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]
________________________________________________________________________________________
Rubi [A] time = 0.0347514, 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, Rules used = {266, 50, 63, 208} \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )+\frac{1}{2} a \sqrt{a+c x^4}+\frac{1}{6} \left (a+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x} \, dx,x,x^4\right )\\ &=\frac{1}{6} \left (a+c x^4\right )^{3/2}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac{1}{2} a \sqrt{a+c x^4}+\frac{1}{6} \left (a+c x^4\right )^{3/2}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=\frac{1}{2} a \sqrt{a+c x^4}+\frac{1}{6} \left (a+c x^4\right )^{3/2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )}{2 c}\\ &=\frac{1}{2} a \sqrt{a+c x^4}+\frac{1}{6} \left (a+c x^4\right )^{3/2}-\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0209842, 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]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 57, normalized size = 1. \begin{align*}{\frac{c{x}^{4}}{6}\sqrt{c{x}^{4}+a}}+{\frac{2\,a}{3}\sqrt{c{x}^{4}+a}}-{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50878, size = 258, normalized size = 4.37 \begin{align*} \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}}{a}\right ) + \frac{1}{6} \,{\left (c x^{4} + 4 \, a\right )} \sqrt{c x^{4} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.57206, size = 80, normalized size = 1.36 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11862, size = 68, normalized size = 1.15 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]