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

Optimal. Leaf size=63 \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0938859, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.95702, size = 56, normalized size = 0.89 \[ - \frac{3 \sqrt{a} c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{4} + \frac{3 c \sqrt{a + c x^{4}}}{4} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0819188, size = 55, normalized size = 0.87 \[ \left (\frac{c}{2}-\frac{a}{4 x^4}\right ) \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 58, normalized size = 0.9 \[{\frac{c}{2}\sqrt{c{x}^{4}+a}}-{\frac{a}{4\,{x}^{4}}\sqrt{c{x}^{4}+a}}-{\frac{3\,c}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246822, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{a} c x^{4} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + 2 \,{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{8 \, x^{4}}, -\frac{3 \, \sqrt{-a} c x^{4} \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right ) -{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{4 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.94732, size = 95, normalized size = 1.51 \[ - \frac{3 \sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{4} - \frac{a^{2}}{4 \sqrt{c} x^{6} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{a \sqrt{c}}{4 x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{c^{\frac{3}{2}} x^{2}}{2 \sqrt{\frac{a}{c x^{4}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214447, size = 77, normalized size = 1.22 \[ \frac{1}{4} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{c x^{4} + a} - \frac{\sqrt{c x^{4} + a} a}{c x^{4}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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