a b^{2} c - a^{2} c^{2}\right )} B\right )} x^{\frac {3}{2}} + {\left ({\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} A - {\left (a b^{3} - 2 \, a^{2} b c\right )} B\right )} \sqrt {x}}{a^{4} c x^{2} + a^{4} b x + a^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

-2/15*(3*A*a^3/x^(5/2) + 15*((b^3 - 2*a*b*c)*A - (a*b^2 - a^2*c)*B)*sqrt(x) - 15*(B*a^2*b - (a*b^2 - a^2*c)*A)
/sqrt(x) + 5*(B*a^3 - A*a^2*b)/x^(3/2))/a^4 - integrate(-(((b^3*c - 2*a*b*c^2)*A - (a*b^2*c - a^2*c^2)*B)*x^(3
/2) + ((b^4 - 3*a*b^2*c + a^2*c^2)*A - (a*b^3 - 2*a^2*b*c)*B)*sqrt(x))/(a^4*c*x^2 + a^4*b*x + a^5), x)

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mupad [B]  time = 4.15, size = 13983, normalized size = 45.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(7/2)*(a + b*x + c*x^2)),x)

[Out]

atan(((x^(1/2)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48*A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*
c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12*b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5
) + (-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^
3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 +
 B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c -
 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A
^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3
)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/
2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(x^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 + B^2
*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*
c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*
c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 +
6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c
- b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2
*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16
*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 32*B*a^15*c^4 + 64*A*a^14*b*c^4 + 8*A*a^12*b^5*c^2 - 48*A*a^13*b^3*c^3 - 8*B
*a^13*b^4*c^2 + 40*B*a^14*b^2*c^3))*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8
+ 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)
^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A
^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^
4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2)
 - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*
a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*1i + (x^(1/2)*(16*A^2*a^12
*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48*A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 +
32*B^2*a^12*b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^
7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(
4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2
)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a
^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^
3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c
+ 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^
2 - 8*a^8*b^2*c)))^(1/2)*(32*B*a^15*c^4 + x^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 + B^2*a^2*b^7 +
A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*
c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)
^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b
^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(
1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*
A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 -
8*a^8*b^2*c)))^(1/2) - 64*A*a^14*b*c^4 - 8*A*a^12*b^5*c^2 + 48*A*a^13*b^3*c^3 + 8*B*a^13*b^4*c^2 - 40*B*a^14*b
^2*c^3))*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A
^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*
c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^
5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3
- 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b
^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3
)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*1i)/((x^(1/2)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*
A^2*a^9*b^6*c^3 + 48*A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12*b^2*c^4 + 16*A*
B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)
^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2
*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4
 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^
3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*
(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c
- b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(x
^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*
b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c -
b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c +
28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^
3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(
1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*
A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 32*B*a^15*c^4 + 64*A
*a^14*b*c^4 + 8*A*a^12*b^5*c^2 - 48*A*a^13*b^3*c^3 - 8*B*a^13*b^4*c^2 + 40*B*a^14*b^2*c^3))*(-(A^2*b^9 + B^2*a
^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^
3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c
- b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*
A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c -
b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b
^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a
^9*c^2 - 8*a^8*b^2*c)))^(1/2) - (x^(1/2)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48*A^2*a^10*
b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12*b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b
^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A
^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1
/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4
*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2
+ 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A
*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*
c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(32*B*a^15*c^4 + x^(1/2)*(32*a^1
6*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a
^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2)
+ 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c
^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76
*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a
*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*
(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 64*A*a^14*b*c^4 - 8*A*a^12*b^5*c^2
 + 48*A*a^13*b^3*c^3 + 8*B*a^13*b^4*c^2 - 40*B*a^14*b^2*c^3))*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^
2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B
^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c
^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2
)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*
c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*
c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)
+ 16*A^3*a^10*c^7 - 16*A^3*a^9*b^2*c^6 + 16*A*B^2*a^11*c^6 + 16*B^3*a^11*b*c^5 - 32*A*B^2*a^10*b^2*c^5 + 16*A^
2*B*a^9*b^3*c^5))*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c
^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2
*a^4*b^3*c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B
