Optimal. Leaf size=105 \[ -\frac{11 x \sqrt{c-\frac{c}{a x}}}{8 a^2}-\frac{11 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{8 a^3}-\frac{1}{3} x^3 \sqrt{c-\frac{c}{a x}}-\frac{11 x^2 \sqrt{c-\frac{c}{a x}}}{12 a} \]
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Rubi [A] time = 0.217048, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6133, 25, 514, 446, 78, 51, 63, 208} \[ -\frac{11 x \sqrt{c-\frac{c}{a x}}}{8 a^2}-\frac{11 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{8 a^3}-\frac{1}{3} x^3 \sqrt{c-\frac{c}{a x}}-\frac{11 x^2 \sqrt{c-\frac{c}{a x}}}{12 a} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\int \frac{\sqrt{c-\frac{c}{a x}} x^2 (1+a x)}{1-a x} \, dx\\ &=-\frac{c \int \frac{x (1+a x)}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=-\frac{c \int \frac{\left (a+\frac{1}{x}\right ) x^2}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^4 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3+\frac{(11 c) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 a}\\ &=-\frac{11 \sqrt{c-\frac{c}{a x}} x^2}{12 a}-\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3+\frac{(11 c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2}\\ &=-\frac{11 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2}{12 a}-\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3+\frac{(11 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^3}\\ &=-\frac{11 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2}{12 a}-\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3-\frac{11 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{8 a^2}\\ &=-\frac{11 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2}{12 a}-\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3-\frac{11 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{8 a^3}\\ \end{align*}
Mathematica [C] time = 0.0348243, size = 50, normalized size = 0.48 \[ -\frac{\sqrt{c-\frac{c}{a x}} \left (11 \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},1-\frac{1}{a x}\right )+a^3 x^3\right )}{3 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.128, size = 155, normalized size = 1.5 \begin{align*} -{\frac{x}{48}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{5/2}+60\,\sqrt{a{x}^{2}-x}{a}^{5/2}x-30\,\sqrt{a{x}^{2}-x}{a}^{3/2}+96\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}+48\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) -15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a x + 1\right )}^{2} \sqrt{c - \frac{c}{a x}} x^{2}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.883, size = 373, normalized size = 3.55 \begin{align*} \left [-\frac{2 \,{\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} - 33 \, \sqrt{c} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{48 \, a^{3}}, -\frac{{\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} - 33 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x^{3} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28809, size = 173, normalized size = 1.65 \begin{align*} -\frac{1}{24} \, \sqrt{a^{2} c x^{2} - a c x}{\left (2 \, x{\left (\frac{4 \, x{\left | a \right |}}{a^{2} \mathrm{sgn}\left (x\right )} + \frac{11 \,{\left | a \right |}}{a^{3} \mathrm{sgn}\left (x\right )}\right )} + \frac{33 \,{\left | a \right |}}{a^{4} \mathrm{sgn}\left (x\right )}\right )} - \frac{11 \, \sqrt{c} \log \left ({\left | a \right |} \sqrt{{\left | c \right |}}\right ) \mathrm{sgn}\left (x\right )}{16 \, a^{3}} + \frac{11 \, \sqrt{c} \log \left ({\left | -2 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}{\left | a \right |} + a \sqrt{c} \right |}\right )}{16 \, a^{3} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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