3.10.34 \(\int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx\)
Optimal. Leaf size=347 \[ -\frac {\sqrt {2} \left (-\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c^3}-\frac {2 x^{3/2} (b B-A c)}{3 c^2}+\frac {2 B x^{5/2}}{5 c} \]
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Rubi [A] time = 4.54, antiderivative size = 347, 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 =
{824, 826, 1166, 205} \begin {gather*} -\frac {\sqrt {2} \left (-\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c^3}-\frac {2 x^{3/2} (b B-A c)}{3 c^2}+\frac {2 B x^{5/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^(5/2)*(A + B*x))/(a + b*x + c*x^2),x]
[Out]
(2*(b^2*B - A*b*c - a*B*c)*Sqrt[x])/c^3 - (2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B*x^(5/2))/(5*c) - (Sqrt[2]*(b^
3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 - (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4
*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) -
(Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 + (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/
Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*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 824
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 0]
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 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 {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx &=\frac {2 B x^{5/2}}{5 c}+\frac {\int \frac {x^{3/2} (-a B-(b B-A c) x)}{a+b x+c x^2} \, dx}{c}\\ &=-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {\int \frac {\sqrt {x} \left (a (b B-A c)+\left (b^2 B-A b c-a B c\right ) x\right )}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac {2 \left (b^2 B-A b c-a B c\right ) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {\int \frac {-a \left (b^2 B-A b c-a B c\right )-\left (b^3 B-A b^2 c-2 a b B c+a A c^2\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{c^3}\\ &=\frac {2 \left (b^2 B-A b c-a B c\right ) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {2 \operatorname {Subst}\left (\int \frac {-a \left (b^2 B-A b c-a B c\right )+\left (-b^3 B+A b^2 c+2 a b B c-a A c^2\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=\frac {2 \left (b^2 B-A b c-a B c\right ) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\left (b^3 B-A b^2 c-2 a b B c+a A c^2-\frac {b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt {b^2-4 a c}}\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 )}{c^3}-\frac {\left (b^3 B-A b^2 c-2 a b B c+a A c^2+\frac {b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt {b^2-4 a c}}\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 )}{c^3}\\ &=\frac {2 \left (b^2 B-A b c-a B c\right ) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\sqrt {2} \left (b^3 B-A b^2 c-2 a b B c+a A c^2-\frac {b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\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 )}{c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b^3 B-A b^2 c-2 a b B c+a A c^2+\frac {b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\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 )}{c^{7/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 480, normalized size = 1.38 \begin {gather*} \frac {\sqrt {2} B \left (\frac {\left (\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}+2 a b c-b^3\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 (-\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}+2 a b c-b^3\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 )}{c^{7/2}}+\frac {\sqrt {2} A \left (\frac {3 a b c-b^3}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} A \left (\frac {b^3-3 a b c}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 B \sqrt {x} \left (b^2-a c\right )}{c^3}-\frac {2 A b \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}-\frac {2 b B x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^(5/2)*(A + B*x))/(a + b*x + c*x^2),x]
[Out]
(-2*A*b*Sqrt[x])/c^2 + (2*B*(b^2 - a*c)*Sqrt[x])/c^3 - (2*b*B*x^(3/2))/(3*c^2) + (2*A*x^(3/2))/(3*c) + (2*B*x^
(5/2))/(5*c) + (Sqrt[2]*A*(b^2 - a*c + (-b^3 + 3*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sq
rt[b - Sqrt[b^2 - 4*a*c]]])/(c^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*A*(b^2 - a*c + (b^3 - 3*a*b*c)/Sq
rt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(5/2)*Sqrt[b + Sqrt[b^2 - 4
*a*c]]) + (Sqrt[2]*B*(((-b^3 + 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/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]] + ((-b^3 + 2*a*b*c - (b^4 - 4*a*b^2*c +
2*a^2*c^2)/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]]))/c^(7/2)
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IntegrateAlgebraic [A] time = 1.10, size = 510, normalized size = 1.47 \begin {gather*} \frac {\left (2 \sqrt {2} a^2 B c^2-\sqrt {2} a A c^2 \sqrt {b^2-4 a c}+\sqrt {2} A b^2 c \sqrt {b^2-4 a c}+3 \sqrt {2} a A b c^2-4 \sqrt {2} a b^2 B c+2 \sqrt {2} a b B c \sqrt {b^2-4 a c}-\sqrt {2} b^3 B \sqrt {b^2-4 a c}-\sqrt {2} A b^3 c+\sqrt {2} b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{7/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (-2 \sqrt {2} a^2 B c^2-\sqrt {2} a A c^2 \sqrt {b^2-4 a c}+\sqrt {2} A b^2 c \sqrt {b^2-4 a c}-3 \sqrt {2} a A b c^2+4 \sqrt {2} a b^2 B c+2 \sqrt {2} a b B c \sqrt {b^2-4 a c}-\sqrt {2} b^3 B \sqrt {b^2-4 a c}+\sqrt {2} A b^3 c-\sqrt {2} b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 \sqrt {x} \left (-15 a B c-15 A b c+5 A c^2 x+15 b^2 B-5 b B c x+3 B c^2 x^2\right )}{15 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x^(5/2)*(A + B*x))/(a + b*x + c*x^2),x]
[Out]
(2*Sqrt[x]*(15*b^2*B - 15*A*b*c - 15*a*B*c - 5*b*B*c*x + 5*A*c^2*x + 3*B*c^2*x^2))/(15*c^3) + ((Sqrt[2]*b^4*B
- Sqrt[2]*A*b^3*c - 4*Sqrt[2]*a*b^2*B*c + 3*Sqrt[2]*a*A*b*c^2 + 2*Sqrt[2]*a^2*B*c^2 - Sqrt[2]*b^3*B*Sqrt[b^2 -
4*a*c] + Sqrt[2]*A*b^2*c*Sqrt[b^2 - 4*a*c] + 2*Sqrt[2]*a*b*B*c*Sqrt[b^2 - 4*a*c] - Sqrt[2]*a*A*c^2*Sqrt[b^2 -
4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sq
rt[b^2 - 4*a*c]]) + ((-(Sqrt[2]*b^4*B) + Sqrt[2]*A*b^3*c + 4*Sqrt[2]*a*b^2*B*c - 3*Sqrt[2]*a*A*b*c^2 - 2*Sqrt[
