Optimal. Leaf size=176 \[ \frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.267924, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6192, 6193, 88, 207} \[ \frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6192
Rule 6193
Rule 88
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^2} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{\left (a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{x^3}{(-1+a x)^3 (1+a x)^2} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{\left (a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (\frac{1}{4 a^3 (-1+a x)^3}+\frac{1}{2 a^3 (-1+a x)^2}+\frac{1}{8 a^3 (1+a x)^2}+\frac{3}{8 a^3 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{2 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (3 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{8 \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{2 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \tanh ^{-1}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0641193, size = 86, normalized size = 0.49 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (5 a^2 x^2-a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)-2\right )}{8 a^3 c^2 (a x-1)^2 (a x+1) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.142, size = 169, normalized size = 1. \begin{align*}{\frac{3\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-3\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+3\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+10\,{a}^{2}{x}^{2}-3\,ax\ln \left ( ax+1 \right ) +3\,\ln \left ( ax-1 \right ) xa-2\,ax+3\,\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ) -4}{ \left ( 16\,ax-16 \right ) \left ({a}^{2}{x}^{2}-1 \right ){c}^{3}{a}^{4} \left ( ax+1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61313, size = 275, normalized size = 1.56 \begin{align*} -\frac{3 \,{\left (a^{4} x^{3} - a^{3} x^{2} - a^{2} x + a\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c} \sqrt{-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \,{\left (5 \, a^{2} x^{2} - a x - 2\right )} \sqrt{-a^{2} c}}{16 \,{\left (a^{8} c^{3} x^{3} - a^{7} c^{3} x^{2} - a^{6} c^{3} x + a^{5} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]