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

Optimal. Leaf size=52 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a-b x^4}}{4 a x^4} \]

[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 = 0.0787222, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a-b x^4}}{4 a x^4} \]

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 = 7.78037, size = 42, normalized size = 0.81 \[ - \frac{\sqrt{a - b x^{4}}}{4 a x^{4}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a - b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

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 = 0.0984556, size = 52, 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 = 0.018, size = 50, 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 = 0.287458, 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 = 8.47815, size = 129, normalized size = 2.48 \[ \begin{cases} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} - 1}}{4 a x^{2}} - \frac{b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\\frac{i}{4 \sqrt{b} x^{6} \sqrt{- \frac{a}{b x^{4}} + 1}} - \frac{i \sqrt{b}}{4 a x^{2} \sqrt{- \frac{a}{b x^{4}} + 1}} + \frac{i b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(4*a*x**2) - b*acosh(sqrt(a)/(sqrt(b)*x
**2))/(4*a**(3/2)), Abs(a/(b*x**4)) > 1), (I/(4*sqrt(b)*x**6*sqrt(-a/(b*x**4) +
1)) - I*sqrt(b)/(4*a*x**2*sqrt(-a/(b*x**4) + 1)) + I*b*asin(sqrt(a)/(sqrt(b)*x**
2))/(4*a**(3/2)), True))

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GIAC/XCAS [A]  time = 0.214263, size = 69, normalized size = 1.33 \[ \frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{-b x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{-b x^{4} + a}}{a b x^{4}}\right )} \]

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))