3.1913 \(\int \frac{x^3}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

Optimal. Leaf size=74 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{3 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x^2}}}{4 a} \]

[Out]

(-3*b*Sqrt[a + b/x^2]*x^2)/(8*a^2) + (Sqrt[a + b/x^2]*x^4)/(4*a) + (3*b^2*ArcTan
h[Sqrt[a + b/x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.112116, antiderivative size = 74, 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 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{3 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x^2}}}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b/x^2],x]

[Out]

(-3*b*Sqrt[a + b/x^2]*x^2)/(8*a^2) + (Sqrt[a + b/x^2]*x^4)/(4*a) + (3*b^2*ArcTan
h[Sqrt[a + b/x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 9.88857, size = 66, normalized size = 0.89 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{2}}}}{4 a} - \frac{3 b x^{2} \sqrt{a + \frac{b}{x^{2}}}}{8 a^{2}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0684053, size = 90, normalized size = 1.22 \[ \frac{\sqrt{a} x \left (2 a^2 x^4-a b x^2-3 b^2\right )+3 b^2 \sqrt{a x^2+b} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{8 a^{5/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 87, normalized size = 1.2 \[{\frac{1}{8\,x}\sqrt{a{x}^{2}+b} \left ( 2\,{x}^{3}\sqrt{a{x}^{2}+b}{a}^{5/2}-3\,{a}^{3/2}\sqrt{a{x}^{2}+b}xb+3\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) a{b}^{2} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^2)^(1/2),x)

[Out]

1/8*(a*x^2+b)^(1/2)*(2*x^3*(a*x^2+b)^(1/2)*a^(5/2)-3*a^(3/2)*(a*x^2+b)^(1/2)*x*b
+3*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*a*b^2)/((a*x^2+b)/x^2)^(1/2)/x/a^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258802, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{2} \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a^{2} x^{4} - 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, a^{3}}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (2 \, a^{2} x^{4} - 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(a + b/x^2),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 - 3*a*b*x^2)*sqrt((a*x^2 + b)/x^2))/a^3, -1/8*(3*sqrt(-a)*b^2*ar
ctan(sqrt(-a)/sqrt((a*x^2 + b)/x^2)) - (2*a^2*x^4 - 3*a*b*x^2)*sqrt((a*x^2 + b)/
x^2))/a^3]

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Sympy [A]  time = 13.7763, size = 95, normalized size = 1.28 \[ \frac{x^{5}}{4 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{\sqrt{b} x^{3}}{8 a \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{3 b^{\frac{3}{2}} x}{8 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24801, size = 134, normalized size = 1.81 \[ -\frac{1}{8} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{5 \, a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{3 \,{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )}^{2} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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