3.10.36 \(\int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=221 \[ -\frac {\sqrt {2} \left (-\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 B \sqrt {x}}{c} \]

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Rubi [A]  time = 0.81, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, 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 B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 B \sqrt {x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[x])/c - (Sqrt[2]*(b*B - A*c - (b^2*B - A*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*S
qrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b*B - A*c + (b^2*B - A
*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqr
t[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 {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx &=\frac {2 B \sqrt {x}}{c}+\frac {\int \frac {-a B-(b B-A c) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{c}\\ &=\frac {2 B \sqrt {x}}{c}+\frac {2 \operatorname {Subst}\left (\int \frac {-a B+(-b B+A c) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 B \sqrt {x}}{c}-\frac {\left (b B-A c+\frac {b^2 B-A b c-2 a B c}{\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}+\frac {\left (-b B+A c+\frac {b^2 B-A b c-2 a B c}{\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}\\ &=\frac {2 B \sqrt {x}}{c}-\frac {\sqrt {2} \left (b B-A c-\frac {b^2 B-A b c-2 a B c}{\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^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b B-A c+\frac {b^2 B-A b c-2 a B c}{\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^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 264, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {2} \left (-A c \sqrt {b^2-4 a c}+b B \sqrt {b^2-4 a c}+2 a B c+A b c+b^2 (-B)\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (-A c \sqrt {b^2-4 a c}+b B \sqrt {b^2-4 a c}-2 a B c-A b c+b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 B \sqrt {x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.57, size = 302, normalized size = 1.37 \begin {gather*} \frac {\left (\sqrt {2} A c \sqrt {b^2-4 a c}-\sqrt {2} b B \sqrt {b^2-4 a c}-2 \sqrt {2} a B c-\sqrt {2} A b c+\sqrt {2} b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\sqrt {2} A c \sqrt {b^2-4 a c}-\sqrt {2} b B \sqrt {b^2-4 a c}+2 \sqrt {2} a B c+\sqrt {2} A b c-\sqrt {2} b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 B \sqrt {x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.96, size = 2642, normalized size = 11.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*c*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((
B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2
+ 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^
2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2 - (5*B^3*a*b^2 + 2*A*B^2*b^3)*c - (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2
)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 -
2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*
A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*
a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*
a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) - sqrt(2)*c*sq
rt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4
- 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c
)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b
+ A^2*B*b^2)*c^2 - (5*B^3*a*b^2 + 2*A*B^2*b^3)*c - (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4
*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 +
2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c + (b^
2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*
b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^
3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) + sqrt(2)*c*sqrt(-(B^2*b^3 + (
4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a
+ 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a
*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2
- (5*B^3*a*b^2 + 2*A*B^2*b^3)*c + (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4*b^4 + A^4*c^4 -
2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(
b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*
sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4
*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2
+ A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) - sqrt(2)*c*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*
c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3
+ (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3
- 4*a*c^4))*log(-sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2 - (5*B^3*a*b^2 +
2*A*B^2*b^3)*c + (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2
*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^
7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 +
A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3
*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^
4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) + 4*B*sqrt(x))/c

