Optimal. Leaf size=248 \[ -\frac{3 x (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{4 a^2 \sqrt{1-a x}}-\frac{13 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-a x}}-\frac{45 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{5/2} \sqrt{1-a x}}-\frac{x^2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 a \sqrt{1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.292, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6134, 6129, 101, 154, 157, 54, 215, 93, 206} \[ -\frac{3 x (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{4 a^2 \sqrt{1-a x}}-\frac{13 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-a x}}-\frac{45 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{5/2} \sqrt{1-a x}}-\frac{x^2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 a \sqrt{1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6134
Rule 6129
Rule 101
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{3 \tanh ^{-1}(a x)} x^{3/2} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2} (1+a x)^{3/2}}{1-a x} \, dx}{\sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x} \sqrt{1+a x} \left (\frac{3}{2}+\frac{9 a x}{2}\right )}{1-a x} \, dx}{3 a \sqrt{1-a x}}\\ &=-\frac{3 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{4 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt{1-a x}}-\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1+a x} \left (-\frac{9 a}{4}-\frac{39 a^2 x}{4}\right )}{\sqrt{x} (1-a x)} \, dx}{6 a^3 \sqrt{1-a x}}\\ &=-\frac{13 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{3 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{4 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\frac{57 a^2}{8}+\frac{135 a^3 x}{8}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{6 a^4 \sqrt{1-a x}}\\ &=-\frac{13 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{3 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{4 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt{1-a x}}-\frac{\left (45 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{16 a^2 \sqrt{1-a x}}+\frac{\left (4 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a^2 \sqrt{1-a x}}\\ &=-\frac{13 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{3 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{4 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt{1-a x}}-\frac{\left (45 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{8 a^2 \sqrt{1-a x}}+\frac{\left (8 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a^2 \sqrt{1-a x}}\\ &=-\frac{13 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}-\frac{3 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{4 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt{1-a x}}-\frac{45 \sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{a^{5/2} \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0831088, size = 122, normalized size = 0.49 \[ -\frac{\sqrt{c-\frac{c}{a x}} \left (\sqrt{a} x \sqrt{a x+1} \left (8 a^2 x^2+26 a x+57\right )+135 \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-96 \sqrt{2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{24 a^{5/2} \sqrt{1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.138, size = 219, normalized size = 0.9 \begin{align*}{\frac{x\sqrt{2}}{96\,ax-96}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 16\,\sqrt{- \left ( ax+1 \right ) x}{a}^{7/2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}+52\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x+114\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-135\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+192\,\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{7}{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} \sqrt{c - \frac{c}{a x}} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.69645, size = 1073, normalized size = 4.33 \begin{align*} \left [\frac{96 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2}{\left (3 \, 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^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 135 \,{\left (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 (8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \,{\left (a^{4} x - a^{3}\right )}}, -\frac{96 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 135 \,{\left (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 (8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \,{\left (a^{4} x - a^{3}\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{x^{2} \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{\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} \sqrt{c - \frac{c}{a x}} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]