2]*a^2*B*c^2 - Sqrt[2]*b^3*B*Sqrt[b^2 - 4*a*c] + Sqrt[2]*A*b^2*c*Sqrt[b^2 - 4*a*c] + 2*Sqrt[2]*a*b*B*c*Sqrt[b^
2 - 4*a*c] - Sqrt[2]*a*A*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])
/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])
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fricas [B] time = 8.66, size = 7707, normalized size = 22.21
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
1/30*(15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*
b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*s
qrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b
+ 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B
^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 +
28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 +
(37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^
15)))/(b^2*c^7 - 4*a*c^8))*log(sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b
+ 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^
5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7
+ 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c - (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*a*b^2)*c^9
- (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7
+ (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2
+ 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^
3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^
2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^
3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*
b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^
7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 1
2*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*
b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2
*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3
*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2
*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A
^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*
a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A
^2*B^2*a^2*b^6)*c)*sqrt(x)) - 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 1
8*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c
+ (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B
^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*
A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5
+ 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b
^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^1
1)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(-sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4
*B^3*a^5 + 28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 + 58*A
^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3
*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c - (B*b^5*c^7 - 8*A*a^2*c^10 + 6*
(2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a
^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4
)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B
^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*
A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 -
2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (
7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5
+ 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^
3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^
4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 16
0*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 +
24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^1
0 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c
^5 - (7*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^3 - A
^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^
4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*sqrt(x)) + 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*
b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7
*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b
+ 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6
- 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^
4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3
*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5
*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(sqrt(2)*(B^3*b^10 + 4*(A^2*B*a
^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 +
87*A*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A^2*B*a*b^6
+ A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c + (B*b^5
*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*
(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*
a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 +
3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 -
2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9
+ 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^
3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2
*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 +
6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^
4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^
5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*
b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^
10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A
*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^
2 - 11*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*
b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*sqrt(x)) - 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 +
(4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a
*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*
B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b
^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4
*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(3
1*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*
A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(-sqrt(
2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3
)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*
a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2 - (10*B^3*a*b^8
+ 3*A*B^2*b^9)*c + (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^