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giac [B]  time = 1.18, size = 3186, normalized size = 14.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2*B*sqrt(x)/c + 1/4*((2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^2*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(
b^2 - 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c^4)*A*c^2 - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5 + 8*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 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^
2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*B*c^2 - 2*(sqrt(2)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 2*sqrt(2)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*a*b^4*c^3 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sqrt(
2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 16*a^2*b^2*
c^4 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 32*a^3*c^5 - 2*(b^2 - 4*a*c)*a*b^2*c^3 + 8*(b^2 - 4*
a*c)*a^2*c^4)*B*abs(c) - (2*b^4*c^5 - 8*a*b^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*
b^4*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^5 -
2*(b^2 - 4*a*c)*b^2*c^5)*A + (2*b^5*c^4 - 12*a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*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 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)
*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 4*(b^2 - 4*a*c)*a*b*c^5)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((b*c + sqr
t(b^2*c^2 - 4*a*c^3))/c^2))/((a*b^4*c^3 - 8*a^2*b^2*c^4 - 2*a*b^3*c^4 + 16*a^3*c^5 + 8*a^2*b*c^5 + a*b^2*c^5 -
4*a^2*c^6)*c^2) - 1/4*((2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^2*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 -
2*(b^2 - 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c^4)*A*c^2 - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 8*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 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2
*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*B*c^2 + 2*(sqrt(2)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 2*sqrt(2)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 2*a*b^4*c^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sq
rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 + 16*a^2*b
^2*c^4 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 - 32*a^3*c^5 + 2*(b^2 - 4*a*c)*a*b^2*c^3 - 8*(b^2 -
4*a*c)*a^2*c^4)*B*abs(c) - (2*b^4*c^5 - 8*a*b^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*b^4*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^5
- 2*(b^2 - 4*a*c)*b^2*c^5)*A + (2*b^5*c^4 - 12*a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*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 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 4*(b^2 - 4*a*c)*a*b*c^5)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((b*c -
sqrt(b^2*c^2 - 4*a*c^3))/c^2))/((a*b^4*c^3 - 8*a^2*b^2*c^4 - 2*a*b^3*c^4 + 16*a^3*c^5 + 8*a^2*b*c^5 + a*b^2*c^
5 - 4*a^2*c^6)*c^2)

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maple [B]  time = 0.08, size = 581, normalized size = 2.63 \begin {gather*} \frac {\sqrt {2}\, A b \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}}+\frac {\sqrt {2}\, A b \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}}+\frac {2 \sqrt {2}\, B a \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}}+\frac {2 \sqrt {2}\, B a \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}}-\frac {\sqrt {2}\, B \,b^{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}\, c}-\frac {\sqrt {2}\, B \,b^{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}\, c}-\frac {\sqrt {2}\, A \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}}+\frac {\sqrt {2}\, A \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}}+\frac {\sqrt {2}\, B b \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}\, c}-\frac {\sqrt {2}\, B b \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}\, c}+\frac {2 B \sqrt {x}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*x^(1/2)/(c*x^2+b*x+a),x)

[Out]

2*B/c*x^(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+1/(-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-1/c*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/(-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-1/c/(-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^2*B-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+1/(-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+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))*b*B+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-1/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))*b^2*B

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} \sqrt {x}}{c x^{2} + b x + a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.42, size = 6401, normalized size = 28.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)*(A + B*x))/(a + b*x + c*x^2),x)

[Out]

(2*B*x^(1/2))/c - atan(((((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c - (8*x^(1/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^
2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3
- 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c -
b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2
- A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*
a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(
1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (8*x^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 +
A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2
*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^
3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) +
12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c +
(8*x^(1/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*
c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^
(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8
*a*b^2*c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^
3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 1
2*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^
4)))^(1/2) + (8*x^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6
*A*B*a*b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/
2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*
a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(
1/2)*1i)/((((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c - (8*x^(1/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 - A^
2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c
^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2)
+ 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(
4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^
2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*
a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (8*x^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 +
2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c
- b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^
3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*
c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c + (8*x^(1/2)*(b^3*c
^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2)
- 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^
2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/
2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*
b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 +
2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (8*x
^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)
*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c
- 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B
*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (16*(A^3*a
*c^2 - B^3*a^2*b + A*B^2*a*b^2 + A*B^2*a^2*c - 2*A^2*B*a*b*c))/c))*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c
- b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b
^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2
*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i - atan(((((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c - (8*x^(1
/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2
)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) +
12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*
c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2
) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a
^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1
/2) - (8*x^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*
b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*
A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c
^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i
- (((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c + (8*x^(1/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(
-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*
B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*
B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c -
b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3
*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c
^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (8*x^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*
a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^
3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B
^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2
*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c - (8*x^(1/2)*(b^3*c^3 -
4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2
*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*
c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/
c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*
c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A
*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (8*x^(1/
2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)*(-(
B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16
*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c
*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (((8*(4*B*a^2*
c^3 - B*a*b^2*c^2))/c + (8*x^(1/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)
^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2
*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(
16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) +
B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4
*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^
5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (8*x^(1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b
^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^
2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c -
b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^
4*c^3 - 8*a*b^2*c^4)))^(1/2) - (16*(A^3*a*c^2 - B^3*a^2*b + A*B^2*a*b^2 + A*B^2*a^2*c - 2*A^2*B*a*b*c))/c))*(-
(B^2*b^5 + A^2*b^3*c^2 + A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 1
6*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c - B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 - 2*A*B*b*
c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i

