3.364 \(\int \frac{x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=491 \[ \frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right )}{2 c^2}+\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right )}{2 c^2}+\frac{e x \sqrt{d+e x^2}}{2 c}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c} \]

[Out]

(e*x*Sqrt[d + e*x^2])/(2*c) + (Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d - b*
e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[
b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^2*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d - b*e +
(b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2
- 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^2*Sqrt[b +
Sqrt[b^2 - 4*a*c]]) + (d*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c) + (
Sqrt[e]*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[
e]*x)/Sqrt[d + e*x^2]])/(2*c^2) + (Sqrt[e]*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e
)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2)

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Rubi [A]  time = 3.99574, antiderivative size = 491, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ \frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right )}{2 c^2}+\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right )}{2 c^2}+\frac{e x \sqrt{d+e x^2}}{2 c}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]

[Out]

(e*x*Sqrt[d + e*x^2])/(2*c) + (Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d - b*
e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[
b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^2*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d - b*e +
(b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2
- 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^2*Sqrt[b +
Sqrt[b^2 - 4*a*c]]) + (d*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c) + (
Sqrt[e]*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[
e]*x)/Sqrt[d + e*x^2]])/(2*c^2) + (Sqrt[e]*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e
)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.953538, size = 0, normalized size = 0. \[ \int \frac{x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(x^2*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]

[Out]

Integrate[(x^2*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4), x]

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Maple [C]  time = 0.042, size = 382, normalized size = 0.8 \[ -{\frac{{x}^{2}}{4\,c}{e}^{{\frac{3}{2}}}}+{\frac{ex}{4\,c}\sqrt{e{x}^{2}+d}}-{\frac{d}{8\,c}\sqrt{e}}+{\frac{1}{2\,{c}^{2}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( ac{e}^{2}-{b}^{2}{e}^{2}+2\,bcde-{c}^{2}{d}^{2} \right ){{\it \_R}}^{2}+2\, \left ( -2\,ab{e}^{3}+3\,acd{e}^{2}+{b}^{2}d{e}^{2}-2\,bc{d}^{2}e+{c}^{2}{d}^{3} \right ){\it \_R}+ac{d}^{2}{e}^{2}-{b}^{2}{d}^{2}{e}^{2}+2\,bc{d}^{3}e-{c}^{2}{d}^{4}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }}+{\frac{{d}^{2}}{8\,c}\sqrt{e} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-2}}+{\frac{b}{{c}^{2}}{e}^{{\frac{3}{2}}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) }-{\frac{3\,d}{2\,c}\sqrt{e}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x)

[Out]

-1/4*e^(3/2)/c*x^2+1/4*e*x*(e*x^2+d)^(1/2)/c-1/8*e^(1/2)/c*d+1/2*e^(1/2)/c^2*sum
(((a*c*e^2-b^2*e^2+2*b*c*d*e-c^2*d^2)*_R^2+2*(-2*a*b*e^3+3*a*c*d*e^2+b^2*d*e^2-2
*b*c*d^2*e+c^2*d^3)*_R+a*c*d^2*e^2-b^2*d^2*e^2+2*b*c*d^3*e-c^2*d^4)/(_R^3*c+3*_R
^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)*ln(((e*x^2+d)^
(1/2)-x*e^(1/2))^2-_R),_R=RootOf(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c
*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+c*d^4))+1/8*e^(1/2)/c*d^2/((e*x^2+d)^(1/2)-x*e
^(1/2))^2+e^(3/2)/c^2*ln((e*x^2+d)^(1/2)-x*e^(1/2))*b-3/2*e^(1/2)/c*ln((e*x^2+d)
^(1/2)-x*e^(1/2))*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{2}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x^2/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^(3/2)*x^2/(c*x^4 + b*x^2 + a), x)

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Fricas [A]  time = 99.5536, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x^2/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

