3.1889 \(\int \sqrt{a+\frac{b}{x^2}} x^3 \, dx\)

Optimal. Leaf size=71 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.115383, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]*x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.55543, size = 60, normalized size = 0.85 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{2}}}}{4} + \frac{b x^{2} \sqrt{a + \frac{b}{x^{2}}}}{8 a} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**(1/2)*x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0734636, size = 88, normalized size = 1.24 \[ x \sqrt{a+\frac{b}{x^2}} \left (\frac{b x}{8 a}+\frac{x^3}{4}\right )-\frac{b^2 x \sqrt{a+\frac{b}{x^2}} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{8 a^{3/2} \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 82, normalized size = 1.2 \[{\frac{x}{8}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( 2\,x \left ( a{x}^{2}+b \right ) ^{3/2}\sqrt{a}-\sqrt{a}\sqrt{a{x}^{2}+b}xb-\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){b}^{2} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)*x^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253864, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} b^{2} \log \left (2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a^{2} x^{4} + a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a^{2}}, \frac{\sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (2 \, a^{2} x^{4} + a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(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 + a*b*x^2)*sqrt((a*x^2 + b)/x^2))/a^2, 1/8*(sqrt(-a)*b^2*arctan(sqr
t(-a)/sqrt((a*x^2 + b)/x^2)) + (2*a^2*x^4 + a*b*x^2)*sqrt((a*x^2 + b)/x^2))/a^2]

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Sympy [A]  time = 13.0924, size = 92, normalized size = 1.3 \[ \frac{a x^{5}}{4 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 \sqrt{b} x^{3}}{8 \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{b^{\frac{3}{2}} x}{8 a \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(1/2)*x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.231018, size = 95, normalized size = 1.34 \[ \frac{1}{8} \, \sqrt{a x^{2} + b}{\left (2 \, x^{2}{\rm sign}\left (x\right ) + \frac{b{\rm sign}\left (x\right )}{a}\right )} x - \frac{b^{2}{\rm ln}\left (\sqrt{b}\right ){\rm sign}\left (x\right )}{8 \, a^{\frac{3}{2}}} + \frac{b^{2}{\rm ln}\left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\right ){\rm sign}\left (x\right )}{8 \, a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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