4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^
2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*
b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3
*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4
*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*s
qrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2
*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*a^
4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 +
52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*
a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*
b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A
*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c
^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3
*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B
^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*sqrt(x)) + 4*(3*B
*c^2*x^2 + 15*B*b^2 - 15*(B*a + A*b)*c - 5*(B*b*c - A*c^2)*x)*sqrt(x))/c^3
________________________________________________________________________________________
giac [B] time = 1.36, size = 5319, normalized size = 15.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x, algorithm="giac")
[Out]
1/4*((2*b^6*c^3 - 18*a*b^4*c^4 + 48*a^2*b^2*c^5 - 32*a^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*b^6*c + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + 2*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c
)*a^2*b^2*c^3 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^
3*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^5
- 2*(b^2 - 4*a*c)*b^4*c^3 + 10*(b^2 - 4*a*c)*a*b^2*c^4 - 8*(b^2 - 4*a*c)*a^2*c^5)*A*c^2 - (2*b^7*c^2 - 20*a*b^
5*c^3 + 64*a^2*b^3*c^4 - 64*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^7 + 10*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b
^2 - 4*a*c)*c)*b^6*c - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 - 12*sqrt(2)*s
qrt(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^2 + 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 + 16*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a*b^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^4 - 2*(b^2 - 4*a*c)*b^5
*c^2 + 12*(b^2 - 4*a*c)*a*b^3*c^3 - 16*(b^2 - 4*a*c)*a^2*b*c^4)*B*c^2 + 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a*b^5*c^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a*b^4*c^4 + 2*a*b^5*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^5 + 8*sqrt(2)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^2*b^2*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^5 - 16*a^2*b^3*c^5 - 4*sqrt(2)
*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^6 + 32*a^3*b*c^6 - 2*(b^2 - 4*a*c)*a*b^3*c^4 + 8*(b^2 - 4*a*c)*a^2*b*
c^5)*A*abs(c) - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c^2 - 9*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*
c)*a^2*b^4*c^3 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^3 + 2*a*b^6*c^3 + 24*sqrt(2)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^3*b^2*c^4 + 10*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^4 + sqrt(2)*sqrt(b*c - sq
rt(b^2 - 4*a*c)*c)*a*b^4*c^4 - 18*a^2*b^4*c^4 - 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^5 - 8*sqrt(2)
*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^5 - 5*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 + 48*a^3*b^
2*c^5 + 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^6 - 32*a^4*c^6 - 2*(b^2 - 4*a*c)*a*b^4*c^3 + 10*(b^2 -
4*a*c)*a^2*b^2*c^4 - 8*(b^2 - 4*a*c)*a^3*c^5)*B*abs(c) - (2*b^6*c^5 - 14*a*b^4*c^6 + 24*a^2*b^2*c^7 - sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c^3 + 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*a*b^4*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^4 - 12*sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 - 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*
a*c)*c)*a*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^5 + 3*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^6 - 2*(b^2 - 4*a*c)*b^4*c^5 + 6*(b^2 - 4*a*c)*a*b^2*c^6)*A + (2*
b^7*c^4 - 16*a*b^5*c^5 + 36*a^2*b^3*c^6 - 16*a^3*b*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*b^7*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c^3 - 18*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^
2*b^3*c^4 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^
5 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt
(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^5 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^6
- 2*(b^2 - 4*a*c)*b^5*c^4 + 8*(b^2 - 4*a*c)*a*b^3*c^5 - 4*(b^2 - 4*a*c)*a^2*b*c^6)*B)*arctan(2*sqrt(1/2)*sqrt(
x)/sqrt((b*c^5 + sqrt(b^2*c^10 - 4*a*c^11))/c^6))/((a*b^4*c^5 - 8*a^2*b^2*c^6 - 2*a*b^3*c^6 + 16*a^3*c^7 + 8*a
^2*b*c^7 + a*b^2*c^7 - 4*a^2*c^8)*c^2) - 1/4*((2*b^6*c^3 - 18*a*b^4*c^4 + 48*a^2*b^2*c^5 - 32*a^3*c^6 - sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a*b^4*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 24*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a*b^3*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 16*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^2*b*c^4 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 - 2*(b^2 - 4*a*c)*b^4*c^3 + 10*(b^2 - 4*a*c)*a*b^2*c^4 - 8*(b^2 - 4*a
*c)*a^2*c^5)*A*c^2 - (2*b^7*c^2 - 20*a*b^5*c^3 + 64*a^2*b^3*c^4 - 64*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*b^7 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c + 2*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b
^2 - 4*a*c)*c)*a^2*b^3*c^2 - 12*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^2 + 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^3*b*c^3 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 6*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^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^4 - 2*(b^2 - 4*a*c)*b^5*c^2 + 12*(b^2 - 4*a*c)*a*b^3*c^3 - 16*(b^2 - 4*a*c)*a^2*b*c^4)*B*c^2
- 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^
4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^4 - 2*a*b^5*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^3*b*c^5 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a*b^3*c^5 + 16*a^2*b^3*c^5 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^6 - 32*a^3*b*c^6 + 2*(b^2 -