[-1/4*(sqrt(1/2)*c^2*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c -
3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 + (b^2*c^4 - 4*a*c^5)*sqrt
((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11
*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c
- 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2
*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log((2*a*c^4*d^6 - 8*a*b*c^3*d^5*e + 4*(3
*a*b^2*c^2 - a^2*c^3)*d^4*e^2 - 4*(2*a*b^3*c - a^2*b*c^2)*d^3*e^3 + 2*(a*b^4 + a
^2*b^2*c - 3*a^3*c^2)*d^2*e^4 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e^5 + ((b^2*c^5 - 4*a*
c^6)*d^3 - (b^3*c^4 - 4*a*b*c^5)*d^2*e + (a*b^2*c^4 - 4*a^2*c^5)*d*e^2)*x^2*sqrt
((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11
*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c
- 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2
*c^8 - 4*a*c^9)) - (b*c^4*d^6 - 4*(b^2*c^3 + a*c^4)*d^5*e + 2*(3*b^3*c^2 + 7*a*b
*c^3)*d^4*e^2 - 2*(2*b^4*c + 11*a*b^2*c^2 - 4*a^2*c^3)*d^3*e^3 + (b^5 + 17*a*b^3
*c - 11*a^2*b*c^2)*d^2*e^4 - (5*a*b^4 + 2*a^2*b^2*c - 12*a^3*c^2)*d*e^5 + 4*(a^2
*b^3 - 2*a^3*b*c)*e^6)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((b^3*c^5 - 4*a*b*c^6)
*d - (b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*e)*x*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*
(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c
^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d
*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - ((b^2*c^4 -
4*a*c^5)*d^4 - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e + 3*(2*b^4*c^2 - 9*a*b^2*c^3 + 4*a
^2*c^4)*d^2*e^2 - (4*b^5*c - 21*a*b^3*c^2 + 20*a^2*b*c^3)*d*e^3 + (b^6 - 6*a*b^4
*c + 8*a^2*b^2*c^2)*e^4)*x)*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(
b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 + (b^2*c^4 - 4*a*c^
5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c
^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*
(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^
6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) - sqrt(1/2)*c^2*sqrt(-(b*c^3
*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^
2*c + 2*a^2*c^2)*e^3 + (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*
b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2
- 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^
5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*
c^5))*log((2*a*c^4*d^6 - 8*a*b*c^3*d^5*e + 4*(3*a*b^2*c^2 - a^2*c^3)*d^4*e^2 - 4
*(2*a*b^3*c - a^2*b*c^2)*d^3*e^3 + 2*(a*b^4 + a^2*b^2*c - 3*a^3*c^2)*d^2*e^4 - 2
*(a^2*b^3 - 2*a^3*b*c)*d*e^5 + ((b^2*c^5 - 4*a*c^6)*d^3 - (b^3*c^4 - 4*a*b*c^5)*
d^2*e + (a*b^2*c^4 - 4*a^2*c^5)*d*e^2)*x^2*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*
b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2
- 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^
5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - (b*c^4*d^6 - 4
*(b^2*c^3 + a*c^4)*d^5*e + 2*(3*b^3*c^2 + 7*a*b*c^3)*d^4*e^2 - 2*(2*b^4*c + 11*a
*b^2*c^2 - 4*a^2*c^3)*d^3*e^3 + (b^5 + 17*a*b^3*c - 11*a^2*b*c^2)*d^2*e^4 - (5*a
*b^4 + 2*a^2*b^2*c - 12*a^3*c^2)*d*e^5 + 4*(a^2*b^3 - 2*a^3*b*c)*e^6)*x^2 - 2*sq
rt(1/2)*sqrt(e*x^2 + d)*(((b^3*c^5 - 4*a*b*c^6)*d - (b^4*c^4 - 6*a*b^2*c^5 + 8*a
^2*c^6)*e)*x*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2
*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^
2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b
^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - ((b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c^3 - 4*a*
b*c^4)*d^3*e + 3*(2*b^4*c^2 - 9*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - (4*b^5*c - 21*a
*b^3*c^2 + 20*a^2*b*c^3)*d*e^3 + (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e^4)*x)*sqrt(
-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 -
4*a*b^2*c + 2*a^2*c^2)*e^3 + (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e
+ 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b
^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^
3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4
- 4*a*c^5)))/x^2) - sqrt(1/2)*c^2*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*
e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 - (b^2*c^4 -
