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]
________________________________________________________________________________________
Rubi [A] time = 0.0263731, 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, Rules used = {266, 47, 63, 208} \[ -\frac{\sqrt{a+c x^4}}{4 x^4}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^4}}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{a+c x^4}}{4 x^4}+\frac{1}{8} c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{a+c x^4}}{4 x^4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )\\ &=-\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}}\\ \end{align*}
Mathematica [A] time = 0.0323087, size = 59, normalized size = 1.26 \[ -\frac{c x^4 \sqrt{\frac{c x^4}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^4}{a}+1}\right )+a+c x^4}{4 x^4 \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 63, normalized size = 1.3 \begin{align*} -{\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}} \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 = 1.53498, size = 258, normalized size = 5.49 \begin{align*} \left [\frac{\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 \, \sqrt{c x^{4} + a} a}{8 \, a x^{4}}, \frac{\sqrt{-a} c x^{4} \arctan \left (\frac{\sqrt{c x^{4} + a} \sqrt{-a}}{a}\right ) - \sqrt{c x^{4} + a} a}{4 \, a x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.69864, size = 46, normalized size = 0.98 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11243, size = 58, normalized size = 1.23 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]