Optimal. Leaf size=31 \[ \frac {2 (a x+1) (c-a c x)^{3/2} e^{3 \coth ^{-1}(a x)}}{5 a} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6174} \[ \frac {2 (a x+1) (c-a c x)^{3/2} e^{3 \coth ^{-1}(a x)}}{5 a} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac {2 e^{3 \coth ^{-1}(a x)} (1+a x) (c-a c x)^{3/2}}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 1.39 \[ \frac {2 x \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 61, normalized size = 1.97 \[ -\frac {2 \, {\left (a^{3} c x^{3} + 3 \, a^{2} c x^{2} + 3 \, a c x + c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 56, normalized size = 1.81 \[ -\frac {2 \, {\left (\frac {4 \, \sqrt {2} \sqrt {-c} c}{\mathrm {sgn}\relax (c)} + \frac {{\left (a c x + c\right )}^{2} \sqrt {-a c x - c}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.13 \[ \frac {2 \left (a x +1\right ) \left (-a c x +c \right )^{\frac {3}{2}}}{5 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 41, normalized size = 1.32 \[ -\frac {2 \, {\left (a^{2} \sqrt {-c} c x^{2} - \sqrt {-c} c\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{5 \, {\left (a x - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 81, normalized size = 2.61 \[ -\frac {2\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^2\,x^2+4\,a\,x+7\right )}{5\,a}-\frac {16\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{5\,a\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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