4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(1
0*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e
^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*
c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log((2*a*c^4*d^6 - 8*a*b*c^
3*d^5*e + 4*(3*a*b^2*c^2 - a^2*c^3)*d^4*e^2 - 4*(2*a*b^3*c - a^2*b*c^2)*d^3*e^3
+ 2*(a*b^4 + a^2*b^2*c - 3*a^3*c^2)*d^2*e^4 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e^5 - ((
b^2*c^5 - 4*a*c^6)*d^3 - (b^3*c^4 - 4*a*b*c^5)*d^2*e + (a*b^2*c^4 - 4*a^2*c^5)*d
*e^2)*x^2*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(1
0*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e
^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*
c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - (b*c^4*d^6 - 4*(b^2*c^3 + a*c^4)*d^5*e + 2*(3*b
^3*c^2 + 7*a*b*c^3)*d^4*e^2 - 2*(2*b^4*c + 11*a*b^2*c^2 - 4*a^2*c^3)*d^3*e^3 + (
b^5 + 17*a*b^3*c - 11*a^2*b*c^2)*d^2*e^4 - (5*a*b^4 + 2*a^2*b^2*c - 12*a^3*c^2)*
d*e^5 + 4*(a^2*b^3 - 2*a^3*b*c)*e^6)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((b^3*c^
5 - 4*a*b*c^6)*d - (b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*e)*x*sqrt((c^6*d^6 - 6*b*
c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^
3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 +
2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9))
+ ((b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e + 3*(2*b^4*c^2 - 9*a
*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - (4*b^5*c - 21*a*b^3*c^2 + 20*a^2*b*c^3)*d*e^3 +
(b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e^4)*x)*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^
3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 - (b^
2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2
- 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4
)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a
^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) + sqrt(1/2)*c^
2*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 -
(b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 - (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5
*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 +
3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a
^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(
b^2*c^4 - 4*a*c^5))*log((2*a*c^4*d^6 - 8*a*b*c^3*d^5*e + 4*(3*a*b^2*c^2 - a^2*c^
3)*d^4*e^2 - 4*(2*a*b^3*c - a^2*b*c^2)*d^3*e^3 + 2*(a*b^4 + a^2*b^2*c - 3*a^3*c^
2)*d^2*e^4 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e^5 - ((b^2*c^5 - 4*a*c^6)*d^3 - (b^3*c^4
- 4*a*b*c^5)*d^2*e + (a*b^2*c^4 - 4*a^2*c^5)*d*e^2)*x^2*sqrt((c^6*d^6 - 6*b*c^5
*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 +
3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a
^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) -
(b*c^4*d^6 - 4*(b^2*c^3 + a*c^4)*d^5*e + 2*(3*b^3*c^2 + 7*a*b*c^3)*d^4*e^2 - 2*(
2*b^4*c + 11*a*b^2*c^2 - 4*a^2*c^3)*d^3*e^3 + (b^5 + 17*a*b^3*c - 11*a^2*b*c^2)*
d^2*e^4 - (5*a*b^4 + 2*a^2*b^2*c - 12*a^3*c^2)*d*e^5 + 4*(a^2*b^3 - 2*a^3*b*c)*e
^6)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((b^3*c^5 - 4*a*b*c^6)*d - (b^4*c^4 - 6*a
*b^2*c^5 + 8*a^2*c^6)*e)*x*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^
5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 +
3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b
^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) + ((b^2*c^4 - 4*a*c^5)*d^4 - 4*(
b^3*c^3 - 4*a*b*c^4)*d^3*e + 3*(2*b^4*c^2 - 9*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - (
4*b^5*c - 21*a*b^3*c^2 + 20*a^2*b*c^3)*d*e^3 + (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)
*e^4)*x)*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*
d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 - (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 -
6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^
3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^
2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c
^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) - 2*sqrt(e*x^2 + d)*c*e*x + (3*c*d - 2*b*e)*sqr
t(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d))/c^2, -1/4*(sqrt(1/2)*c^2*s
qrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b
