Optimal. Leaf size=141 \[ \frac{256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a c x}}{21 a}+\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a} \]
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Rubi [A] time = 0.104202, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 657, 649} \[ \frac{256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a c x}}{21 a}+\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 657
Rule 649
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=c \int (c-a c x)^{5/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac{1}{3} \left (4 c^2\right ) \int (c-a c x)^{3/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{8 c^3 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac{1}{21} \left (32 c^3\right ) \int \sqrt{c-a c x} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac{1}{105} \left (128 c^4\right ) \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac{64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt{c-a c x}}+\frac{8 c^3 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac{2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}\\ \end{align*}
Mathematica [A] time = 0.0413035, size = 62, normalized size = 0.44 \[ -\frac{2 c^3 (a x+1)^{3/2} \left (35 a^3 x^3-165 a^2 x^2+321 a x-319\right ) \sqrt{c-a c x}}{315 a \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.033, size = 63, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( 35\,{x}^{3}{a}^{3}-165\,{a}^{2}{x}^{2}+321\,ax-319 \right ) \left ( ax+1 \right ) ^{2}}{315\, \left ( ax-1 \right ) ^{3}a} \left ( -acx+c \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02646, size = 173, normalized size = 1.23 \begin{align*} -\frac{2 \,{\left (5 \, a^{5} c^{\frac{7}{2}} x^{5} - 20 \, a^{4} c^{\frac{7}{2}} x^{4} + 32 \, a^{3} c^{\frac{7}{2}} x^{3} - 34 \, a^{2} c^{\frac{7}{2}} x^{2} + 91 \, a c^{\frac{7}{2}} x + 182 \, c^{\frac{7}{2}}\right )}}{45 \, \sqrt{a x + 1} a} - \frac{2 \,{\left (5 \, a^{4} c^{\frac{7}{2}} x^{4} - 22 \, a^{3} c^{\frac{7}{2}} x^{3} + 44 \, a^{2} c^{\frac{7}{2}} x^{2} - 106 \, a c^{\frac{7}{2}} x - 177 \, c^{\frac{7}{2}}\right )}}{35 \, \sqrt{a x + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58907, size = 176, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (35 \, a^{4} c^{3} x^{4} - 130 \, a^{3} c^{3} x^{3} + 156 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{315 \,{\left (a^{2} x - a\right )}} \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 [A] time = 1.3117, size = 99, normalized size = 0.7 \begin{align*} -\frac{2 \,{\left (256 \, \sqrt{2} c^{\frac{5}{2}} + \frac{35 \,{\left (a c x + c\right )}^{\frac{9}{2}} - 270 \,{\left (a c x + c\right )}^{\frac{7}{2}} c + 756 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{2} - 840 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{3}}{c^{2}}\right )} c^{2}}{315 \, a{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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