3.378 \(\int \frac{x^8}{\sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=479 \[ -\frac{\left (-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{c^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{c^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c^3 \sqrt{e}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 e^{3/2}}-\frac{b x \sqrt{d+e x^2}}{2 c^2 e}+\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c e^{5/2}}-\frac{3 d x \sqrt{d+e x^2}}{8 c e^2}+\frac{x^3 \sqrt{d+e x^2}}{4 c e} \]

[Out]

(-3*d*x*Sqrt[d + e*x^2])/(8*c*e^2) - (b*x*Sqrt[d + e*x^2])/(2*c^2*e) + (x^3*Sqrt
[d + e*x^2])/(4*c*e) - ((b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2
- 4*a*c]]*Sqrt[d + e*x^2])])/(c^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b -
Sqrt[b^2 - 4*a*c])*e]) - ((b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^
2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^
2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]) + (3*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*c*e^(
5/2)) + (b*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2*e^(3/2)) + ((b^2 - a*c
)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c^3*Sqrt[e])

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Rubi [A]  time = 3.98145, antiderivative size = 479, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{\left (-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{c^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{c^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c^3 \sqrt{e}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 e^{3/2}}-\frac{b x \sqrt{d+e x^2}}{2 c^2 e}+\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c e^{5/2}}-\frac{3 d x \sqrt{d+e x^2}}{8 c e^2}+\frac{x^3 \sqrt{d+e x^2}}{4 c e} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

(-3*d*x*Sqrt[d + e*x^2])/(8*c*e^2) - (b*x*Sqrt[d + e*x^2])/(2*c^2*e) + (x^3*Sqrt
[d + e*x^2])/(4*c*e) - ((b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2
- 4*a*c]]*Sqrt[d + e*x^2])])/(c^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b -
Sqrt[b^2 - 4*a*c])*e]) - ((b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^
2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^
2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]) + (3*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*c*e^(
5/2)) + (b*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2*e^(3/2)) + ((b^2 - a*c
)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c^3*Sqrt[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(x**8/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.897436, size = 0, normalized size = 0. \[ \int \frac{x^8}{\sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^8/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

Integrate[x^8/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)), x]

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Maple [C]  time = 0.04, size = 377, normalized size = 0.8 \[{\frac{{b}^{2}}{{c}^{3}}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{{x}^{3}}{4\,ce}\sqrt{e{x}^{2}+d}}-{\frac{3\,dx}{8\,{e}^{2}c}\sqrt{e{x}^{2}+d}}+{\frac{3\,{d}^{2}}{8\,c}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}}-{\frac{a}{{c}^{2}}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}-{\frac{bx}{2\,{c}^{2}e}\sqrt{e{x}^{2}+d}}+{\frac{bd}{2\,{c}^{2}}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}-{\frac{1}{2\,{c}^{3}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{b \left ( 2\,ac-{b}^{2} \right ){{\it \_R}}^{2}+2\, \left ( 2\,{a}^{2}ce-2\,a{b}^{2}e-2\,abcd+{b}^{3}d \right ){\it \_R}+2\,abc{d}^{2}-{b}^{3}{d}^{2}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x)

[Out]

1/c^3*b^2*ln(x*e^(1/2)+(e*x^2+d)^(1/2))/e^(1/2)+1/4*x^3*(e*x^2+d)^(1/2)/c/e-3/8*
d*x*(e*x^2+d)^(1/2)/e^2/c+3/8/c*d^2/e^(5/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))-1/c^2*
a*ln(x*e^(1/2)+(e*x^2+d)^(1/2))/e^(1/2)-1/2*b*x*(e*x^2+d)^(1/2)/c^2/e+1/2/c^2*b*
d/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))-1/2/c^3*e^(1/2)*sum((b*(2*a*c-b^2)*_R^2+
2*(2*a^2*c*e-2*a*b^2*e-2*a*b*c*d+b^3*d)*_R+2*a*b*c*d^2-b^3*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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt{e x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)),x, algorithm="maxima")

[Out]

integrate(x^8/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Integral(x**8/(sqrt(d + e*x**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(x^8/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)),x, algorithm="giac")

[Out]

Timed out