^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 + (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^
5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*
(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*
b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2
*c^4 - 4*a*c^5))*log((2*a*c^4*d^6 - 8*a*b*c^3*d^5*e + 4*(3*a*b^2*c^2 - a^2*c^3)*
d^4*e^2 - 4*(2*a*b^3*c - a^2*b*c^2)*d^3*e^3 + 2*(a*b^4 + a^2*b^2*c - 3*a^3*c^2)*
d^2*e^4 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e^5 + ((b^2*c^5 - 4*a*c^6)*d^3 - (b^3*c^4 -
4*a*b*c^5)*d^2*e + (a*b^2*c^4 - 4*a^2*c^5)*d*e^2)*x^2*sqrt((c^6*d^6 - 6*b*c^5*d^
5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*
(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*
b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - (b*
c^4*d^6 - 4*(b^2*c^3 + a*c^4)*d^5*e + 2*(3*b^3*c^2 + 7*a*b*c^3)*d^4*e^2 - 2*(2*b
^4*c + 11*a*b^2*c^2 - 4*a^2*c^3)*d^3*e^3 + (b^5 + 17*a*b^3*c - 11*a^2*b*c^2)*d^2
*e^4 - (5*a*b^4 + 2*a^2*b^2*c - 12*a^3*c^2)*d*e^5 + 4*(a^2*b^3 - 2*a^3*b*c)*e^6)
*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((b^3*c^5 - 4*a*b*c^6)*d - (b^4*c^4 - 6*a*b^
2*c^5 + 8*a^2*c^6)*e)*x*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*
d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*
a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*
c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - ((b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3
*c^3 - 4*a*b*c^4)*d^3*e + 3*(2*b^4*c^2 - 9*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - (4*b
^5*c - 21*a*b^3*c^2 + 20*a^2*b*c^3)*d*e^3 + (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e^
4)*x)*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e
^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 + (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b
*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e
^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 +
2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)
))/(b^2*c^4 - 4*a*c^5)))/x^2) - sqrt(1/2)*c^2*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*
a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 +
(b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4
*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2
*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c +
4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log((2*a*c^4*d^6
- 8*a*b*c^3*d^5*e + 4*(3*a*b^2*c^2 - a^2*c^3)*d^4*e^2 - 4*(2*a*b^3*c - a^2*b*c^
2)*d^3*e^3 + 2*(a*b^4 + a^2*b^2*c - 3*a^3*c^2)*d^2*e^4 - 2*(a^2*b^3 - 2*a^3*b*c)
*d*e^5 + ((b^2*c^5 - 4*a*c^6)*d^3 - (b^3*c^4 - 4*a*b*c^5)*d^2*e + (a*b^2*c^4 - 4
*a^2*c^5)*d*e^2)*x^2*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4
*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2
*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c +
4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - (b*c^4*d^6 - 4*(b^2*c^3 + a*c^4)*d^5
*e + 2*(3*b^3*c^2 + 7*a*b*c^3)*d^4*e^2 - 2*(2*b^4*c + 11*a*b^2*c^2 - 4*a^2*c^3)*
d^3*e^3 + (b^5 + 17*a*b^3*c - 11*a^2*b*c^2)*d^2*e^4 - (5*a*b^4 + 2*a^2*b^2*c - 1
2*a^3*c^2)*d*e^5 + 4*(a^2*b^3 - 2*a^3*b*c)*e^6)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d
)*(((b^3*c^5 - 4*a*b*c^6)*d - (b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*e)*x*sqrt((c^6
*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*
c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a
*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8
- 4*a*c^9)) - ((b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e + 3*(2*b^
4*c^2 - 9*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - (4*b^5*c - 21*a*b^3*c^2 + 20*a^2*b*c^
3)*d*e^3 + (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e^4)*x)*sqrt(-(b*c^3*d^3 - 3*(b^2*c
^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2
)*e^3 + (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c
^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3
+ 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*
b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) - s
qrt(1/2)*c^2*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c
^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 - (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^
6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4