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sympy [B]  time = 22.13, size = 14158, normalized size = 64.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x**(1/2)/(c*x**2+b*x+a),x)

[Out]

Piecewise((-I*A*log(-I*sqrt(b)*sqrt(1/c) + sqrt(x))/(sqrt(b)*c*sqrt(1/c)) + I*A*log(I*sqrt(b)*sqrt(1/c) + sqrt
(x))/(sqrt(b)*c*sqrt(1/c)) + I*B*sqrt(b)*log(-I*sqrt(b)*sqrt(1/c) + sqrt(x))/(c**2*sqrt(1/c)) - I*B*sqrt(b)*lo
g(I*sqrt(b)*sqrt(1/c) + sqrt(x))/(c**2*sqrt(1/c)) + 2*B*sqrt(x)/c, Eq(a, 0)), (-8*I*A*sqrt(b)*c**2*sqrt(x)*sqr
t(1/c)/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 2*sqrt(2)*A*b*c*log(-sqrt(2)*I*sqrt(b)*s
qrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) - 2*sqrt(2)*A*b*c*log(sqrt(
2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 4*sqrt(2)*A
*c**2*x*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1
/c)) - 4*sqrt(2)*A*c**2*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt
(b)*c**3*x*sqrt(1/c)) + 12*I*B*b**(3/2)*c*sqrt(x)*sqrt(1/c)/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*
sqrt(1/c)) + 16*I*B*sqrt(b)*c**2*x**(3/2)*sqrt(1/c)/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c
)) - 3*sqrt(2)*B*b**2*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)
*c**3*x*sqrt(1/c)) + 3*sqrt(2)*B*b**2*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c
) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) - 6*sqrt(2)*B*b*c*x*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3
/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 6*sqrt(2)*B*b*c*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqr
t(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)), Eq(a, b**2/(4*c))), (I*A*sqrt(a)*log(-I*sq
rt(a)*sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b)) - I*A*sqrt(a)*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b))
+ 2*A*sqrt(x)/b - I*B*a**(3/2)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**3*sqrt(1/b)) + I*B*a**(3/2)*log(I*sqrt(
a)*sqrt(1/b) + sqrt(x))/(b**3*sqrt(1/b)) - 2*B*a*sqrt(x)/b**2 + 2*B*x**(3/2)/(3*b), Eq(c, 0)), (2*sqrt(2)*A*a*
*2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a**2
*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-
4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c +
b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-
4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c)) - 2*sqrt(2)*A*a**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4
*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b*
*2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)
*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b
**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*
sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 2*sqrt(2)*A*a**2*c**3*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - s
qrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(
-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*
b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*s
qrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - 2*sqrt(2)*A*a**2*c**3*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c +
b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqr
t(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*
c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqr
t(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*A*a*b**2*c**
2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*
sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c
+ b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)
/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/
c)) + sqrt(2)*A*a*b**2*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**
3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(
-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c
)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-
b/c + sqrt(-4*a*c + b**2)/c)) - 4*sqrt(2)*A*a*b**2*c**2*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(
2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a
*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c*
*3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(
-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - s
qrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 4*sqrt(2)*A*a*b**2*c**2*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b*
*2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(
-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(
-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*A*a*b*c**2*sqr
t(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2
)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt
(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c
+ sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(
-4*a*c + b**2)/c)) + sqrt(2)*A*a*b*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + s
qrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(
-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*
b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*s
qrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - 2*sqrt(2)*A*a*b*c**2*sqrt(-4*a*c + b**2)*sqrt(
-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b
/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)
/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqr
t(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)
/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 2
*sqrt(2)*A*a*b*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - s
qrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) -
5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c +
b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a
*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**
2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*A*b**4*c*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) -
sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt
(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a
*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*
sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/
c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*A*b**4*c*sqrt(-b/c + sqrt(-4*a*c + b*
*2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2
)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4
*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4
*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*A*b**3*c*sqrt(-4
*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4
*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - s
qrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*
a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + s
qrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a
*c + b**2)/c)) - sqrt(2)*A*b**3*c*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)
*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3
*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b
/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqr
t(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*B*a**2*b*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2
