3.813 \(\int \frac{1}{x^5 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^4}}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.0722205, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^4}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[a + b*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.03555, size = 41, normalized size = 0.82 \[ - \frac{\sqrt{a + b x^{4}}}{4 a x^{4}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0753576, size = 50, normalized size = 1. \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^4}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 48, normalized size = 1. \[ -{\frac{1}{4\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{b}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^4+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251624, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{4} \log \left (\frac{{\left (b x^{4} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{4} + a} a}{x^{4}}\right ) - 2 \, \sqrt{b x^{4} + a} \sqrt{a}}{8 \, a^{\frac{3}{2}} x^{4}}, -\frac{b x^{4} \arctan \left (\frac{a}{\sqrt{b x^{4} + a} \sqrt{-a}}\right ) + \sqrt{b x^{4} + a} \sqrt{-a}}{4 \, \sqrt{-a} a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.10246, size = 46, normalized size = 0.92 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} + 1}}{4 a x^{2}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21636, size = 65, normalized size = 1.3 \[ -\frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{4} + a}}{a b x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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