3.384 \(\int \frac{1}{x^4 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=341 \[ \frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{b \sqrt{d+e x^2}}{a^2 d x}+\frac{2 e \sqrt{d+e x^2}}{3 a d^2 x}-\frac{\sqrt{d+e x^2}}{3 a d x^3} \]

[Out]

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

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Rubi [A]  time = 1.77365, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{b \sqrt{d+e x^2}}{a^2 d x}+\frac{2 e \sqrt{d+e x^2}}{3 a d^2 x}-\frac{\sqrt{d+e x^2}}{3 a d x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

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

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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(1/x**4/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.76783, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/(x^4*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

Integrate[1/(x^4*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)), x]

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Maple [C]  time = 0.05, size = 248, normalized size = 0.7 \[ -{\frac{1}{3\,ad{x}^{3}}\sqrt{e{x}^{2}+d}}+{\frac{2\,e}{3\,a{d}^{2}x}\sqrt{e{x}^{2}+d}}-{\frac{1}{2\,{a}^{2}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{bc{{\it \_R}}^{2}+2\, \left ( -2\,ace+2\,{b}^{2}e-bcd \right ){\it \_R}+bc{d}^{2}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }}+{\frac{b}{{a}^{2}dx}\sqrt{e{x}^{2}+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x)

[Out]

-1/3*(e*x^2+d)^(1/2)/a/d/x^3+2/3*e*(e*x^2+d)^(1/2)/a/d^2/x-1/2/a^2*e^(1/2)*sum((
b*c*_R^2+2*(-2*a*c*e+2*b^2*e-b*c*d)*_R+b*c*d^2)/(_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))+b*(e*x^2+d)^(1/2)/a^2/d/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt{e x^{2} + d} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^4), x)

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Fricas [A]  time = 58.1071, size = 11052, normalized size = 32.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^4),x, algorithm="fricas")

[Out]

