3.1928 \(\int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=93 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^3}+\frac{5 x^4 \sqrt{a+\frac{b}{x^2}}}{4 a^2}-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

(-15*b*Sqrt[a + b/x^2]*x^2)/(8*a^3) - x^4/(a*Sqrt[a + b/x^2]) + (5*Sqrt[a + b/x^
2]*x^4)/(4*a^2) + (15*b^2*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.148227, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^3}+\frac{5 x^4 \sqrt{a+\frac{b}{x^2}}}{4 a^2}-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^2)^(3/2),x]

[Out]

(-15*b*Sqrt[a + b/x^2]*x^2)/(8*a^3) - x^4/(a*Sqrt[a + b/x^2]) + (5*Sqrt[a + b/x^
2]*x^4)/(4*a^2) + (15*b^2*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 13.3396, size = 85, normalized size = 0.91 \[ - \frac{x^{4}}{a \sqrt{a + \frac{b}{x^{2}}}} + \frac{5 x^{4} \sqrt{a + \frac{b}{x^{2}}}}{4 a^{2}} - \frac{15 b x^{2} \sqrt{a + \frac{b}{x^{2}}}}{8 a^{3}} + \frac{15 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4/(a*sqrt(a + b/x**2)) + 5*x**4*sqrt(a + b/x**2)/(4*a**2) - 15*b*x**2*sqrt(a
+ b/x**2)/(8*a**3) + 15*b**2*atanh(sqrt(a + b/x**2)/sqrt(a))/(8*a**(7/2))

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Mathematica [A]  time = 0.0749547, size = 90, normalized size = 0.97 \[ \frac{\sqrt{a} x \left (2 a^2 x^4-5 a b x^2-15 b^2\right )+15 b^2 \sqrt{a x^2+b} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{8 a^{7/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*x*(-15*b^2 - 5*a*b*x^2 + 2*a^2*x^4) + 15*b^2*Sqrt[b + a*x^2]*Log[a*x +
Sqrt[a]*Sqrt[b + a*x^2]])/(8*a^(7/2)*Sqrt[a + b/x^2]*x)

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Maple [A]  time = 0.015, size = 87, normalized size = 0.9 \[{\frac{a{x}^{2}+b}{8\,{x}^{3}} \left ( 2\,{x}^{5}{a}^{7/2}-5\,{a}^{5/2}{x}^{3}b-15\,{a}^{3/2}x{b}^{2}+15\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \sqrt{a{x}^{2}+b}a{b}^{2} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^2)^(3/2),x)

[Out]

1/8*(a*x^2+b)*(2*x^5*a^(7/2)-5*a^(5/2)*x^3*b-15*a^(3/2)*x*b^2+15*ln(a^(1/2)*x+(a
*x^2+b)^(1/2))*(a*x^2+b)^(1/2)*a*b^2)/((a*x^2+b)/x^2)^(3/2)/x^3/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^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256666, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{a} \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a^{3} x^{6} - 5 \, a^{2} b x^{4} - 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \,{\left (a^{5} x^{2} + a^{4} b\right )}}, -\frac{15 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (2 \, a^{3} x^{6} - 5 \, a^{2} b x^{4} - 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \,{\left (a^{5} x^{2} + a^{4} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(15*(a*b^2*x^2 + b^3)*sqrt(a)*log(-2*a*x^2*sqrt((a*x^2 + b)/x^2) - (2*a*x^
2 + b)*sqrt(a)) + 2*(2*a^3*x^6 - 5*a^2*b*x^4 - 15*a*b^2*x^2)*sqrt((a*x^2 + b)/x^
2))/(a^5*x^2 + a^4*b), -1/8*(15*(a*b^2*x^2 + b^3)*sqrt(-a)*arctan(sqrt(-a)/sqrt(
(a*x^2 + b)/x^2)) - (2*a^3*x^6 - 5*a^2*b*x^4 - 15*a*b^2*x^2)*sqrt((a*x^2 + b)/x^
2))/(a^5*x^2 + a^4*b)]

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Sympy [A]  time = 19.0428, size = 100, normalized size = 1.08 \[ \frac{x^{5}}{4 a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{5 \sqrt{b} x^{3}}{8 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{15 b^{\frac{3}{2}} x}{8 a^{3} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5/(4*a*sqrt(b)*sqrt(a*x**2/b + 1)) - 5*sqrt(b)*x**3/(8*a**2*sqrt(a*x**2/b + 1
)) - 15*b**(3/2)*x/(8*a**3*sqrt(a*x**2/b + 1)) + 15*b**2*asinh(sqrt(a)*x/sqrt(b)
)/(8*a**(7/2))

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GIAC/XCAS [A]  time = 0.263866, size = 158, normalized size = 1.7 \[ -\frac{1}{8} \, b^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{8}{a^{3} \sqrt{\frac{a x^{2} + b}{x^{2}}}} - \frac{9 \, a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{7 \,{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )}^{2} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*b^2*(15*arctan(sqrt((a*x^2 + b)/x^2)/sqrt(-a))/(sqrt(-a)*a^3) + 8/(a^3*sqrt
((a*x^2 + b)/x^2)) - (9*a*sqrt((a*x^2 + b)/x^2) - 7*(a*x^2 + b)*sqrt((a*x^2 + b)
/x^2)/x^2)/((a - (a*x^2 + b)/x^2)^2*a^3))