3.1897 \(\int \left (a+\frac{b}{x^2}\right )^{3/2} x^3 \, dx\)

Optimal. Leaf size=68 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{3}{8} b x^2 \sqrt{a+\frac{b}{x^2}}+\frac{1}{4} x^4 \left (a+\frac{b}{x^2}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.104661, antiderivative size = 68, 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 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{3}{8} b x^2 \sqrt{a+\frac{b}{x^2}}+\frac{1}{4} x^4 \left (a+\frac{b}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)*x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.52387, size = 61, normalized size = 0.9 \[ \frac{3 b x^{2} \sqrt{a + \frac{b}{x^{2}}}}{8} + \frac{x^{4} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{4} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b*x**2*sqrt(a + b/x**2)/8 + x**4*(a + b/x**2)**(3/2)/4 + 3*b**2*atanh(sqrt(a +
b/x**2)/sqrt(a))/(8*sqrt(a))

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Mathematica [A]  time = 0.101756, size = 70, normalized size = 1.03 \[ \frac{1}{8} x \sqrt{a+\frac{b}{x^2}} \left (\frac{3 b^2 \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a} \sqrt{a x^2+b}}+2 a x^3+5 b x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^2]*x*(5*b*x + 2*a*x^3 + (3*b^2*Log[a*x + Sqrt[a]*Sqrt[b + a*x^2]])
/(Sqrt[a]*Sqrt[b + a*x^2])))/8

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Maple [A]  time = 0.008, size = 84, normalized size = 1.2 \[{\frac{{x}^{3}}{8} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 2\,x \left ( a{x}^{2}+b \right ) ^{3/2}\sqrt{a}+3\,\sqrt{a}\sqrt{a{x}^{2}+b}xb+3\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)*x^3,x)

[Out]

1/8*((a*x^2+b)/x^2)^(3/2)*x^3*(2*x*(a*x^2+b)^(3/2)*a^(1/2)+3*a^(1/2)*(a*x^2+b)^(
1/2)*x*b+3*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*b^2)/(a*x^2+b)^(3/2)/a^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249188, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{2} \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a^{2} x^{4} + 5 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (2 \, a^{2} x^{4} + 5 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.75, size = 70, normalized size = 1.03 \[ \frac{a \sqrt{b} x^{3} \sqrt{\frac{a x^{2}}{b} + 1}}{4} + \frac{5 b^{\frac{3}{2}} x \sqrt{\frac{a x^{2}}{b} + 1}}{8} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236494, size = 93, normalized size = 1.37 \[ \frac{3 \, b^{2}{\rm ln}\left (\sqrt{b}\right ){\rm sign}\left (x\right )}{8 \, \sqrt{a}} - \frac{3 \, b^{2}{\rm ln}\left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\right ){\rm sign}\left (x\right )}{8 \, \sqrt{a}} + \frac{1}{8} \,{\left (2 \, a x^{2}{\rm sign}\left (x\right ) + 5 \, b{\rm sign}\left (x\right )\right )} \sqrt{a x^{2} + b} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*b^2*ln(sqrt(b))*sign(x)/sqrt(a) - 3/8*b^2*ln(abs(-sqrt(a)*x + sqrt(a*x^2 + b
)))*sign(x)/sqrt(a) + 1/8*(2*a*x^2*sign(x) + 5*b*sign(x))*sqrt(a*x^2 + b)*x