3.1905 \(\int \left (a+\frac{b}{x^2}\right )^{5/2} x^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.136751, antiderivative size = 86, 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 \[ -\frac{15}{8} b^2 \sqrt{a+\frac{b}{x^2}}+\frac{15}{8} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )+\frac{5}{8} b x^2 \left (a+\frac{b}{x^2}\right )^{3/2}+\frac{1}{4} x^4 \left (a+\frac{b}{x^2}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)*x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.8629, size = 78, normalized size = 0.91 \[ \frac{15 \sqrt{a} b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8} - \frac{15 b^{2} \sqrt{a + \frac{b}{x^{2}}}}{8} + \frac{5 b x^{2} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{8} + \frac{x^{4} \left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**(5/2)*x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0853884, size = 93, normalized size = 1.08 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b} \left (2 a^2 x^4+9 a b x^2-8 b^2\right )+15 \sqrt{a} b^2 x \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )\right )}{8 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(5/2)*x^3,x]

[Out]

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

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Maple [A]  time = 0.011, size = 127, normalized size = 1.5 \[{\frac{{x}^{4}}{8\,b} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( 8\,{a}^{3/2} \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{2}+10\,{a}^{3/2} \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{2}b+15\,{a}^{3/2}\sqrt{a{x}^{2}+b}{x}^{2}{b}^{2}-8\, \left ( a{x}^{2}+b \right ) ^{7/2}\sqrt{a}+15\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) xa{b}^{3} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)*x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262544, size = 1, normalized size = 0.01 \[ \left [\frac{15}{16} \, \sqrt{a} b^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) + \frac{1}{8} \,{\left (2 \, a^{2} x^{4} + 9 \, a b x^{2} - 8 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}, \frac{15}{8} \, \sqrt{-a} b^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) + \frac{1}{8} \,{\left (2 \, a^{2} x^{4} + 9 \, a b x^{2} - 8 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^(5/2)*x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 21.6335, size = 117, normalized size = 1.36 \[ \frac{15 \sqrt{a} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8} + \frac{a^{3} x^{5}}{4 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{11 a^{2} \sqrt{b} x^{3}}{8 \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{a b^{\frac{3}{2}} x}{8 \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{b^{\frac{5}{2}}}{x \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(5/2)*x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.278371, size = 128, normalized size = 1.49 \[ -\frac{15}{16} \, \sqrt{a} b^{2}{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \, \sqrt{a} b^{3}{\rm sign}\left (x\right )}{{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b} + \frac{1}{8} \,{\left (2 \, a^{2} x^{2}{\rm sign}\left (x\right ) + 9 \, a b{\rm sign}\left (x\right )\right )} \sqrt{a x^{2} + b} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^(5/2)*x^3,x, algorithm="giac")

[Out]

-15/16*sqrt(a)*b^2*ln((sqrt(a)*x - sqrt(a*x^2 + b))^2)*sign(x) + 2*sqrt(a)*b^3*s
ign(x)/((sqrt(a)*x - sqrt(a*x^2 + b))^2 - b) + 1/8*(2*a^2*x^2*sign(x) + 9*a*b*si
gn(x))*sqrt(a*x^2 + b)*x