3.1906 \(\int \left (a+\frac{b}{x^2}\right )^{5/2} x^2 \, dx\)

Optimal. Leaf size=88 \[ -\frac{5}{2} a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )-\frac{5 b^2 \sqrt{a+\frac{b}{x^2}}}{2 x}+\frac{5}{3} b x \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.124649, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5}{2} a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )-\frac{5 b^2 \sqrt{a+\frac{b}{x^2}}}{2 x}+\frac{5}{3} b x \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.6446, size = 78, normalized size = 0.89 \[ - \frac{5 a b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2} - \frac{5 b^{2} \sqrt{a + \frac{b}{x^{2}}}}{2 x} + \frac{5 b x \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{3} \]

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

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Mathematica [A]  time = 0.1091, size = 107, normalized size = 1.22 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b} \left (2 a^2 x^4+14 a b x^2-3 b^2\right )+15 a b^{3/2} x^2 \log (x)-15 a b^{3/2} x^2 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )\right )}{6 x \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 122, normalized size = 1.4 \[ -{\frac{{x}^{3}}{6\,b} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{2}a+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}a+3\, \left ( a{x}^{2}+b \right ) ^{7/2}-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{2}ab-15\,\sqrt{a{x}^{2}+b}{x}^{2}a{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)*x^2,x)

[Out]

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

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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 = 0.259164, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a b^{\frac{3}{2}} x \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \,{\left (2 \, a^{2} x^{4} + 14 \, a b x^{2} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \, x}, -\frac{15 \, a \sqrt{-b} b x \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (2 \, a^{2} x^{4} + 14 \, a b x^{2} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.9147, size = 112, normalized size = 1.27 \[ \frac{a^{2} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{7 a b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{5 a b^{\frac{3}{2}} \log{\left (\frac{a x^{2}}{b} \right )}}{4} - \frac{5 a b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a x^{2}}{b} + 1} + 1 \right )}}{2} - \frac{b^{\frac{5}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.262812, size = 101, normalized size = 1.15 \[ \frac{1}{6} \,{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} + 12 \, \sqrt{a x^{2} + b} b - \frac{3 \, \sqrt{a x^{2} + b} b^{2}}{a x^{2}}\right )} a{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(15*b^2*arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) + 2*(a*x^2 + b)^(3/2) + 12
*sqrt(a*x^2 + b)*b - 3*sqrt(a*x^2 + b)*b^2/(a*x^2))*a*sign(x)