3.1909 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=72 \[ a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-a^2 \sqrt{a+\frac{b}{x^2}}-\frac{1}{3} a \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{1}{5} \left (a+\frac{b}{x^2}\right )^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.125726, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-a^2 \sqrt{a+\frac{b}{x^2}}-\frac{1}{3} a \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{1}{5} \left (a+\frac{b}{x^2}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9748, size = 60, normalized size = 0.83 \[ a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )} - a^{2} \sqrt{a + \frac{b}{x^{2}}} - \frac{a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0940606, size = 82, normalized size = 1.14 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\frac{15 a^{5/2} x^5 \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a x^2+b}}-23 a^2 x^4-11 a b x^2-3 b^2\right )}{15 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^2]*(-3*b^2 - 11*a*b*x^2 - 23*a^2*x^4 + (15*a^(5/2)*x^5*Log[a*x + S
qrt[a]*Sqrt[b + a*x^2]])/Sqrt[b + a*x^2]))/(15*x^4)

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Maple [B]  time = 0.015, size = 166, normalized size = 2.3 \[{\frac{1}{15\,{b}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( 8\,{a}^{7/2} \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{6}+10\,{a}^{7/2} \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{6}b+15\,{a}^{7/2}\sqrt{a{x}^{2}+b}{x}^{6}{b}^{2}-8\,{a}^{5/2} \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{4}-2\,{a}^{3/2} \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}b+15\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){x}^{5}{a}^{3}{b}^{3}-3\, \left ( a{x}^{2}+b \right ) ^{7/2}{b}^{2}\sqrt{a} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)/x,x)

[Out]

1/15*((a*x^2+b)/x^2)^(5/2)*(8*a^(7/2)*(a*x^2+b)^(5/2)*x^6+10*a^(7/2)*(a*x^2+b)^(
3/2)*x^6*b+15*a^(7/2)*(a*x^2+b)^(1/2)*x^6*b^2-8*a^(5/2)*(a*x^2+b)^(7/2)*x^4-2*a^
(3/2)*(a*x^2+b)^(7/2)*x^2*b+15*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*x^5*a^3*b^3-3*(a*x^
2+b)^(7/2)*b^2*a^(1/2))/(a*x^2+b)^(5/2)/b^3/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)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252471, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{5}{2}} x^{4} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) - 2 \,{\left (23 \, a^{2} x^{4} + 11 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{30 \, x^{4}}, \frac{15 \, \sqrt{-a} a^{2} x^{4} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (23 \, a^{2} x^{4} + 11 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{15 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*a^(5/2)*x^4*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) -
2*(23*a^2*x^4 + 11*a*b*x^2 + 3*b^2)*sqrt((a*x^2 + b)/x^2))/x^4, 1/15*(15*sqrt(-a
)*a^2*x^4*arctan(a/(sqrt(-a)*sqrt((a*x^2 + b)/x^2))) - (23*a^2*x^4 + 11*a*b*x^2
+ 3*b^2)*sqrt((a*x^2 + b)/x^2))/x^4]

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Sympy [A]  time = 15.6275, size = 105, normalized size = 1.46 \[ - \frac{23 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{15} - \frac{a^{\frac{5}{2}} \log{\left (\frac{b}{a x^{2}} \right )}}{2} + a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )} - \frac{11 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{2}}}}{15 x^{2}} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23*a**(5/2)*sqrt(1 + b/(a*x**2))/15 - a**(5/2)*log(b/(a*x**2))/2 + a**(5/2)*log
(sqrt(1 + b/(a*x**2)) + 1) - 11*a**(3/2)*b*sqrt(1 + b/(a*x**2))/(15*x**2) - sqrt
(a)*b**2*sqrt(1 + b/(a*x**2))/(5*x**4)

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GIAC/XCAS [A]  time = 0.605609, size = 243, normalized size = 3.38 \[ -\frac{1}{2} \, a^{\frac{5}{2}}{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \,{\left (45 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{5}{2}} b{\rm sign}\left (x\right ) - 90 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{6} a^{\frac{5}{2}} b^{2}{\rm sign}\left (x\right ) + 140 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{5}{2}} b^{3}{\rm sign}\left (x\right ) - 70 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} a^{\frac{5}{2}} b^{4}{\rm sign}\left (x\right ) + 23 \, a^{\frac{5}{2}} b^{5}{\rm sign}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^(5/2)*ln((sqrt(a)*x - sqrt(a*x^2 + b))^2)*sign(x) + 2/15*(45*(sqrt(a)*x -
sqrt(a*x^2 + b))^8*a^(5/2)*b*sign(x) - 90*(sqrt(a)*x - sqrt(a*x^2 + b))^6*a^(5/
2)*b^2*sign(x) + 140*(sqrt(a)*x - sqrt(a*x^2 + b))^4*a^(5/2)*b^3*sign(x) - 70*(s
qrt(a)*x - sqrt(a*x^2 + b))^2*a^(5/2)*b^4*sign(x) + 23*a^(5/2)*b^5*sign(x))/((sq
rt(a)*x - sqrt(a*x^2 + b))^2 - b)^5