1/12*(3*sqrt(1/2)*a^2*d^2*x^3*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^3)*d - (b^
6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e - ((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a
^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6*a*b^6*c^3
+ 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^
2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c
^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(
a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^
2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a^5*b^2*c -
4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2))*log(-(((
a^5*b^2*c^4 - 4*a^6*c^5)*d^3 - (a^5*b^3*c^3 - 4*a^6*b*c^4)*d^2*e + (a^6*b^2*c^3
- 4*a^7*c^4)*d*e^2)*x^2*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^
2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4
+ 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4
*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*
d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*
c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) + 2*(a*b^4*c^4 - 3*a^2*b^2*c^5 + a^3*c^6)
*d^2 - 2*(a*b^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*d*e - ((b^5*c^4 - 3*a*b^3*c^5
+ a^2*b*c^6)*d^2 - (b^6*c^3 - 9*a^2*b^2*c^5 + 4*a^3*c^6)*d*e + 4*(a*b^5*c^3 - 4
*a^2*b^3*c^4 + 3*a^3*b*c^5)*e^2)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^6*b^4*c^
2 - 6*a^7*b^2*c^3 + 8*a^8*c^4)*d^3 - (2*a^6*b^5*c - 13*a^7*b^3*c^2 + 20*a^8*b*c^
3)*d^2*e + (a^6*b^6 - 6*a^7*b^4*c + 6*a^8*b^2*c^2 + 8*a^9*c^3)*d*e^2 - (a^7*b^5
- 7*a^8*b^3*c + 12*a^9*b*c^2)*e^3)*x*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c
^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13
*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^
4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4
*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^
3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) + ((a*b^7*c^2 - 7*a^2*b^5*c^
3 + 13*a^3*b^3*c^4 - 4*a^4*b*c^5)*d^2 - (2*a*b^8*c - 16*a^2*b^6*c^2 + 39*a^3*b^4
*c^3 - 29*a^4*b^2*c^4 + 4*a^5*c^5)*d*e + (a*b^9 - 9*a^2*b^7*c + 27*a^3*b^5*c^2 -
31*a^4*b^3*c^3 + 12*a^5*b*c^4)*e^2)*x)*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^
3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e - ((a^5*b^2*c - 4*a^6*c^2
)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6*
a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c
^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22
*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)
*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c
^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a
^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)
))/x^2) - 3*sqrt(1/2)*a^2*d^2*x^3*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^3)*d -
(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e - ((a^5*b^2*c - 4*a^6*c^2)*d^2
- (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6*a*b^6*
c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 1
6*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b
^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 -
2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^
2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a^5*b^2
*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2))*log(
-(((a^5*b^2*c^4 - 4*a^6*c^5)*d^3 - (a^5*b^3*c^3 - 4*a^6*b*c^4)*d^2*e + (a^6*b^2*
c^3 - 4*a^7*c^4)*d*e^2)*x^2*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^
3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*
c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9
*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c
^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^1
2*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) + 2*(a*b^4*c^4 - 3*a^2*b^2*c^5 + a^3*
c^6)*d^2 - 2*(a*b^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*d*e - ((b^5*c^4 - 3*a*b^3
*c^5 + a^2*b*c^6)*d^2 - (b^6*c^3 - 9*a^2*b^2*c^5 + 4*a^3*c^6)*d*e + 4*(a*b^5*c^3
- 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*e^2)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^6*b^
4*c^2 - 6*a^7*b^2*c^3 + 8*a^8*c^4)*d^3 - (2*a^6*b^5*c - 13*a^7*b^3*c^2 + 20*a^8*
b*c^3)*d^2*e + (a^6*b^6 - 6*a^7*b^4*c + 6*a^8*b^2*c^2 + 8*a^9*c^3)*d*e^2 - (a^7*
b^5 - 7*a^8*b^3*c + 12*a^9*b*c^2)*e^3)*x*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b
^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3
- 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^
3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c
- 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^1
1*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) + ((a*b^7*c^2 - 7*a^2*b^
5*c^3 + 13*a^3*b^3*c^4 - 4*a^4*b*c^5)*d^2 - (2*a*b^8*c - 16*a^2*b^6*c^2 + 39*a^3
*b^4*c^3 - 29*a^4*b^2*c^4 + 4*a^5*c^5)*d*e + (a*b^9 - 9*a^2*b^7*c + 27*a^3*b^5*c
^2 - 31*a^4*b^3*c^3 + 12*a^5*b*c^4)*e^2)*x)*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*
b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e - ((a^5*b^2*c - 4*a^6
*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2
- 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b
^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c
+ 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*
c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^
12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))
/((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*
e^2)))/x^2) + 3*sqrt(1/2)*a^2*d^2*x^3*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^3)
*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e + ((a^5*b^2*c - 4*a^6*c^2)*
d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6*a*
b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2
+ 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a
^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d
^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2
)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a^5
*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2))*
log((((a^5*b^2*c^4 - 4*a^6*c^5)*d^3 - (a^5*b^3*c^3 - 4*a^6*b*c^4)*d^2*e + (a^6*b
^2*c^3 - 4*a^7*c^4)*d*e^2)*x^2*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6
*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b
^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3
+ 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*
b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*
a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) - 2*(a*b^4*c^4 - 3*a^2*b^2*c^5 + a
^3*c^6)*d^2 + 2*(a*b^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*d*e + ((b^5*c^4 - 3*a*
b^3*c^5 + a^2*b*c^6)*d^2 - (b^6*c^3 - 9*a^2*b^2*c^5 + 4*a^3*c^6)*d*e + 4*(a*b^5*
c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*e^2)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^6
*b^4*c^2 - 6*a^7*b^2*c^3 + 8*a^8*c^4)*d^3 - (2*a^6*b^5*c - 13*a^7*b^3*c^2 + 20*a
^8*b*c^3)*d^2*e + (a^6*b^6 - 6*a^7*b^4*c + 6*a^8*b^2*c^2 + 8*a^9*c^3)*d*e^2 - (a
^7*b^5 - 7*a^8*b^3*c + 12*a^9*b*c^2)*e^3)*x*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^
2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c
^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24
*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^
3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(
a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) - ((a*b^7*c^2 - 7*a^2
*b^5*c^3 + 13*a^3*b^3*c^4 - 4*a^4*b*c^5)*d^2 - (2*a*b^8*c - 16*a^2*b^6*c^2 + 39*
a^3*b^4*c^3 - 29*a^4*b^2*c^4 + 4*a^5*c^5)*d*e + (a*b^9 - 9*a^2*b^7*c + 27*a^3*b^
5*c^2 - 31*a^4*b^3*c^3 + 12*a^5*b*c^4)*e^2)*x)*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a
^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e + ((a^5*b^2*c - 4*
a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c
^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*
a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8
*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^
11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8
*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4
)))/((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*
c)*e^2)))/x^2) - 3*sqrt(1/2)*a^2*d^2*x^3*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c
^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e + ((a^5*b^2*c - 4*a^6*c^
2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6
*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*
c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 2
2*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3
)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*
c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((
a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2
))*log((((a^5*b^2*c^4 - 4*a^6*c^5)*d^3 - (a^5*b^3*c^3 - 4*a^6*b*c^4)*d^2*e + (a^
6*b^2*c^3 - 4*a^7*c^4)*d*e^2)*x^2*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4
- 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^
3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c
^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^
11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 -
4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) - 2*(a*b^4*c^4 - 3*a^2*b^2*c^5
+ a^3*c^6)*d^2 + 2*(a*b^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*d*e + ((b^5*c^4 - 3
*a*b^3*c^5 + a^2*b*c^6)*d^2 - (b^6*c^3 - 9*a^2*b^2*c^5 + 4*a^3*c^6)*d*e + 4*(a*b
^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*e^2)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((
a^6*b^4*c^2 - 6*a^7*b^2*c^3 + 8*a^8*c^4)*d^3 - (2*a^6*b^5*c - 13*a^7*b^3*c^2 + 2
0*a^8*b*c^3)*d^2*e + (a^6*b^6 - 6*a^7*b^4*c + 6*a^8*b^2*c^2 + 8*a^9*c^3)*d*e^2 -
(a^7*b^5 - 7*a^8*b^3*c + 12*a^9*b*c^2)*e^3)*x*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11
*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^
5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 -
24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10
*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 -
2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)) - ((a*b^7*c^2 - 7*
a^2*b^5*c^3 + 13*a^3*b^3*c^4 - 4*a^4*b*c^5)*d^2 - (2*a*b^8*c - 16*a^2*b^6*c^2 +
39*a^3*b^4*c^3 - 29*a^4*b^2*c^4 + 4*a^5*c^5)*d*e + (a*b^9 - 9*a^2*b^7*c + 27*a^3
*b^5*c^2 - 31*a^4*b^3*c^3 + 12*a^5*b*c^4)*e^2)*x)*sqrt(-((b^5*c - 5*a*b^3*c^2 +
5*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e + ((a^5*b^2*c -
4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^
8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c -
7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*
b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4
*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c
- 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*
e^4)))/((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a
^7*c)*e^2)))/x^2) + 4*((3*b*d + 2*a*e)*x^2 - a*d)*sqrt(e*x^2 + d))/(a^2*d^2*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Timed out

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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(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^4),x, algorithm="giac")

[Out]

Timed out