3.2073 \(\int \left (a+\frac{b}{x^4}\right )^{5/2} x \, dx\)

Optimal. Leaf size=91 \[ -\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2} \]

[Out]

(-15*a*b*Sqrt[a + b/x^4])/(16*x^2) - (5*b*(a + b/x^4)^(3/2))/(8*x^2) + ((a + b/x
^4)^(5/2)*x^2)/2 - (15*a^2*Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^4]*x^2)])/16

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Rubi [A]  time = 0.179598, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)*x,x]

[Out]

(-15*a*b*Sqrt[a + b/x^4])/(16*x^2) - (5*b*(a + b/x^4)^(3/2))/(8*x^2) + ((a + b/x
^4)^(5/2)*x^2)/2 - (15*a^2*Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^4]*x^2)])/16

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Rubi in Sympy [A]  time = 11.5405, size = 85, normalized size = 0.93 \[ - \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{16} - \frac{15 a b \sqrt{a + \frac{b}{x^{4}}}}{16 x^{2}} - \frac{5 b \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{8 x^{2}} + \frac{x^{2} \left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**4)**(5/2)*x,x)

[Out]

-15*a**2*sqrt(b)*atanh(sqrt(b)/(x**2*sqrt(a + b/x**4)))/16 - 15*a*b*sqrt(a + b/x
**4)/(16*x**2) - 5*b*(a + b/x**4)**(3/2)/(8*x**2) + x**2*(a + b/x**4)**(5/2)/2

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Mathematica [A]  time = 0.131894, size = 94, normalized size = 1.03 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (\sqrt{a x^4+b} \left (-8 a^2 x^8+9 a b x^4+2 b^2\right )+15 a^2 \sqrt{b} x^8 \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )\right )}{16 x^6 \sqrt{a x^4+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(5/2)*x,x]

[Out]

-(Sqrt[a + b/x^4]*(Sqrt[b + a*x^4]*(2*b^2 + 9*a*b*x^4 - 8*a^2*x^8) + 15*a^2*Sqrt
[b]*x^8*ArcTanh[Sqrt[b + a*x^4]/Sqrt[b]]))/(16*x^6*Sqrt[b + a*x^4])

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Maple [A]  time = 0.027, size = 108, normalized size = 1.2 \[ -{\frac{{x}^{2}}{16} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,\sqrt{b}{a}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{8}-8\,{a}^{2}{x}^{8}\sqrt{a{x}^{4}+b}+9\,ab\sqrt{a{x}^{4}+b}{x}^{4}+2\,{b}^{2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)*x,x)

[Out]

-1/16*((a*x^4+b)/x^4)^(5/2)*x^2*(15*b^(1/2)*a^2*ln(2*(b^(1/2)*(a*x^4+b)^(1/2)+b)
/x^2)*x^8-8*a^2*x^8*(a*x^4+b)^(1/2)+9*a*b*(a*x^4+b)^(1/2)*x^4+2*b^2*(a*x^4+b)^(1
/2))/(a*x^4+b)^(5/2)

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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^4)^(5/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253666, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x^{6} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) + 2 \,{\left (8 \, a^{2} x^{8} - 9 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{32 \, x^{6}}, -\frac{15 \, a^{2} \sqrt{-b} x^{6} \arctan \left (\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-b}}\right ) -{\left (8 \, a^{2} x^{8} - 9 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{16 \, x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(5/2)*x,x, algorithm="fricas")

[Out]

[1/32*(15*a^2*sqrt(b)*x^6*log((a*x^4 - 2*sqrt(b)*x^2*sqrt((a*x^4 + b)/x^4) + 2*b
)/x^4) + 2*(8*a^2*x^8 - 9*a*b*x^4 - 2*b^2)*sqrt((a*x^4 + b)/x^4))/x^6, -1/16*(15
*a^2*sqrt(-b)*x^6*arctan(x^2*sqrt((a*x^4 + b)/x^4)/sqrt(-b)) - (8*a^2*x^8 - 9*a*
b*x^4 - 2*b^2)*sqrt((a*x^4 + b)/x^4))/x^6]

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Sympy [A]  time = 22.2637, size = 124, normalized size = 1.36 \[ \frac{a^{\frac{5}{2}} x^{2}}{2 \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{a^{\frac{3}{2}} b}{16 x^{2} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{11 \sqrt{a} b^{2}}{16 x^{6} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{16} - \frac{b^{3}}{8 \sqrt{a} x^{10} \sqrt{1 + \frac{b}{a x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(5/2)*x,x)

[Out]

a**(5/2)*x**2/(2*sqrt(1 + b/(a*x**4))) - a**(3/2)*b/(16*x**2*sqrt(1 + b/(a*x**4)
)) - 11*sqrt(a)*b**2/(16*x**6*sqrt(1 + b/(a*x**4))) - 15*a**2*sqrt(b)*asinh(sqrt
(b)/(sqrt(a)*x**2))/16 - b**3/(8*sqrt(a)*x**10*sqrt(1 + b/(a*x**4)))

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GIAC/XCAS [A]  time = 0.232008, size = 103, normalized size = 1.13 \[ \frac{1}{16} \,{\left (\frac{15 \, b \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 8 \, \sqrt{a x^{4} + b} - \frac{9 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} b - 7 \, \sqrt{a x^{4} + b} b^{2}}{a^{2} x^{8}}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(5/2)*x,x, algorithm="giac")

[Out]

1/16*(15*b*arctan(sqrt(a*x^4 + b)/sqrt(-b))/sqrt(-b) + 8*sqrt(a*x^4 + b) - (9*(a
*x^4 + b)^(3/2)*b - 7*sqrt(a*x^4 + b)*b^2)/(a^2*x^8))*a^2