Optimal. Leaf size=195 \[ \frac{7 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{2} \sqrt{1-a x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}+\frac{(a x+1)^{3/2}}{a \sqrt{1-a x} \sqrt{c-\frac{c}{a x}}}+\frac{2 \sqrt{1-a x} \sqrt{a x+1}}{a \sqrt{c-\frac{c}{a x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.170096, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6134, 6129, 97, 154, 157, 54, 215, 93, 206} \[ \frac{7 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{2} \sqrt{1-a x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}}}+\frac{(a x+1)^{3/2}}{a \sqrt{1-a x} \sqrt{c-\frac{c}{a x}}}+\frac{2 \sqrt{1-a x} \sqrt{a x+1}}{a \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6134
Rule 6129
Rule 97
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\frac{\sqrt{1-a x} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{x}}{\sqrt{1-a x}} \, dx}{\sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{\sqrt{1-a x} \int \frac{\sqrt{x} (1+a x)^{3/2}}{(1-a x)^2} \, dx}{\sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{(1+a x)^{3/2}}{a \sqrt{c-\frac{c}{a x}} \sqrt{1-a x}}-\frac{\sqrt{1-a x} \int \frac{\sqrt{1+a x} \left (\frac{1}{2}+2 a x\right )}{\sqrt{x} (1-a x)} \, dx}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{2 \sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}+\frac{(1+a x)^{3/2}}{a \sqrt{c-\frac{c}{a x}} \sqrt{1-a x}}+\frac{\sqrt{1-a x} \int \frac{-\frac{3 a}{2}-\frac{7 a^2 x}{2}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a^2 \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{2 \sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}+\frac{(1+a x)^{3/2}}{a \sqrt{c-\frac{c}{a x}} \sqrt{1-a x}}+\frac{\left (7 \sqrt{1-a x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a \sqrt{c-\frac{c}{a x}} \sqrt{x}}-\frac{\left (5 \sqrt{1-a x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{2 \sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}+\frac{(1+a x)^{3/2}}{a \sqrt{c-\frac{c}{a x}} \sqrt{1-a x}}+\frac{\left (7 \sqrt{1-a x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}-\frac{\left (10 \sqrt{1-a x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ &=\frac{2 \sqrt{1-a x} \sqrt{1+a x}}{a \sqrt{c-\frac{c}{a x}}}+\frac{(1+a x)^{3/2}}{a \sqrt{c-\frac{c}{a x}} \sqrt{1-a x}}+\frac{7 \sqrt{1-a x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{c-\frac{c}{a x}} \sqrt{x}}-\frac{5 \sqrt{2} \sqrt{1-a x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{a^{3/2} \sqrt{c-\frac{c}{a x}} \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0990348, size = 120, normalized size = 0.62 \[ \frac{\sqrt{a} \sqrt{x} \sqrt{a x+1} (3-a x)+(7-7 a x) \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+5 \sqrt{2} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{x} \sqrt{1-a x} \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.146, size = 276, normalized size = 1.4 \begin{align*}{\frac{x\sqrt{2}}{4\,c \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x-7\,{a}^{2}\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}x-6\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}+10\,{a}^{3/2}\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ) x+7\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}-10\,\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ) \sqrt{a} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{-{a}^{-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{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.63296, size = 1131, normalized size = 5.8 \begin{align*} \left [\frac{5 \, \sqrt{2}{\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt{-\frac{1}{c}} \log \left (-\frac{17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 4 \, \sqrt{2}{\left (3 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - 7 \,{\left (a^{2} x^{2} - 2 \, a x + 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 x^{2} - 2 \, a^{2} c x + a c\right )}}, \frac{7 \,{\left (a^{2} x^{2} - 2 \, a x + 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}} - \frac{5 \, \sqrt{2}{\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}}}{{\left (3 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt{c}}\right )}{\sqrt{c}}}{2 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x + 1\right )^{3}}{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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 x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]