Optimal. Leaf size=293 \[ \frac{103 \sqrt{a x+1} (1-a x)^{5/2}}{32 a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{43 (a x+1)^{3/2} (1-a x)^{3/2}}{32 a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{11 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{249 (1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{16 \sqrt{2} a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{13 (a x+1)^{3/2} \sqrt{1-a x}}{24 a^2 x \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(a x+1)^{3/2}}{3 a \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.231221, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6134, 6129, 97, 149, 154, 157, 54, 215, 93, 206} \[ \frac{103 \sqrt{a x+1} (1-a x)^{5/2}}{32 a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{43 (a x+1)^{3/2} (1-a x)^{3/2}}{32 a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{11 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{249 (1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{16 \sqrt{2} a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{13 (a x+1)^{3/2} \sqrt{1-a x}}{24 a^2 x \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(a x+1)^{3/2}}{3 a \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 97
Rule 149
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx &=\frac{(1-a x)^{5/2} \int \frac{e^{3 \tanh ^{-1}(a x)} x^{5/2}}{(1-a x)^{5/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2} \int \frac{x^{5/2} (1+a x)^{3/2}}{(1-a x)^4} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{(1-a x)^{5/2} \int \frac{x^{3/2} \sqrt{1+a x} \left (\frac{5}{2}+4 a x\right )}{(1-a x)^3} \, dx}{3 a \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{13 \sqrt{1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}-\frac{(1-a x)^{5/2} \int \frac{\sqrt{x} \sqrt{1+a x} \left (-\frac{39 a}{4}-\frac{45 a^2 x}{2}\right )}{(1-a x)^2} \, dx}{12 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{13 \sqrt{1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}-\frac{(1-a x)^{5/2} \int \frac{\sqrt{1+a x} \left (\frac{129 a^2}{8}+\frac{309 a^3 x}{4}\right )}{\sqrt{x} (1-a x)} \, dx}{24 a^5 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{103 (1-a x)^{5/2} \sqrt{1+a x}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{13 \sqrt{1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1-a x)^{5/2} \int \frac{-\frac{219 a^3}{4}-132 a^4 x}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{24 a^6 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{103 (1-a x)^{5/2} \sqrt{1+a x}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{13 \sqrt{1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{\left (11 (1-a x)^{5/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}-\frac{\left (249 (1-a x)^{5/2}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{103 (1-a x)^{5/2} \sqrt{1+a x}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{13 \sqrt{1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{\left (11 (1-a x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}-\frac{\left (249 (1-a x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{16 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{103 (1-a x)^{5/2} \sqrt{1+a x}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1+a x)^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{5/2} \sqrt{1-a x}}-\frac{13 \sqrt{1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac{c}{a x}\right )^{5/2} x}+\frac{43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{11 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}-\frac{249 (1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{16 \sqrt{2} a^{7/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.160495, size = 147, normalized size = 0.5 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{a x+1} \left (-48 a^3 x^3+415 a^2 x^2-554 a x+219\right )-1056 (a x-1)^3 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+747 \sqrt{2} (a x-1)^3 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{96 a^{3/2} c^2 \sqrt{x} (1-a x)^{5/2} \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.177, size = 504, normalized size = 1.7 \begin{align*}{\frac{x\sqrt{2}}{192\,{c}^{3} \left ( ax-1 \right ) ^{4}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 96\,{a}^{9/2}\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}{x}^{3}-830\,\sqrt{- \left ( ax+1 \right ) x}{a}^{7/2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}+747\,{a}^{7/2}\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}^{3}-528\,{a}^{4}\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}{x}^{3}+1108\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x-2241\,{a}^{5/2}\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}^{2}+1584\,{a}^{3}\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}-438\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-1584\,{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+2241\,{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+528\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}-747\,\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.
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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}}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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}}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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