4*a*c)*a*b^3*c^4 - 8*(b^2 - 4*a*c)*a^2*b*c^5)*A*abs(c) + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c^2
- 9*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^3
- 2*a*b^6*c^3 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^4 + 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*
c)*c)*a^2*b^3*c^4 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^4 + 18*a^2*b^4*c^4 - 16*sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^4*c^5 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^5 - 5*sqrt(2)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*a^2*b^2*c^5 - 48*a^3*b^2*c^5 + 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^6 + 32*a^4*c^
6 + 2*(b^2 - 4*a*c)*a*b^4*c^3 - 10*(b^2 - 4*a*c)*a^2*b^2*c^4 + 8*(b^2 - 4*a*c)*a^3*c^5)*B*abs(c) - (2*b^6*c^5
- 14*a*b^4*c^6 + 24*a^2*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^3 + 7*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*b^5*c^4 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 - 6*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*b^4*c^5 + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^6 - 2*(b^2 - 4*a*c)*b^4
*c^5 + 6*(b^2 - 4*a*c)*a*b^2*c^6)*A + (2*b^7*c^4 - 16*a*b^5*c^5 + 36*a^2*b^3*c^6 - 16*a^3*b*c^7 - sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*
c)*c)*a*b^5*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^3 - 18*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^4 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a*b^4*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^5 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b
^2*c^5 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^5 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^6 - 2*(b^2 - 4*a*c)*b^5*c^4 + 8*(b^2 - 4*a*c)*a*b^3*c^5 - 4*(b^2 - 4*a*
c)*a^2*b*c^6)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((b*c^5 - sqrt(b^2*c^10 - 4*a*c^11))/c^6))/((a*b^4*c^5 - 8*a^2
*b^2*c^6 - 2*a*b^3*c^6 + 16*a^3*c^7 + 8*a^2*b*c^7 + a*b^2*c^7 - 4*a^2*c^8)*c^2) + 2/15*(3*B*c^4*x^(5/2) - 5*B*
b*c^3*x^(3/2) + 5*A*c^4*x^(3/2) + 15*B*b^2*c^2*sqrt(x) - 15*B*a*c^3*sqrt(x) - 15*A*b*c^3*sqrt(x))/c^5
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maple [B] time = 0.15, size = 1141, normalized size = 3.29
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x)
[Out]
2/5*B/c*x^(5/2)+2/3*A/c*x^(3/2)-2/3/c^2*B*x^(3/2)*b-2/c^2*A*b*x^(1/2)-2*B*a/c^2*x^(1/2)+2/c^3*b^2*B*x^(1/2)-1/
c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*a+1/c^
2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^2-3/
c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))
*c)^(1/2))*a*A*b+1/c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((
b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3+2/c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/
((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*B-1/c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2
)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*B-2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*B+4/c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*B-1/c^3/(-4*a*c+b^2)^(1/2)*2
^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4*B+1/c*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*a-1/c^
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^2
-3/c/(-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*A*b+1/c^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^3-2/c^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*B+1/c^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))*b^3*B-2/c/(-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^2*B+4/c^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^2*B-1/c^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))*b^4*B
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left (3 \, B c x^{\frac {5}{2}} - 5 \, {\left (B b - A c\right )} x^{\frac {3}{2}}\right )}}{15 \, c^{2}} - \int \frac {{\left (A b c - {\left (b^{2} - a c\right )} B\right )} x^{\frac {3}{2}} - {\left (B a b - A a c\right )} \sqrt {x}}{c^{3} x^{2} + b c^{2} x + a c^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
2/15*(3*B*c*x^(5/2) - 5*(B*b - A*c)*x^(3/2))/c^2 - integrate(((A*b*c - (b^2 - a*c)*B)*x^(3/2) - (B*a*b - A*a*c
)*sqrt(x))/(c^3*x^2 + b*c^2*x + a*c^2), x)
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mupad [B] time = 3.19, size = 14120, normalized size = 40.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^(5/2)*(A + B*x))/(a + b*x + c*x^2),x)
[Out]
x^(3/2)*((2*A)/(3*c) - (2*B*b)/(3*c^2)) - x^(1/2)*((b*((2*A)/c - (2*B*b)/c^2))/c + (2*B*a)/c^2) + atan(((((8*(
4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b*c^6 + B*a*b^4*c^4 - 5*B*a^2*b^2*c^5))/c^5 - (8*x^(1/2)*(b^3*c^7 - 4*a*b*
c^8)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*
c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2)
- B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5
+ 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*
B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5
*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1
/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^
3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B
^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5
- 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3
)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(
-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c -
b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (
8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3*c^5 + A^2*b^6*c^2 + 2*B^2*a^4*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^
2*b^4*c^2 - 16*B^2*a^3*b^2*c^3 - 8*B^2*a*b^6*c - 6*A^2*a*b^4*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*
A*B*a^3*b*c^4))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^
3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c
- b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 2
0*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*
a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6
