3.770 \(\int \frac{\sqrt{a+c x^4}}{x^3} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0665315, antiderivative size = 49, 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 \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.07481, size = 41, normalized size = 0.84 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{2} - \frac{\sqrt{a + c x^{4}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0607901, size = 49, normalized size = 1. \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^3,x]

[Out]

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

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Maple [A]  time = 0.015, size = 60, normalized size = 1.2 \[ -{\frac{1}{2\,a{x}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{c{x}^{2}}{2\,a}\sqrt{c{x}^{4}+a}}+{\frac{1}{2}\sqrt{c}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267745, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c} x^{2} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) - 2 \, \sqrt{c x^{4} + a}}{4 \, x^{2}}, \frac{\sqrt{-c} x^{2} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + a} \sqrt{-c}}\right ) - \sqrt{c x^{4} + a}}{2 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.94525, size = 66, normalized size = 1.35 \[ - \frac{\sqrt{a}}{2 x^{2} \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{\sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2} - \frac{c x^{2}}{2 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219232, size = 49, normalized size = 1. \[ -\frac{c \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c}} - \frac{1}{2} \, \sqrt{c + \frac{a}{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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