Optimal. Leaf size=130 \[ \frac{25 x^2 \sqrt{c-\frac{c}{a x}}}{32 a^2}+\frac{75 x \sqrt{c-\frac{c}{a x}}}{64 a^3}+\frac{75 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{64 a^4}+\frac{1}{4} x^4 \sqrt{c-\frac{c}{a x}}+\frac{5 x^3 \sqrt{c-\frac{c}{a x}}}{8 a} \]
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Rubi [A] time = 0.377355, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6133, 25, 514, 446, 78, 51, 63, 208} \[ \frac{25 x^2 \sqrt{c-\frac{c}{a x}}}{32 a^2}+\frac{75 x \sqrt{c-\frac{c}{a x}}}{64 a^3}+\frac{75 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{64 a^4}+\frac{1}{4} x^4 \sqrt{c-\frac{c}{a x}}+\frac{5 x^3 \sqrt{c-\frac{c}{a x}}}{8 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)}{1-a x} \, dx\\ &=\frac{c \int \frac{x^2 (1+a x)}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=\frac{c \int \frac{\left (a+\frac{1}{x}\right ) x^3}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^5 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{(15 c) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=\frac{5 \sqrt{c-\frac{c}{a x}} x^3}{8 a}+\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{(25 c) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^2}\\ &=\frac{25 \sqrt{c-\frac{c}{a x}} x^2}{32 a^2}+\frac{5 \sqrt{c-\frac{c}{a x}} x^3}{8 a}+\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{(75 c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{64 a^3}\\ &=\frac{75 \sqrt{c-\frac{c}{a x}} x}{64 a^3}+\frac{25 \sqrt{c-\frac{c}{a x}} x^2}{32 a^2}+\frac{5 \sqrt{c-\frac{c}{a x}} x^3}{8 a}+\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4-\frac{(75 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{128 a^4}\\ &=\frac{75 \sqrt{c-\frac{c}{a x}} x}{64 a^3}+\frac{25 \sqrt{c-\frac{c}{a x}} x^2}{32 a^2}+\frac{5 \sqrt{c-\frac{c}{a x}} x^3}{8 a}+\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4+\frac{75 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{64 a^3}\\ &=\frac{75 \sqrt{c-\frac{c}{a x}} x}{64 a^3}+\frac{25 \sqrt{c-\frac{c}{a x}} x^2}{32 a^2}+\frac{5 \sqrt{c-\frac{c}{a x}} x^3}{8 a}+\frac{1}{4} \sqrt{c-\frac{c}{a x}} x^4+\frac{75 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{64 a^4}\\ \end{align*}
Mathematica [C] time = 0.0380654, size = 50, normalized size = 0.38 \[ \frac{\sqrt{c-\frac{c}{a x}} \left (15 \text{Hypergeometric2F1}\left (\frac{1}{2},4,\frac{3}{2},1-\frac{1}{a x}\right )+a^4 x^4\right )}{4 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.166, size = 172, normalized size = 1.3 \begin{align*}{\frac{x}{128}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 32\,x \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{7/2}+112\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{5/2}+212\,\sqrt{a{x}^{2}-x}{a}^{5/2}x-106\,\sqrt{a{x}^{2}-x}{a}^{3/2}+256\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}+128\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) -53\,\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{9}{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 )} \sqrt{c - \frac{c}{a x}} x^{3}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59547, size = 409, normalized size = 3.15 \begin{align*} \left [\frac{2 \,{\left (16 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 75 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} + 75 \, \sqrt{c} \log \left (-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{128 \, a^{4}}, \frac{{\left (16 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 75 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} - 75 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{64 \, a^{4}}\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^{3} \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2295, size = 193, normalized size = 1.48 \begin{align*} \frac{1}{64} \, \sqrt{a^{2} c x^{2} - a c x}{\left (2 \,{\left (4 \, x{\left (\frac{2 \, x{\left | a \right |}}{a^{2} \mathrm{sgn}\left (x\right )} + \frac{5 \,{\left | a \right |}}{a^{3} \mathrm{sgn}\left (x\right )}\right )} + \frac{25 \,{\left | a \right |}}{a^{4} \mathrm{sgn}\left (x\right )}\right )} x + \frac{75 \,{\left | a \right |}}{a^{5} \mathrm{sgn}\left (x\right )}\right )} + \frac{75 \, \sqrt{c} \log \left ({\left | a \right |} \sqrt{{\left | c \right |}}\right ) \mathrm{sgn}\left (x\right )}{128 \, a^{4}} - \frac{75 \, \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 )}{128 \, a^{4} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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