3.1908 \(\int \left (a+\frac{b}{x^2}\right )^{5/2} \, dx\)

Optimal. Leaf size=86 \[ -\frac{15}{8} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )+x \left (a+\frac{b}{x^2}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^2}\right )^{3/2}}{4 x}-\frac{15 a b \sqrt{a+\frac{b}{x^2}}}{8 x} \]

[Out]

(-15*a*b*Sqrt[a + b/x^2])/(8*x) - (5*b*(a + b/x^2)^(3/2))/(4*x) + (a + b/x^2)^(5
/2)*x - (15*a^2*Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/8

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Rubi [A]  time = 0.122049, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{15}{8} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )+x \left (a+\frac{b}{x^2}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^2}\right )^{3/2}}{4 x}-\frac{15 a b \sqrt{a+\frac{b}{x^2}}}{8 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2),x]

[Out]

(-15*a*b*Sqrt[a + b/x^2])/(8*x) - (5*b*(a + b/x^2)^(3/2))/(4*x) + (a + b/x^2)^(5
/2)*x - (15*a^2*Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/8

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Rubi in Sympy [A]  time = 7.87058, size = 76, normalized size = 0.88 \[ - \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{8} - \frac{15 a b \sqrt{a + \frac{b}{x^{2}}}}{8 x} - \frac{5 b \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{4 x} + x \left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*a**2*sqrt(b)*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/8 - 15*a*b*sqrt(a + b/x**2)
/(8*x) - 5*b*(a + b/x**2)**(3/2)/(4*x) + x*(a + b/x**2)**(5/2)

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Mathematica [A]  time = 0.106025, size = 111, normalized size = 1.29 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b} \left (8 a^2 x^4-9 a b x^2-2 b^2\right )+15 a^2 \sqrt{b} x^4 \log (x)-15 a^2 \sqrt{b} x^4 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )\right )}{8 x^3 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(5/2),x]

[Out]

(Sqrt[a + b/x^2]*(Sqrt[b + a*x^2]*(-2*b^2 - 9*a*b*x^2 + 8*a^2*x^4) + 15*a^2*Sqrt
[b]*x^4*Log[x] - 15*a^2*Sqrt[b]*x^4*Log[b + Sqrt[b]*Sqrt[b + a*x^2]]))/(8*x^3*Sq
rt[b + a*x^2])

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Maple [B]  time = 0.011, size = 144, normalized size = 1.7 \[ -{\frac{x}{8\,{b}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{4}{a}^{2}+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{4}{a}^{2}+3\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}a-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{4}{a}^{2}b-15\,\sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}{b}^{2}+2\, \left ( a{x}^{2}+b \right ) ^{7/2}b \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255689, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x^{3} \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \,{\left (8 \, a^{2} x^{4} - 9 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, x^{3}}, -\frac{15 \, a^{2} \sqrt{-b} x^{3} \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (8 \, a^{2} x^{4} - 9 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(15*a^2*sqrt(b)*x^3*log(-(a*x^2 - 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b)
/x^2) + 2*(8*a^2*x^4 - 9*a*b*x^2 - 2*b^2)*sqrt((a*x^2 + b)/x^2))/x^3, -1/8*(15*a
^2*sqrt(-b)*x^3*arctan(b/(sqrt(-b)*x*sqrt((a*x^2 + b)/x^2))) - (8*a^2*x^4 - 9*a*
b*x^2 - 2*b^2)*sqrt((a*x^2 + b)/x^2))/x^3]

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Sympy [A]  time = 16.474, size = 117, normalized size = 1.36 \[ \frac{a^{\frac{5}{2}} x}{\sqrt{1 + \frac{b}{a x^{2}}}} - \frac{a^{\frac{3}{2}} b}{8 x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{11 \sqrt{a} b^{2}}{8 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8} - \frac{b^{3}}{4 \sqrt{a} x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(5/2),x)

[Out]

a**(5/2)*x/sqrt(1 + b/(a*x**2)) - a**(3/2)*b/(8*x*sqrt(1 + b/(a*x**2))) - 11*sqr
t(a)*b**2/(8*x**3*sqrt(1 + b/(a*x**2))) - 15*a**2*sqrt(b)*asinh(sqrt(b)/(sqrt(a)
*x))/8 - b**3/(4*sqrt(a)*x**5*sqrt(1 + b/(a*x**2)))

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GIAC/XCAS [A]  time = 0.263365, size = 105, normalized size = 1.22 \[ \frac{1}{8} \,{\left (\frac{15 \, b \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 8 \, \sqrt{a x^{2} + b} - \frac{9 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b - 7 \, \sqrt{a x^{2} + b} b^{2}}{a^{2} x^{4}}\right )} a^{2}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(15*b*arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) + 8*sqrt(a*x^2 + b) - (9*(a*
x^2 + b)^(3/2)*b - 7*sqrt(a*x^2 + b)*b^2)/(a^2*x^4))*a^2*sign(x)