Optimal. Leaf size=35 \[ \frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a (c-a c x)^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6127, 649} \[ \frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 6127
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a (c-a c x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.06 \[ \frac {2 (a x+1)^{5/2} (c-a c x)^{3/2}}{5 a (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 49, normalized size = 1.40 \[ -\frac {2 \, {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{5 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 34, normalized size = 0.97 \[ \frac {2 \left (a x +1\right )^{4} \left (-a c x +c \right )^{\frac {3}{2}}}{5 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 121, normalized size = 3.46 \[ -\frac {2 \, c^{\frac {3}{2}}}{\sqrt {a x + 1} a} + \frac {2 \, {\left (a^{3} c^{\frac {3}{2}} x^{3} - 2 \, a^{2} c^{\frac {3}{2}} x^{2} + 8 \, a c^{\frac {3}{2}} x + 16 \, c^{\frac {3}{2}}\right )}}{5 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (a^{2} c^{\frac {3}{2}} x^{2} - 4 \, a c^{\frac {3}{2}} x - 8 \, c^{\frac {3}{2}}\right )}}{\sqrt {a x + 1} a} + \frac {6 \, {\left (a c^{\frac {3}{2}} x + 2 \, c^{\frac {3}{2}}\right )}}{\sqrt {a x + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 49, normalized size = 1.40 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {6\,c\,x}{5}+\frac {2\,c}{5\,a}+\frac {2\,a^2\,c\,x^3}{5}+\frac {6\,a\,c\,x^2}{5}\right )}{\sqrt {1-a^2\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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