3.853 \(\int \frac{1}{x^5 \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{3 \sqrt{a+b x^4}}{4 a^2 x^4}+\frac{1}{2 a x^4 \sqrt{a+b x^4}} \]

[Out]

1/(2*a*x^4*Sqrt[a + b*x^4]) - (3*Sqrt[a + b*x^4])/(4*a^2*x^4) + (3*b*ArcTanh[Sqr
t[a + b*x^4]/Sqrt[a]])/(4*a^(5/2))

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Rubi [A]  time = 0.099341, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{3 \sqrt{a+b x^4}}{4 a^2 x^4}+\frac{1}{2 a x^4 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*x^4*Sqrt[a + b*x^4]) - (3*Sqrt[a + b*x^4])/(4*a^2*x^4) + (3*b*ArcTanh[Sqr
t[a + b*x^4]/Sqrt[a]])/(4*a^(5/2))

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Rubi in Sympy [A]  time = 9.88013, size = 65, normalized size = 0.92 \[ \frac{1}{2 a x^{4} \sqrt{a + b x^{4}}} - \frac{3 \sqrt{a + b x^{4}}}{4 a^{2} x^{4}} + \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**4+a)**(3/2),x)

[Out]

1/(2*a*x**4*sqrt(a + b*x**4)) - 3*sqrt(a + b*x**4)/(4*a**2*x**4) + 3*b*atanh(sqr
t(a + b*x**4)/sqrt(a))/(4*a**(5/2))

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Mathematica [A]  time = 0.130668, size = 58, normalized size = 0.82 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{a+3 b x^4}{4 a^2 x^4 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^4)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.018, size = 63, normalized size = 0.9 \[ -{\frac{1}{4\,a{x}^{4}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{3\,b}{4\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{3\,b}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^4+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.384068, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{b x^{4} + a} b x^{4} \log \left (\frac{{\left (b x^{4} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{4} + a} a}{x^{4}}\right ) - 2 \,{\left (3 \, b x^{4} + a\right )} \sqrt{a}}{8 \, \sqrt{b x^{4} + a} a^{\frac{5}{2}} x^{4}}, -\frac{3 \, \sqrt{b x^{4} + a} b x^{4} \arctan \left (\frac{a}{\sqrt{b x^{4} + a} \sqrt{-a}}\right ) +{\left (3 \, b x^{4} + a\right )} \sqrt{-a}}{4 \, \sqrt{b x^{4} + a} \sqrt{-a} a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^5),x, algorithm="fricas")

[Out]

[1/8*(3*sqrt(b*x^4 + a)*b*x^4*log(((b*x^4 + 2*a)*sqrt(a) + 2*sqrt(b*x^4 + a)*a)/
x^4) - 2*(3*b*x^4 + a)*sqrt(a))/(sqrt(b*x^4 + a)*a^(5/2)*x^4), -1/4*(3*sqrt(b*x^
4 + a)*b*x^4*arctan(a/(sqrt(b*x^4 + a)*sqrt(-a))) + (3*b*x^4 + a)*sqrt(-a))/(sqr
t(b*x^4 + a)*sqrt(-a)*a^2*x^4)]

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Sympy [A]  time = 12.1225, size = 76, normalized size = 1.07 \[ - \frac{1}{4 a \sqrt{b} x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{3 \sqrt{b}}{4 a^{2} x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**4+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.2153, size = 89, normalized size = 1.25 \[ -\frac{1}{4} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \, b x^{4} + a}{{\left ({\left (b x^{4} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{4} + a} a\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^5),x, algorithm="giac")

[Out]

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