3.1953 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^6} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{b^{5/2}}+\frac{1}{b^2 x \sqrt{a+\frac{b}{x^2}}}+\frac{1}{3 b x^3 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.10844, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{b^{5/2}}+\frac{1}{b^2 x \sqrt{a+\frac{b}{x^2}}}+\frac{1}{3 b x^3 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(5/2)*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7652, size = 58, normalized size = 0.85 \[ \frac{1}{3 b x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} + \frac{1}{b^{2} x \sqrt{a + \frac{b}{x^{2}}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(5/2)/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.12355, size = 97, normalized size = 1.43 \[ \frac{\sqrt{b} \left (3 a x^2+4 b\right )+3 \log (x) \left (a x^2+b\right )^{3/2}-3 \left (a x^2+b\right )^{3/2} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{3 b^{5/2} x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(5/2)*x^6),x]

[Out]

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

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Maple [A]  time = 0.01, size = 77, normalized size = 1.1 \[{\frac{a{x}^{2}+b}{3\,{x}^{5}} \left ( 3\,{b}^{3/2}{x}^{2}a+4\,{b}^{5/2}-3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}b \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^6,x)

[Out]

1/3*(a*x^2+b)*(3*b^(3/2)*x^2*a+4*b^(5/2)-3*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*(
a*x^2+b)^(3/2)*b)/((a*x^2+b)/x^2)^(5/2)/x^5/b^(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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258944, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{b} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (3 \, a b x^{3} + 4 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}}, \frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a b x^{3} + 4 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^6),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 21.7834, size = 740, normalized size = 10.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(5/2)/x**6,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^6),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(5/2)*x^6), x)