*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*
a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i - (((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*
A*a^2*b*c^6 + B*a*b^4*c^4 - 5*B*a^2*b^2*c^5))/c^5 + (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^9 + A^2*b^7*c^2
+ B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) +
42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^
3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2
*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)
^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) +
8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7
- 8*a*b^2*c^8)))^(1/2))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^
2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(
-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5
*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 +
76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A
*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c
^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3
*c^5 + A^2*b^6*c^2 + 2*B^2*a^4*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16*B^2*a^3*b^2*c^3
- 8*B^2*a*b^6*c - 6*A^2*a*b^4*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4))/c^5)*(-(B^2*b^
9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c -
b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*
(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b
*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-
(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c -
b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^
2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i)/((((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b*c^6 + B*a*b^4*c^4 - 5*
B*a^2*b^2*c^5))/c^5 - (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(
1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a
^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11
*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1
/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*
a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2
)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/c^5)*(
-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-
(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*
a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B
^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*
b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(
4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(
2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3*c^5 + A^2*b^6*c^2 + 2*B^2*a^4
*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16*B^2*a^3*b^2*c^3 - 8*B^2*a*b^6*c - 6*A^2*a*b^4
*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4
*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^
5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*
A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*
a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2
*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^
2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)
))^(1/2) - (16*(A^3*a^4*c^3 + B^3*a^4*b^3 - A^3*a^3*b^2*c^2 - 2*B^3*a^5*b*c - A*B^2*a^3*b^4 + A*B^2*a^5*c^2 +
A*B^2*a^4*b^2*c + 2*A^2*B*a^3*b^3*c - 3*A^2*B*a^4*b*c^2))/c^5 + (((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b*c^
6 + B*a*b^4*c^4 - 5*B*a^2*b^2*c^5))/c^5 + (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*
(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^
2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) -
16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(
-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3
*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^
3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*
c^8)))^(1/2))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*
c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c -
b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*
A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^
3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c
^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*
c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3*c^5 + A^2
*b^6*c^2 + 2*B^2*a^4*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16*B^2*a^3*b^2*c^3 - 8*B^2*a
*b^6*c - 6*A^2*a*b^4*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4))/c^5)*(-(B^2*b^9 + A^2*b^
7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(
1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c -
b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B
^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b
^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1
/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^
4*c^7 - 8*a*b^2*c^8)))^(1/2)))*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*
A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2
*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b
^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3
+ 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20
*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b
*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*2i + atan(((((8*(4*B*a^3*c^6 -
A*a*b^3*c^5 + 4*A*a^2*b*c^6 + B*a*b^4*c^4 - 5*B*a^2*b^2*c^5))/c^5 - (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^
9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c -
b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*
(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b
*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-
(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c -
b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^
2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A
*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3
- A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7
*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A
*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)
^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2)
+ 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*x^(1/2)*(B^2
