Optimal. Leaf size=54 \[ -\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x F_1\left (-\frac {1}{2};\frac {1}{2} (-3+n),-\frac {n}{2};\frac {1}{2};a x,-a x\right )}{(1-a x)^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6269, 6264,
138} \begin {gather*} -\frac {2 x \left (c-\frac {c}{a x}\right )^{3/2} F_1\left (-\frac {1}{2};\frac {n-3}{2},-\frac {n}{2};\frac {1}{2};a x,-a x\right )}{(1-a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{n \tanh ^{-1}(a x)} (1-a x)^{3/2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {(1-a x)^{\frac {3}{2}-\frac {n}{2}} (1+a x)^{n/2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x F_1\left (-\frac {1}{2};\frac {1}{2} (-3+n),-\frac {n}{2};\frac {1}{2};a x,-a x\right )}{(1-a x)^{3/2}}\\ \end {align*}
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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} \left (c -\frac {c}{a x}\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a\,x}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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