3.2088 \(\int \frac{x^3}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 x^4 \sqrt{a+\frac{b}{x^4}}}{4 a^2}-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

-x^4/(2*a*Sqrt[a + b/x^4]) + (3*Sqrt[a + b/x^4]*x^4)/(4*a^2) - (3*b*ArcTanh[Sqrt
[a + b/x^4]/Sqrt[a]])/(4*a^(5/2))

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Rubi [A]  time = 0.121718, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 x^4 \sqrt{a+\frac{b}{x^4}}}{4 a^2}-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^4)^(3/2),x]

[Out]

-x^4/(2*a*Sqrt[a + b/x^4]) + (3*Sqrt[a + b/x^4]*x^4)/(4*a^2) - (3*b*ArcTanh[Sqrt
[a + b/x^4]/Sqrt[a]])/(4*a^(5/2))

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Rubi in Sympy [A]  time = 9.79503, size = 63, normalized size = 0.89 \[ - \frac{x^{4}}{2 a \sqrt{a + \frac{b}{x^{4}}}} + \frac{3 x^{4} \sqrt{a + \frac{b}{x^{4}}}}{4 a^{2}} - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4/(2*a*sqrt(a + b/x**4)) + 3*x**4*sqrt(a + b/x**4)/(4*a**2) - 3*b*atanh(sqrt
(a + b/x**4)/sqrt(a))/(4*a**(5/2))

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Mathematica [A]  time = 0.0579726, size = 80, normalized size = 1.13 \[ \frac{\sqrt{a} x^2 \left (a x^4+3 b\right )-3 b \sqrt{a x^4+b} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )}{4 a^{5/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*x^2*(3*b + a*x^4) - 3*b*Sqrt[b + a*x^4]*Log[a*x^2 + Sqrt[a]*Sqrt[b + a*
x^4]])/(4*a^(5/2)*Sqrt[a + b/x^4]*x^2)

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Maple [A]  time = 0.022, size = 80, normalized size = 1.1 \[ -{\frac{a{x}^{4}+b}{4\,{x}^{6}} \left ( -{x}^{6}{a}^{{\frac{7}{2}}}-3\,{x}^{2}b{a}^{5/2}+3\,b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){a}^{2}\sqrt{a{x}^{4}+b} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^4)^(3/2),x)

[Out]

-1/4*(a*x^4+b)*(-x^6*a^(7/2)-3*x^2*b*a^(5/2)+3*b*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))
*a^2*(a*x^4+b)^(1/2))/((a*x^4+b)/x^4)^(3/2)/x^6/a^(9/2)

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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^3/(a + b/x^4)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255975, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a b x^{4} + b^{2}\right )} \sqrt{a} \log \left (2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right ) + 2 \,{\left (a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{8 \,{\left (a^{4} x^{4} + a^{3} b\right )}}, \frac{3 \,{\left (a b x^{4} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{4 \,{\left (a^{4} x^{4} + a^{3} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*(a*b*x^4 + b^2)*sqrt(a)*log(2*a*x^4*sqrt((a*x^4 + b)/x^4) - (2*a*x^4 + b
)*sqrt(a)) + 2*(a^2*x^8 + 3*a*b*x^4)*sqrt((a*x^4 + b)/x^4))/(a^4*x^4 + a^3*b), 1
/4*(3*(a*b*x^4 + b^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt((a*x^4 + b)/x^4)) + (a^2*x^8
+ 3*a*b*x^4)*sqrt((a*x^4 + b)/x^4))/(a^4*x^4 + a^3*b)]

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Sympy [A]  time = 12.4341, size = 75, normalized size = 1.06 \[ \frac{x^{6}}{4 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} + \frac{3 \sqrt{b} x^{2}}{4 a^{2} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6/(4*a*sqrt(b)*sqrt(a*x**4/b + 1)) + 3*sqrt(b)*x**2/(4*a**2*sqrt(a*x**4/b + 1
)) - 3*b*asinh(sqrt(a)*x**2/sqrt(b))/(4*a**(5/2))

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GIAC/XCAS [A]  time = 0.31306, size = 131, normalized size = 1.85 \[ \frac{1}{4} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x^{4} + b\right )}}{x^{4}}}{{\left (a \sqrt{\frac{a x^{4} + b}{x^{4}}} - \frac{{\left (a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{4}}\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b*(3*arctan(sqrt((a*x^4 + b)/x^4)/sqrt(-a))/(sqrt(-a)*a^2) + (2*a - 3*(a*x^4
+ b)/x^4)/((a*sqrt((a*x^4 + b)/x^4) - (a*x^4 + b)*sqrt((a*x^4 + b)/x^4)/x^4)*a^
2))