3.766 \(\int \frac{\sqrt{a+c x^4}}{x^5} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\sqrt{a+c x^4}}{4 x^4}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]

[Out]

-Sqrt[a + c*x^4]/(4*x^4) - (c*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(4*Sqrt[a])

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Rubi [A]  time = 0.0683244, antiderivative size = 47, 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 \[ -\frac{\sqrt{a+c x^4}}{4 x^4}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^5,x]

[Out]

-Sqrt[a + c*x^4]/(4*x^4) - (c*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(4*Sqrt[a])

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Rubi in Sympy [A]  time = 6.93961, size = 41, normalized size = 0.87 \[ - \frac{\sqrt{a + c x^{4}}}{4 x^{4}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + c*x**4)/(4*x**4) - c*atanh(sqrt(a + c*x**4)/sqrt(a))/(4*sqrt(a))

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Mathematica [A]  time = 0.0769847, size = 47, normalized size = 1. \[ -\frac{\sqrt{a+c x^4}}{4 x^4}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^5,x]

[Out]

-Sqrt[a + c*x^4]/(4*x^4) - (c*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(4*Sqrt[a])

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Maple [A]  time = 0.014, size = 63, normalized size = 1.3 \[ -{\frac{1}{4\,a{x}^{4}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{c}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{c}{4\,a}\sqrt{c{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266395, size = 1, normalized size = 0.02 \[ \left [\frac{c x^{4} \log \left (\frac{{\left (c x^{4} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{4} + a} a}{x^{4}}\right ) - 2 \, \sqrt{c x^{4} + a} \sqrt{a}}{8 \, \sqrt{a} x^{4}}, \frac{c x^{4} \arctan \left (\frac{a}{\sqrt{c x^{4} + a} \sqrt{-a}}\right ) - \sqrt{c x^{4} + a} \sqrt{-a}}{4 \, \sqrt{-a} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.79009, size = 46, normalized size = 0.98 \[ - \frac{\sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{4 x^{2}} - \frac{c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216746, size = 58, normalized size = 1.23 \[ \frac{1}{4} \, c{\left (\frac{\arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{c x^{4} + a}}{c x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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