)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/
c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a
*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b*
*2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a
*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*B*a**2*b*c**2*sqrt
(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-
b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2
)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sq
rt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2
)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) -
5*sqrt(2)*B*a**2*b*c**2*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**
2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sq
rt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sq
rt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c
+ sqrt(-4*a*c + b**2)/c)) + 5*sqrt(2)*B*a**2*b*c**2*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*s
qrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*s
qrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(
-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 8*B*a**2*c**3*sqrt(x)*sqrt(-b/c - sqrt(-4*a*c + b**2)/
c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*
sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/
c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt
(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*B*a**2*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - s
qrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqr
t(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt
(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c +
sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b*
*3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*B
*a**2*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*
c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*
c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sq
rt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2
)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqr
t(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*B*a**2*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*
log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sq
rt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c
) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c +
b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*B*a**2*c**2*sqrt(-4*a*c
+ b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**
2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(
-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c
+ b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(
-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c)) + 5*sqrt(2)*B*a*b**3*c*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4
*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b*
*2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)
*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b
**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*
sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - 5*sqrt(2)*B*a*b**3*c*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sq
rt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-
4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b
*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sq
rt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - 10*B*a*b**2*c**2*sqrt(x)*sqrt(-b/c - sqrt(-4*a*
c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sq
rt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3
*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**
2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-
b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 3*sqrt(2)*B*a*b**2*c*sqrt(-4*a*c + b**2)*sq
rt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt
(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b*
*2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*
sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b*
*2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c))
- 3*sqrt(2)*B*a*b**2*c*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c
- sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)
- 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*
c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-
4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c +
b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - 6*B*a*b*c**2*sqrt(x)*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*
c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sq
rt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3
*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**
2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-
b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)*B*b**5*sqrt(-b/c + sqrt(-4*a*c + b*
*2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2
)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4
*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4
*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*B*b**5*sqrt(-b/c
+ sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c -
sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*
sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b
/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)
+ b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 2*B*b
**4*c*sqrt(x)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/(4*a**2*c**4*sqrt(-b/c - s
qrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sq
rt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c
+ sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) +
b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) - sqrt(2)
*B*b**4*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c +
b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**
3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(
-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c
)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-
b/c + sqrt(-4*a*c + b**2)/c)) + sqrt(2)*B*b**4*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt
(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c
+ sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)
- 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**4
*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt(-4*a*c + b**2)*sq
rt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)) + 2*B*b**3*c*sqrt(x)*sqrt(-4*a*c + b**2)*
sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/(4*a**2*c**4*sqrt(-b/c - sqrt(-4*a*c + b
**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) - 5*a*b**2*c**3*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt
(-4*a*c + b**2)/c) - 3*a*b*c**3*sqrt(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c) + b**4*c**2*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c) + b**3*c**2*sqrt
(-4*a*c + b**2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)), True))

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