3.2074 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=77 \[ \frac{1}{2} a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a^2 \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} a \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{1}{10} \left (a+\frac{b}{x^4}\right )^{5/2} \]

[Out]

-(a^2*Sqrt[a + b/x^4])/2 - (a*(a + b/x^4)^(3/2))/6 - (a + b/x^4)^(5/2)/10 + (a^(
5/2)*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/2

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Rubi [A]  time = 0.133058, antiderivative size = 77, 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 \[ \frac{1}{2} a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a^2 \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} a \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{1}{10} \left (a+\frac{b}{x^4}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)/x,x]

[Out]

-(a^2*Sqrt[a + b/x^4])/2 - (a*(a + b/x^4)^(3/2))/6 - (a + b/x^4)^(5/2)/10 + (a^(
5/2)*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/2

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Rubi in Sympy [A]  time = 10.9971, size = 63, normalized size = 0.82 \[ \frac{a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2} - \frac{a^{2} \sqrt{a + \frac{b}{x^{4}}}}{2} - \frac{a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{6} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*atanh(sqrt(a + b/x**4)/sqrt(a))/2 - a**2*sqrt(a + b/x**4)/2 - a*(a + b/
x**4)**(3/2)/6 - (a + b/x**4)**(5/2)/10

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Mathematica [A]  time = 0.137116, size = 81, normalized size = 1.05 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (\frac{15 a^{5/2} x^{10} \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )}{\sqrt{a x^4+b}}-23 a^2 x^8-11 a b x^4-3 b^2\right )}{30 x^8} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*(-3*b^2 - 11*a*b*x^4 - 23*a^2*x^8 + (15*a^(5/2)*x^10*ArcTanh[(S
qrt[a]*x^2)/Sqrt[b + a*x^4]])/Sqrt[b + a*x^4]))/(30*x^8)

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Maple [A]  time = 0.028, size = 99, normalized size = 1.3 \[{\frac{1}{30} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,{a}^{5/2}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{10}-23\,{a}^{2}{x}^{8}\sqrt{a{x}^{4}+b}-11\,ab\sqrt{a{x}^{4}+b}{x}^{4}-3\,{b}^{2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)/x,x)

[Out]

1/30*((a*x^4+b)/x^4)^(5/2)*(15*a^(5/2)*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^10-23*a
^2*x^8*(a*x^4+b)^(1/2)-11*a*b*(a*x^4+b)^(1/2)*x^4-3*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 = 0.264653, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{5}{2}} x^{8} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - 2 \,{\left (23 \, a^{2} x^{8} + 11 \, a b x^{4} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{60 \, x^{8}}, \frac{15 \, \sqrt{-a} a^{2} x^{8} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) -{\left (23 \, a^{2} x^{8} + 11 \, a b x^{4} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{30 \, x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/60*(15*a^(5/2)*x^8*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) -
2*(23*a^2*x^8 + 11*a*b*x^4 + 3*b^2)*sqrt((a*x^4 + b)/x^4))/x^8, 1/30*(15*sqrt(-a
)*a^2*x^8*arctan(a/(sqrt(-a)*sqrt((a*x^4 + b)/x^4))) - (23*a^2*x^8 + 11*a*b*x^4
+ 3*b^2)*sqrt((a*x^4 + b)/x^4))/x^8]

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Sympy [A]  time = 19.1706, size = 107, normalized size = 1.39 \[ - \frac{23 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{30} - \frac{a^{\frac{5}{2}} \log{\left (\frac{b}{a x^{4}} \right )}}{4} + \frac{a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{2} - \frac{11 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{4}}}}{30 x^{4}} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{4}}}}{10 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23*a**(5/2)*sqrt(1 + b/(a*x**4))/30 - a**(5/2)*log(b/(a*x**4))/4 + a**(5/2)*log
(sqrt(1 + b/(a*x**4)) + 1)/2 - 11*a**(3/2)*b*sqrt(1 + b/(a*x**4))/(30*x**4) - sq
rt(a)*b**2*sqrt(1 + b/(a*x**4))/(10*x**8)

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GIAC/XCAS [A]  time = 0.23573, size = 86, normalized size = 1.12 \[ -\frac{a^{3} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{10} \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} - \frac{1}{6} \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} a - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^3*arctan(sqrt(a + b/x^4)/sqrt(-a))/sqrt(-a) - 1/10*(a + b/x^4)^(5/2) - 1/
6*(a + b/x^4)^(3/2)*a - 1/2*sqrt(a + b/x^4)*a^2