Optimal. Leaf size=157 \[ -\frac{\sqrt{1-a^2 x^2}}{32 a c^2 (c-a c x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{32 \sqrt{2} a c^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}} \]
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Rubi [A] time = 0.130479, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6127, 663, 673, 661, 208} \[ -\frac{\sqrt{1-a^2 x^2}}{32 a c^2 (c-a c x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{32 \sqrt{2} a c^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c (c-a c x)^{5/2}}+\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}} \]
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^{\tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{9/2}} \, dx\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}}-\frac{\int \frac{1}{(c-a c x)^{5/2} \sqrt{1-a^2 x^2}} \, dx}{6 c}\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c (c-a c x)^{5/2}}-\frac{\int \frac{1}{(c-a c x)^{3/2} \sqrt{1-a^2 x^2}} \, dx}{16 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c (c-a c x)^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^2 (c-a c x)^{3/2}}-\frac{\int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{64 c^3}\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c (c-a c x)^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^2 (c-a c x)^{3/2}}+\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 )}{32 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a (c-a c x)^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c (c-a c x)^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{32 a c^2 (c-a c x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{32 \sqrt{2} a c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.01961, size = 57, normalized size = 0.36 \[ \frac{(a x+1)^{3/2} \sqrt{c-a c x} \text{Hypergeometric2F1}\left (\frac{3}{2},4,\frac{5}{2},\frac{1}{2} (a x+1)\right )}{24 a c^4 \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.105, size = 208, normalized size = 1.3 \begin{align*}{\frac{1}{192\, \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+20\,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+50\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{9}{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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (-a c x + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14033, size = 822, normalized size = 5.24 \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} - 10 \, a x - 25\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{384 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\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} - 10 \, a x - 25\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\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{a x + 1}{\left (- c \left (a x - 1\right )\right )^{\frac{7}{2}} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31744, 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^{2}} + \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^{2}}}{192 \, a{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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