3.1907 \(\int \left (a+\frac{b}{x^2}\right )^{5/2} x \, dx\)

Optimal. Leaf size=80 \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^2}\right )^{5/2}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}} \]

[Out]

(-5*a*b*Sqrt[a + b/x^2])/2 - (5*b*(a + b/x^2)^(3/2))/6 + ((a + b/x^2)^(5/2)*x^2)
/2 + (5*a^(3/2)*b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/2

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Rubi [A]  time = 0.125316, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^2}\right )^{5/2}-\frac{5}{6} b \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{5}{2} a b \sqrt{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)*x,x]

[Out]

(-5*a*b*Sqrt[a + b/x^2])/2 - (5*b*(a + b/x^2)^(3/2))/6 + ((a + b/x^2)^(5/2)*x^2)
/2 + (5*a^(3/2)*b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/2

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Rubi in Sympy [A]  time = 10.8488, size = 73, normalized size = 0.91 \[ \frac{5 a^{\frac{3}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2} - \frac{5 a b \sqrt{a + \frac{b}{x^{2}}}}{2} - \frac{5 b \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{6} + \frac{x^{2} \left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(3/2)*b*atanh(sqrt(a + b/x**2)/sqrt(a))/2 - 5*a*b*sqrt(a + b/x**2)/2 - 5*b*
(a + b/x**2)**(3/2)/6 + x**2*(a + b/x**2)**(5/2)/2

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Mathematica [A]  time = 0.0930837, size = 83, normalized size = 1.04 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\frac{15 a^{3/2} b x^3 \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a x^2+b}}+3 a^2 x^4-14 a b x^2-2 b^2\right )}{6 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^2]*(-2*b^2 - 14*a*b*x^2 + 3*a^2*x^4 + (15*a^(3/2)*b*x^3*Log[a*x +
Sqrt[a]*Sqrt[b + a*x^2]])/Sqrt[b + a*x^2]))/(6*x^2)

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Maple [B]  time = 0.012, size = 149, normalized size = 1.9 \[{\frac{{x}^{2}}{6\,{b}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( 8\,{a}^{5/2} \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{4}+10\,{a}^{5/2} \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{4}b+15\,{a}^{5/2}\sqrt{a{x}^{2}+b}{x}^{4}{b}^{2}-8\,{a}^{3/2} \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}-2\, \left ( a{x}^{2}+b \right ) ^{7/2}b\sqrt{a}+15\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){x}^{3}{a}^{2}{b}^{3} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)*x,x)

[Out]

1/6*((a*x^2+b)/x^2)^(5/2)*x^2*(8*a^(5/2)*(a*x^2+b)^(5/2)*x^4+10*a^(5/2)*(a*x^2+b
)^(3/2)*x^4*b+15*a^(5/2)*(a*x^2+b)^(1/2)*x^4*b^2-8*a^(3/2)*(a*x^2+b)^(7/2)*x^2-2
*(a*x^2+b)^(7/2)*b*a^(1/2)+15*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*x^3*a^2*b^3)/(a*x^2+
b)^(5/2)/b^2/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 = 0.251322, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{3}{2}} b x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) + 2 \,{\left (3 \, a^{2} x^{4} - 14 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \, x^{2}}, \frac{15 \, \sqrt{-a} a b x^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a^{2} x^{4} - 14 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(15*a^(3/2)*b*x^2*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b)
+ 2*(3*a^2*x^4 - 14*a*b*x^2 - 2*b^2)*sqrt((a*x^2 + b)/x^2))/x^2, 1/6*(15*sqrt(-a
)*a*b*x^2*arctan(a/(sqrt(-a)*sqrt((a*x^2 + b)/x^2))) + (3*a^2*x^4 - 14*a*b*x^2 -
2*b^2)*sqrt((a*x^2 + b)/x^2))/x^2]

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Sympy [A]  time = 15.901, size = 112, normalized size = 1.4 \[ \frac{a^{\frac{5}{2}} x^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{2} - \frac{7 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{2}}}}{3} - \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b}{a x^{2}} \right )}}{4} + \frac{5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*x**2*sqrt(1 + b/(a*x**2))/2 - 7*a**(3/2)*b*sqrt(1 + b/(a*x**2))/3 - 5*a
**(3/2)*b*log(b/(a*x**2))/4 + 5*a**(3/2)*b*log(sqrt(1 + b/(a*x**2)) + 1)/2 - sqr
t(a)*b**2*sqrt(1 + b/(a*x**2))/(3*x**2)

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GIAC/XCAS [A]  time = 0.376626, size = 192, normalized size = 2.4 \[ \frac{1}{2} \, \sqrt{a x^{2} + b} a^{2} x{\rm sign}\left (x\right ) - \frac{5}{4} \, a^{\frac{3}{2}} b{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \,{\left (9 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{3}{2}} b^{2}{\rm sign}\left (x\right ) - 12 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} a^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right ) + 7 \, a^{\frac{3}{2}} b^{4}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

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*sqrt(a*x^2 + b)*a^2*x*sign(x) - 5/4*a^(3/2)*b*ln((sqrt(a)*x - sqrt(a*x^2 + b
))^2)*sign(x) + 2/3*(9*(sqrt(a)*x - sqrt(a*x^2 + b))^4*a^(3/2)*b^2*sign(x) - 12*
(sqrt(a)*x - sqrt(a*x^2 + b))^2*a^(3/2)*b^3*sign(x) + 7*a^(3/2)*b^4*sign(x))/((s
qrt(a)*x - sqrt(a*x^2 + b))^2 - b)^3