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

Optimal. Leaf size=59 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )+\frac{1}{2} a \sqrt{a+c x^4}+\frac{1}{6} \left (a+c x^4\right )^{3/2} \]

[Out]

(a*Sqrt[a + c*x^4])/2 + (a + c*x^4)^(3/2)/6 - (a^(3/2)*ArcTanh[Sqrt[a + c*x^4]/S
qrt[a]])/2

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Rubi [A]  time = 0.0910448, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )+\frac{1}{2} a \sqrt{a+c x^4}+\frac{1}{6} \left (a+c x^4\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x,x]

[Out]

(a*Sqrt[a + c*x^4])/2 + (a + c*x^4)^(3/2)/6 - (a^(3/2)*ArcTanh[Sqrt[a + c*x^4]/S
qrt[a]])/2

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Rubi in Sympy [A]  time = 8.63257, size = 48, normalized size = 0.81 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{2} + \frac{a \sqrt{a + c x^{4}}}{2} + \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0957863, size = 51, normalized size = 0.86 \[ \frac{1}{6} \left (\sqrt{a+c x^4} \left (4 a+c x^4\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^4]*(4*a + c*x^4) - 3*a^(3/2)*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/6

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Maple [A]  time = 0.023, size = 57, normalized size = 1. \[ -{\frac{1}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) }+{\frac{c{x}^{4}}{6}\sqrt{c{x}^{4}+a}}+{\frac{2\,a}{3}\sqrt{c{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x,x)

[Out]

-1/2*a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)+1/6*c*x^4*(c*x^4+a)^(1/2)+2
/3*a*(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((c*x^4 + a)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261046, size = 1, normalized size = 0.02 \[ \left [\frac{1}{4} \, a^{\frac{3}{2}} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + \frac{1}{6} \,{\left (c x^{4} + 4 \, a\right )} \sqrt{c x^{4} + a}, -\frac{1}{2} \, \sqrt{-a} a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right ) + \frac{1}{6} \,{\left (c x^{4} + 4 \, a\right )} \sqrt{c x^{4} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*a^(3/2)*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) + 1/6*(c*x^4 + 4
*a)*sqrt(c*x^4 + a), -1/2*sqrt(-a)*a*arctan(sqrt(c*x^4 + a)/sqrt(-a)) + 1/6*(c*x
^4 + 4*a)*sqrt(c*x^4 + a)]

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Sympy [A]  time = 6.84488, size = 80, normalized size = 1.36 \[ \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{4}}{a}}}{3} + \frac{a^{\frac{3}{2}} \log{\left (\frac{c x^{4}}{a} \right )}}{4} - \frac{a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{c x^{4}}{a}} + 1 \right )}}{2} + \frac{\sqrt{a} c x^{4} \sqrt{1 + \frac{c x^{4}}{a}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218739, size = 68, normalized size = 1.15 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} + \frac{1}{6} \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{c x^{4} + a} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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