Optimal. Leaf size=131 \[ -\frac{3 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{3 \sqrt{a x+1} (1-a x)^{3/2}}{a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 (1-a x)^{3/2}}{a \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.193546, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6134, 6128, 848, 47, 50, 54, 215} \[ -\frac{3 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{3 \sqrt{a x+1} (1-a x)^{3/2}}{a^2 x \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 (1-a x)^{3/2}}{a \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6134
Rule 6128
Rule 848
Rule 47
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=\frac{(1-a x)^{3/2} \int \frac{e^{-3 \tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{(1-a x)^{3/2} \int \frac{x^{3/2} (1-a x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{(1-a x)^{3/2} \int \frac{x^{3/2}}{(1+a x)^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{2 (1-a x)^{3/2}}{a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1+a x}}+\frac{\left (3 (1-a x)^{3/2}\right ) \int \frac{\sqrt{x}}{\sqrt{1+a x}} \, dx}{a \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{2 (1-a x)^{3/2}}{a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1+a x}}+\frac{3 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{\left (3 (1-a x)^{3/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{2 (1-a x)^{3/2}}{a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1+a x}}+\frac{3 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{\left (3 (1-a x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{2 (1-a x)^{3/2}}{a \left (c-\frac{c}{a x}\right )^{3/2} \sqrt{1+a x}}+\frac{3 (1-a x)^{3/2} \sqrt{1+a x}}{a^2 \left (c-\frac{c}{a x}\right )^{3/2} x}-\frac{3 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{5/2} \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0612196, size = 44, normalized size = 0.34 \[ \frac{2 x (1-a x)^{3/2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},-a x\right )}{5 \left (c-\frac{c}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.137, size = 143, normalized size = 1.1 \begin{align*} -{\frac{x}{2\,{c}^{2} \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+3\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) xa+6\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}+3\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.13781, size = 609, normalized size = 4.65 \begin{align*} \left [-\frac{3 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{3} c^{2} x^{2} - a c^{2}\right )}}, \frac{3 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{3} c^{2} x^{2} - a c^{2}\right )}}\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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]