3.1910 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{x^2} \, dx\)

Optimal. Leaf size=92 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{16 x}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{24 x}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{6 x} \]

[Out]

(-5*a^2*Sqrt[a + b/x^2])/(16*x) - (5*a*(a + b/x^2)^(3/2))/(24*x) - (a + b/x^2)^(
5/2)/(6*x) - (5*a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(16*Sqrt[b])

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Rubi [A]  time = 0.103693, antiderivative size = 92, 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 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{16 x}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{24 x}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{6 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)/x^2,x]

[Out]

(-5*a^2*Sqrt[a + b/x^2])/(16*x) - (5*a*(a + b/x^2)^(3/2))/(24*x) - (a + b/x^2)^(
5/2)/(6*x) - (5*a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(16*Sqrt[b])

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Rubi in Sympy [A]  time = 7.67312, size = 80, normalized size = 0.87 \[ - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{16 \sqrt{b}} - \frac{5 a^{2} \sqrt{a + \frac{b}{x^{2}}}}{16 x} - \frac{5 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{24 x} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{6 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**3*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(16*sqrt(b)) - 5*a**2*sqrt(a + b/x**
2)/(16*x) - 5*a*(a + b/x**2)**(3/2)/(24*x) - (a + b/x**2)**(5/2)/(6*x)

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Mathematica [A]  time = 0.119784, size = 112, normalized size = 1.22 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-15 a^3 x^6 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+15 a^3 x^6 \log (x)-\sqrt{b} \sqrt{a x^2+b} \left (33 a^2 x^4+26 a b x^2+8 b^2\right )\right )}{48 \sqrt{b} x^5 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^2]*(-(Sqrt[b]*Sqrt[b + a*x^2]*(8*b^2 + 26*a*b*x^2 + 33*a^2*x^4)) +
15*a^3*x^6*Log[x] - 15*a^3*x^6*Log[b + Sqrt[b]*Sqrt[b + a*x^2]]))/(48*Sqrt[b]*x
^5*Sqrt[b + a*x^2])

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Maple [B]  time = 0.018, size = 166, normalized size = 1.8 \[ -{\frac{1}{48\,{b}^{3}x} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{6}{a}^{3}+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{6}{a}^{3}+3\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{4}{a}^{2}-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{6}{a}^{3}b-15\,\sqrt{a{x}^{2}+b}{x}^{6}{a}^{3}{b}^{2}+2\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}ab+8\, \left ( a{x}^{2}+b \right ) ^{7/2}{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)/x^2,x)

[Out]

-1/48*((a*x^2+b)/x^2)^(5/2)/x*(-3*(a*x^2+b)^(5/2)*x^6*a^3+15*b^(5/2)*ln(2*(b^(1/
2)*(a*x^2+b)^(1/2)+b)/x)*x^6*a^3+3*(a*x^2+b)^(7/2)*x^4*a^2-5*(a*x^2+b)^(3/2)*x^6
*a^3*b-15*(a*x^2+b)^(1/2)*x^6*a^3*b^2+2*(a*x^2+b)^(7/2)*x^2*a*b+8*(a*x^2+b)^(7/2
)*b^2)/(a*x^2+b)^(5/2)/b^3

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263277, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \sqrt{b} x^{5} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (33 \, a^{2} b x^{4} + 26 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{96 \, b x^{5}}, \frac{15 \, a^{3} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (33 \, a^{2} b x^{4} + 26 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{48 \, b x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(15*a^3*sqrt(b)*x^5*log((2*b*x*sqrt((a*x^2 + b)/x^2) - (a*x^2 + 2*b)*sqrt(
b))/x^2) - 2*(33*a^2*b*x^4 + 26*a*b^2*x^2 + 8*b^3)*sqrt((a*x^2 + b)/x^2))/(b*x^5
), 1/48*(15*a^3*sqrt(-b)*x^5*arctan(sqrt(-b)/(x*sqrt((a*x^2 + b)/x^2))) - (33*a^
2*b*x^4 + 26*a*b^2*x^2 + 8*b^3)*sqrt((a*x^2 + b)/x^2))/(b*x^5)]

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Sympy [A]  time = 18.1762, size = 99, normalized size = 1.08 \[ - \frac{11 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{16 x} - \frac{13 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{2}}}}{24 x^{3}} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{6 x^{5}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{16 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*a**(5/2)*sqrt(1 + b/(a*x**2))/(16*x) - 13*a**(3/2)*b*sqrt(1 + b/(a*x**2))/(2
4*x**3) - sqrt(a)*b**2*sqrt(1 + b/(a*x**2))/(6*x**5) - 5*a**3*asinh(sqrt(b)/(sqr
t(a)*x))/(16*sqrt(b))

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GIAC/XCAS [A]  time = 0.283501, size = 104, normalized size = 1.13 \[ \frac{1}{48} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{33 \,{\left (a x^{2} + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x^{2} + b} b^{2}}{a^{3} x^{6}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*a^3*(15*arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) - (33*(a*x^2 + b)^(5/2) -
40*(a*x^2 + b)^(3/2)*b + 15*sqrt(a*x^2 + b)*b^2)/(a^3*x^6))*sign(x)