3.275 \(\int \frac{x^5}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{c^{3/2}}-\frac{x^2}{c \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.156033, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{c^{3/2}}-\frac{x^2}{c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.9388, size = 46, normalized size = 0.84 \[ - \frac{x^{2}}{c \sqrt{b x^{2} + c x^{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**4+b*x**2)**(3/2),x)

[Out]

-x**2/(c*sqrt(b*x**2 + c*x**4)) + atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4))/c**(
3/2)

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Mathematica [A]  time = 0.0464279, size = 65, normalized size = 1.18 \[ \frac{x \left (\sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-\sqrt{c} x\right )}{c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(-(Sqrt[c]*x) + Sqrt[b + c*x^2]*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(c^(3/2)
*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.011, size = 62, normalized size = 1.1 \[{{x}^{3} \left ( c{x}^{2}+b \right ) \left ( -x{c}^{{\frac{3}{2}}}+\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) c\sqrt{c{x}^{2}+b} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^4+b*x^2)^(3/2),x)

[Out]

x^3*(c*x^2+b)*(-x*c^(3/2)+ln(x*c^(1/2)+(c*x^2+b)^(1/2))*c*(c*x^2+b)^(1/2))/(c*x^
4+b*x^2)^(3/2)/c^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275846, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c x^{2} + b\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} c}{2 \,{\left (c^{3} x^{2} + b c^{2}\right )}}, -\frac{{\left (c x^{2} + b\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} c}{c^{3} x^{2} + b c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2*((c*x^2 + b)*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) - 2*sqrt(c*x^4 + b*x^2)*c)
- 2*sqrt(c*x^4 + b*x^2)*c)/(c^3*x^2 + b*c^2), -((c*x^2 + b)*sqrt(-c)*arctan(sqrt
(-c)*x^2/sqrt(c*x^4 + b*x^2)) + sqrt(c*x^4 + b*x^2)*c)/(c^3*x^2 + b*c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**5/(x**2*(b + c*x**2))**(3/2), x)

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GIAC/XCAS [A]  time = 0.284797, size = 55, normalized size = 1. \[ -\frac{\arctan \left (\frac{\sqrt{c + \frac{b}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{1}{\sqrt{c + \frac{b}{x^{2}}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

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