3.2075 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=92 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{32 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x^4}}}{32 x^2}-\frac{5 a \left (a+\frac{b}{x^4}\right )^{3/2}}{48 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{12 x^2} \]

[Out]

(-5*a^2*Sqrt[a + b/x^4])/(32*x^2) - (5*a*(a + b/x^4)^(3/2))/(48*x^2) - (a + b/x^
4)^(5/2)/(12*x^2) - (5*a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^4]*x^2)])/(32*Sqrt[b])

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Rubi [A]  time = 0.165975, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{32 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x^4}}}{32 x^2}-\frac{5 a \left (a+\frac{b}{x^4}\right )^{3/2}}{48 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{12 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)/x^3,x]

[Out]

(-5*a^2*Sqrt[a + b/x^4])/(32*x^2) - (5*a*(a + b/x^4)^(3/2))/(48*x^2) - (a + b/x^
4)^(5/2)/(12*x^2) - (5*a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^4]*x^2)])/(32*Sqrt[b])

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Rubi in Sympy [A]  time = 10.0874, size = 87, normalized size = 0.95 \[ - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{32 \sqrt{b}} - \frac{5 a^{2} \sqrt{a + \frac{b}{x^{4}}}}{32 x^{2}} - \frac{5 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{48 x^{2}} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{12 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**4)**(5/2)/x**3,x)

[Out]

-5*a**3*atanh(sqrt(b)/(x**2*sqrt(a + b/x**4)))/(32*sqrt(b)) - 5*a**2*sqrt(a + b/
x**4)/(32*x**2) - 5*a*(a + b/x**4)**(3/2)/(48*x**2) - (a + b/x**4)**(5/2)/(12*x*
*2)

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Mathematica [A]  time = 0.170989, size = 81, normalized size = 0.88 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (-\frac{15 a^3 x^{12} \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{a x^4+b}}-33 a^2 x^8-26 a b x^4-8 b^2\right )}{96 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(5/2)/x^3,x]

[Out]

(Sqrt[a + b/x^4]*(-8*b^2 - 26*a*b*x^4 - 33*a^2*x^8 - (15*a^3*x^12*ArcTanh[Sqrt[b
+ a*x^4]/Sqrt[b]])/(Sqrt[b]*Sqrt[b + a*x^4])))/(96*x^10)

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Maple [A]  time = 0.031, size = 113, normalized size = 1.2 \[ -{\frac{1}{96\,{x}^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,{a}^{3}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{12}+33\,{a}^{2}\sqrt{a{x}^{4}+b}\sqrt{b}{x}^{8}+26\,{b}^{3/2}a\sqrt{a{x}^{4}+b}{x}^{4}+8\,{b}^{5/2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)/x^3,x)

[Out]

-1/96*((a*x^4+b)/x^4)^(5/2)/x^2*(15*a^3*ln(2*(b^(1/2)*(a*x^4+b)^(1/2)+b)/x^2)*x^
12+33*a^2*(a*x^4+b)^(1/2)*b^(1/2)*x^8+26*b^(3/2)*a*(a*x^4+b)^(1/2)*x^4+8*b^(5/2)
*(a*x^4+b)^(1/2))/(a*x^4+b)^(5/2)/b^(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^4)^(5/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252581, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \sqrt{b} x^{10} \log \left (-\frac{2 \, b x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + 2 \, b\right )} \sqrt{b}}{x^{4}}\right ) - 2 \,{\left (33 \, a^{2} b x^{8} + 26 \, a b^{2} x^{4} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{192 \, b x^{10}}, -\frac{15 \, a^{3} \sqrt{-b} x^{10} \arctan \left (\frac{b}{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (33 \, a^{2} b x^{8} + 26 \, a b^{2} x^{4} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{96 \, b x^{10}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(5/2)/x^3,x, algorithm="fricas")

[Out]

[1/192*(15*a^3*sqrt(b)*x^10*log(-(2*b*x^2*sqrt((a*x^4 + b)/x^4) - (a*x^4 + 2*b)*
sqrt(b))/x^4) - 2*(33*a^2*b*x^8 + 26*a*b^2*x^4 + 8*b^3)*sqrt((a*x^4 + b)/x^4))/(
b*x^10), -1/96*(15*a^3*sqrt(-b)*x^10*arctan(b/(sqrt(-b)*x^2*sqrt((a*x^4 + b)/x^4
))) + (33*a^2*b*x^8 + 26*a*b^2*x^4 + 8*b^3)*sqrt((a*x^4 + b)/x^4))/(b*x^10)]

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Sympy [A]  time = 22.0288, size = 102, normalized size = 1.11 \[ - \frac{11 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{32 x^{2}} - \frac{13 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{4}}}}{48 x^{6}} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{4}}}}{12 x^{10}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{32 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(5/2)/x**3,x)

[Out]

-11*a**(5/2)*sqrt(1 + b/(a*x**4))/(32*x**2) - 13*a**(3/2)*b*sqrt(1 + b/(a*x**4))
/(48*x**6) - sqrt(a)*b**2*sqrt(1 + b/(a*x**4))/(12*x**10) - 5*a**3*asinh(sqrt(b)
/(sqrt(a)*x**2))/(32*sqrt(b))

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GIAC/XCAS [A]  time = 0.23276, size = 101, normalized size = 1.1 \[ \frac{1}{96} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{33 \,{\left (a x^{4} + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x^{4} + b} b^{2}}{a^{3} x^{12}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(5/2)/x^3,x, algorithm="giac")

[Out]

1/96*a^3*(15*arctan(sqrt(a*x^4 + b)/sqrt(-b))/sqrt(-b) - (33*(a*x^4 + b)^(5/2) -
40*(a*x^4 + b)^(3/2)*b + 15*sqrt(a*x^4 + b)*b^2)/(a^3*x^12))