3.1944 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x} \, dx\)

Optimal. Leaf size=59 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{1}{a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{1}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

-1/(3*a*(a + b/x^2)^(3/2)) - 1/(a^2*Sqrt[a + b/x^2]) + ArcTanh[Sqrt[a + b/x^2]/S
qrt[a]]/a^(5/2)

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Rubi [A]  time = 0.106175, antiderivative size = 59, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{1}{a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{1}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(5/2)*x),x]

[Out]

-1/(3*a*(a + b/x^2)^(3/2)) - 1/(a^2*Sqrt[a + b/x^2]) + ArcTanh[Sqrt[a + b/x^2]/S
qrt[a]]/a^(5/2)

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Rubi in Sympy [A]  time = 9.64342, size = 51, normalized size = 0.86 \[ - \frac{1}{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{1}{a^{2} \sqrt{a + \frac{b}{x^{2}}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(5/2)/x,x)

[Out]

-1/(3*a*(a + b/x**2)**(3/2)) - 1/(a**2*sqrt(a + b/x**2)) + atanh(sqrt(a + b/x**2
)/sqrt(a))/a**(5/2)

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Mathematica [A]  time = 0.0687765, size = 86, normalized size = 1.46 \[ \frac{3 \left (a x^2+b\right )^{3/2} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )-\sqrt{a} x \left (4 a x^2+3 b\right )}{3 a^{5/2} x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(5/2)*x),x]

[Out]

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

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Maple [A]  time = 0.013, size = 73, normalized size = 1.2 \[ -{\frac{a{x}^{2}+b}{3\,{x}^{5}} \left ( 4\,{x}^{3}{a}^{5/2}+3\,{a}^{3/2}xb-3\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}a \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x,x)

[Out]

-1/3*(a*x^2+b)*(4*x^3*a^(5/2)+3*a^(3/2)*x*b-3*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*(a*x
^2+b)^(3/2)*a)/((a*x^2+b)/x^2)^(5/2)/x^5/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(1/((a + b/x^2)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258358, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{a} \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) - 2 \,{\left (4 \, a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}, -\frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (4 \, a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*x^4 + 2*a*b*x^2 + b^2)*sqrt(a)*log(-2*a*x^2*sqrt((a*x^2 + b)/x^2) -
(2*a*x^2 + b)*sqrt(a)) - 2*(4*a^2*x^4 + 3*a*b*x^2)*sqrt((a*x^2 + b)/x^2))/(a^5*
x^4 + 2*a^4*b*x^2 + a^3*b^2), -1/3*(3*(a^2*x^4 + 2*a*b*x^2 + b^2)*sqrt(-a)*arcta
n(sqrt(-a)/sqrt((a*x^2 + b)/x^2)) + (4*a^2*x^4 + 3*a*b*x^2)*sqrt((a*x^2 + b)/x^2
))/(a^5*x^4 + 2*a^4*b*x^2 + a^3*b^2)]

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Sympy [A]  time = 12.4281, size = 743, normalized size = 12.59 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(5/2)/x,x)

[Out]

-8*a**7*x**6*sqrt(1 + b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a
**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 3*a**7*x**6*log(b/(a*x**2))/(6*a**(19/2
)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) + 6*a*
*7*x**6*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 +
18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 14*a**6*b*x**4*sqrt(1 + b/(a*x**2))
/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*
b**3) - 9*a**6*b*x**4*log(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 +
18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) + 18*a**6*b*x**4*log(sqrt(1 + b/(a*x*
*2)) + 1)/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a
**(13/2)*b**3) - 6*a**5*b**2*x**2*sqrt(1 + b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a*
*(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 9*a**5*b**2*x**2*l
og(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2
+ 6*a**(13/2)*b**3) + 18*a**5*b**2*x**2*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**(19/
2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3) - 3*a
**4*b**3*log(b/(a*x**2))/(6*a**(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*
b**2*x**2 + 6*a**(13/2)*b**3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x**2)) + 1)/(6*a**
(19/2)*x**6 + 18*a**(17/2)*b*x**4 + 18*a**(15/2)*b**2*x**2 + 6*a**(13/2)*b**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(5/2)*x), x)