)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^
3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4
*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log((2*a*c^4*d^6 - 8*a*b*c^3*d^5*e + 4*(3*a*b^2*c
^2 - a^2*c^3)*d^4*e^2 - 4*(2*a*b^3*c - a^2*b*c^2)*d^3*e^3 + 2*(a*b^4 + a^2*b^2*c
- 3*a^3*c^2)*d^2*e^4 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e^5 - ((b^2*c^5 - 4*a*c^6)*d^3
- (b^3*c^4 - 4*a*b*c^5)*d^2*e + (a*b^2*c^4 - 4*a^2*c^5)*d*e^2)*x^2*sqrt((c^6*d^
6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4
)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^
3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4
*a*c^9)) - (b*c^4*d^6 - 4*(b^2*c^3 + a*c^4)*d^5*e + 2*(3*b^3*c^2 + 7*a*b*c^3)*d^
4*e^2 - 2*(2*b^4*c + 11*a*b^2*c^2 - 4*a^2*c^3)*d^3*e^3 + (b^5 + 17*a*b^3*c - 11*
a^2*b*c^2)*d^2*e^4 - (5*a*b^4 + 2*a^2*b^2*c - 12*a^3*c^2)*d*e^5 + 4*(a^2*b^3 - 2
*a^3*b*c)*e^6)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((b^3*c^5 - 4*a*b*c^6)*d - (b^
4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*e)*x*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c
^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*
a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (
b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) + ((b^2*c^4 - 4*a*c^5
)*d^4 - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e + 3*(2*b^4*c^2 - 9*a*b^2*c^3 + 4*a^2*c^4)*
d^2*e^2 - (4*b^5*c - 21*a*b^3*c^2 + 20*a^2*b*c^3)*d*e^3 + (b^6 - 6*a*b^4*c + 8*a
^2*b^2*c^2)*e^4)*x)*sqrt(-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c -
3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 - (b^2*c^4 - 4*a*c^5)*sqrt(
(c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*
a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c -
3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*
c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) + sqrt(1/2)*c^2*sqrt(-(b*c^3*d^3 - 3
*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*
a^2*c^2)*e^3 - (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4
- 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b
^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6
- 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*lo
g((2*a*c^4*d^6 - 8*a*b*c^3*d^5*e + 4*(3*a*b^2*c^2 - a^2*c^3)*d^4*e^2 - 4*(2*a*b^
3*c - a^2*b*c^2)*d^3*e^3 + 2*(a*b^4 + a^2*b^2*c - 3*a^3*c^2)*d^2*e^4 - 2*(a^2*b^
3 - 2*a^3*b*c)*d*e^5 - ((b^2*c^5 - 4*a*c^6)*d^3 - (b^3*c^4 - 4*a*b*c^5)*d^2*e +
(a*b^2*c^4 - 4*a^2*c^5)*d*e^2)*x^2*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4
- 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b
^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6
- 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)) - (b*c^4*d^6 - 4*(b^2*c^
3 + a*c^4)*d^5*e + 2*(3*b^3*c^2 + 7*a*b*c^3)*d^4*e^2 - 2*(2*b^4*c + 11*a*b^2*c^2
- 4*a^2*c^3)*d^3*e^3 + (b^5 + 17*a*b^3*c - 11*a^2*b*c^2)*d^2*e^4 - (5*a*b^4 + 2
*a^2*b^2*c - 12*a^3*c^2)*d*e^5 + 4*(a^2*b^3 - 2*a^3*b*c)*e^6)*x^2 - 2*sqrt(1/2)*
sqrt(e*x^2 + d)*(((b^3*c^5 - 4*a*b*c^6)*d - (b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*
e)*x*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3
*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 -
6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*
e^6)/(b^2*c^8 - 4*a*c^9)) + ((b^2*c^4 - 4*a*c^5)*d^4 - 4*(b^3*c^3 - 4*a*b*c^4)*d
^3*e + 3*(2*b^4*c^2 - 9*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^2 - (4*b^5*c - 21*a*b^3*c^2
+ 20*a^2*b*c^3)*d*e^3 + (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e^4)*x)*sqrt(-(b*c^3*
d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2
*c + 2*a^2*c^2)*e^3 - (b^2*c^4 - 4*a*c^5)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b
^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 -
10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e^5
+ (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c
^5)))/x^2) - 2*sqrt(e*x^2 + d)*c*e*x - 2*(3*c*d - 2*b*e)*sqrt(-e)*arctan(e*x/(sq
rt(e*x^2 + d)*sqrt(-e))))/c^2]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}}{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**2*(d + e*x**2)**(3/2)/(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x^2/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Timed out