Optimal. Leaf size=157 \[ \frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{16 \sqrt{2} a c^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{4 a (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.133522, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6127, 663, 673, 661, 208} \[ \frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{16 \sqrt{2} a c^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{4 a (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 663
Rule 673
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{11/2}} \, dx\\ &=\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}-\frac{1}{2} c \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{7/2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 a (c-a c x)^{5/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}+\frac{\int \frac{1}{(c-a c x)^{3/2} \sqrt{1-a^2 x^2}} \, dx}{8 c}\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 a (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c (c-a c x)^{3/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}+\frac{\int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{32 c^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 a (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c (c-a c x)^{3/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )}{16 c}\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 a (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{16 a c (c-a c x)^{3/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{9/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{16 \sqrt{2} a c^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0242673, size = 57, normalized size = 0.36 \[ \frac{(a x+1)^{5/2} (c-a c x)^{3/2} \text{Hypergeometric2F1}\left (\frac{5}{2},4,\frac{7}{2},\frac{1}{2} (a x+1)\right )}{40 a c^4 (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.112, size = 208, normalized size = 1.3 \begin{align*} -{\frac{1}{96\, \left ( ax-1 \right ) ^{4}a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{3}{a}^{3}c-9\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c-6\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+9\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac-44\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c-14\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \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 (-a c x + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27386, size = 817, normalized size = 5.2 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \,{\left (3 \, a^{2} x^{2} + 22 \, a x + 7\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, \frac{3 \, \sqrt{2}{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + 2 \,{\left (3 \, a^{2} x^{2} + 22 \, a x + 7\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{96 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\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{\left (a x + 1\right )^{3}}{\left (- c \left (a x - 1\right )\right )^{\frac{5}{2}} \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.
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Giac [A] time = 1.38714, size = 124, normalized size = 0.79 \begin{align*} -\frac{\frac{3 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{2 \,{\left (3 \,{\left (a c x + c\right )}^{\frac{5}{2}} + 16 \,{\left (a c x + c\right )}^{\frac{3}{2}} c - 12 \, \sqrt{a c x + c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c}}{96 \, a{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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