*b^8 - 2*A^2*a^3*c^5 + A^2*b^6*c^2 + 2*B^2*a^4*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16
*B^2*a^3*b^2*c^3 - 8*B^2*a*b^6*c - 6*A^2*a*b^4*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4)
)/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^
2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2
) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^
5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 +
5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b
^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^
(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i - (((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b*c^6 +
B*a*b^4*c^4 - 5*B*a^2*b^2*c^5))/c^5 + (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4
*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^
5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*
A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*
a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2
*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^
2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)
))^(1/2))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4
- A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)
^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*
a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^
2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 +
2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c -
b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3*c^5 + A^2*b^6
*c^2 + 2*B^2*a^4*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16*B^2*a^3*b^2*c^3 - 8*B^2*a*b^6
*c - 6*A^2*a*b^4*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4))/c^5)*(-(B^2*b^9 + A^2*b^7*c^
2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2)
+ 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2
)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a
^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^
3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2)
- 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^
7 - 8*a*b^2*c^8)))^(1/2)*1i)/((((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b*c^6 + B*a*b^4*c^4 - 5*B*a^2*b^2*c^5)
)/c^5 - (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b
^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A
^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c -
9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a
^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^
(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6
*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/c^5)*(-(B^2*b^9 + A^
2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^
3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a
*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 -
6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c
- b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3
)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9
+ b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3*c^5 + A^2*b^6*c^2 + 2*B^2*a^4*c^4 - 2*A*B*b
^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16*B^2*a^3*b^2*c^3 - 8*B^2*a*b^6*c - 6*A^2*a*b^4*c^3 - 28*A*B*
a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)
^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2
*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 -
11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^
(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(
4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b
^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (16
*(A^3*a^4*c^3 + B^3*a^4*b^3 - A^3*a^3*b^2*c^2 - 2*B^3*a^5*b*c - A*B^2*a^3*b^4 + A*B^2*a^5*c^2 + A*B^2*a^4*b^2*
c + 2*A^2*B*a^3*b^3*c - 3*A^2*B*a^4*b*c^2))/c^5 + (((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b*c^6 + B*a*b^4*c^
4 - 5*B*a^2*b^2*c^5))/c^5 + (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^8)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2
)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63
*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^
5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)
^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3
*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c
- b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/
c^5)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*
c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2)
+ B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5
+ 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*
B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5
*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1
/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*x^(1/2)*(B^2*b^8 - 2*A^2*a^3*c^5 + A^2*b^6*c^2 + 2*B
^2*a^4*c^4 - 2*A*B*b^7*c + 9*A^2*a^2*b^2*c^4 + 20*B^2*a^2*b^4*c^2 - 16*B^2*a^3*b^2*c^3 - 8*B^2*a*b^6*c - 6*A^2
*a*b^4*c^3 - 28*A*B*a^2*b^3*c^3 + 14*A*B*a*b^5*c^2 + 14*A*B*a^3*b*c^4))/c^5)*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^
6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*
a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2)
- 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2
*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) +
3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 + 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*
b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^
2*c^8)))^(1/2)))*(-(B^2*b^9 + A^2*b^7*c^2 - B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^
4 - A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 - A^2*b^4*c^2*(-(4*a*c - b^
2)^3)^(1/2) + B^2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^
2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 - 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*
b^2*c^4 + 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2
+ 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*A*B*a^2*b*c^3*(-(4*a*c
- b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*2i + (2*B*x^(5/2))/(5*c)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x+a),x)
[Out]
Timed out
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