^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4
*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(
4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c
 - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*2i + atan(((x^(1/2)*(16*A^2*a^12*c^6 - 16*B^
2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48*A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12*
b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*
(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)
^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) -
 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(
-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3
*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*
b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^
2*c)))^(1/2)*(x^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(
1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*
b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11
*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1
/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*
a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2
)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 32*B*
a^15*c^4 + 64*A*a^14*b*c^4 + 8*A*a^12*b^5*c^2 - 48*A*a^13*b^3*c^3 - 8*B*a^13*b^4*c^2 + 40*B*a^14*b^2*c^3))*(-(
A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c
^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^
4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2
*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^
4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2)
 + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*
(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*1i + (x^(1/2)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*
c^3 + 48*A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12*b^2*c^4 + 16*A*B*a^10*b^5*c
^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) -
2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(
4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*
b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6
6*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b
^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1
/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(32*B*a^15*c^4
 + x^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*
B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*
c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*
c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*
B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^
3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2)
+ 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 64*A*a^14*b*c^4
- 8*A*a^12*b^5*c^2 + 48*A*a^13*b^3*c^3 + 8*B*a^13*b^4*c^2 - 40*B*a^14*b^2*c^3))*(-(A^2*b^9 + B^2*a^2*b^7 - A^2
*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c -
 b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1
/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*
c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2
) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B
*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a
^8*b^2*c)))^(1/2)*1i)/((x^(1/2)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48*A^2*a^10*b^4*c^4 -
 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12*b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 +
80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^
5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*
B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 -
9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*
a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*
(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*
a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(x^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)
*(-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b
^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^
2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20
*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*
a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(
1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))
/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 32*B*a^15*c^4 + 64*A*a^14*b*c^4 + 8*A*a^12*b^5*c^2 - 48*A*a
^13*b^3*c^3 - 8*B*a^13*b^4*c^2 + 40*B*a^14*b^2*c^3))*(-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/
2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^
4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A
^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2
) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*
c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^
3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - (x^(1/2
)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48*A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a
^11*b^4*c^3 + 32*B^2*a^12*b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^
9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A
^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*
(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b
*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-
(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*
A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b
^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(32*B*a^15*c^4 + x^(1/2)*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(A^2*b^9 +
B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a
^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4
*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3
 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a
*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*
a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 +
 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) - 64*A*a^14*b*c^4 - 8*A*a^12*b^5*c^2 + 48*A*a^13*b^3*c^3 + 8*B*a^13*b^4*c^2
 - 40*B*a^14*b^2*c^3))*(-(A^2*b^9 + B^2*a^2*b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*
b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 2
5*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4
- 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*
B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^
5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(
4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2) + 16*A^3*a^10*c^7 - 16*A^3*a^9*b^2*c^6
+ 16*A*B^2*a^11*c^6 + 16*B^3*a^11*b*c^5 - 32*A*B^2*a^10*b^2*c^5 + 16*A^2*B*a^9*b^3*c^5))*(-(A^2*b^9 + B^2*a^2*
b^7 - A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 + A^2*a^3*c^3*(
-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 - B^2*a^4*c^2*(-(4*a*c - b
^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 - 6*A^2
*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 + 5*A^2*a*b^4*c*(-(4*a*c - b^2
)^3)^(1/2) + 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*
c - 8*A*B*a^2*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^7*b^4 + 16*a^9*
c^2 - 8*a^8*b^2*c)))^(1/2)*2i + ((2*x^2*(A*a*c - A*b^2 + B*a*b))/a^3 - (2*A)/(5*a) + (2*x*(A*b - B*a))/(3*a^2)
)/x^(5/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(7/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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