Optimal. Leaf size=81 \[ -\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} (c-a c x)^{5/2} \, _2F_1\left (\frac {5-n}{2},-\frac {n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a c (5-n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6265, 23, 71}
\begin {gather*} -\frac {2^{\frac {n}{2}+1} (c-a c x)^{5/2} (1-a x)^{-n/2} \, _2F_1\left (\frac {5-n}{2},-\frac {n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a c (5-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 71
Rule 6265
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\int (1-a x)^{-n/2} (1+a x)^{n/2} (c-a c x)^{3/2} \, dx\\ &=\left ((1-a x)^{-n/2} (c-a c x)^{n/2}\right ) \int (1+a x)^{n/2} (c-a c x)^{\frac {3}{2}-\frac {n}{2}} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} (c-a c x)^{5/2} \, _2F_1\left (\frac {5-n}{2},-\frac {n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a c (5-n)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 78, normalized size = 0.96 \begin {gather*} \frac {2^{1+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \sqrt {c-a c x} \, _2F_1\left (\frac {5}{2}-\frac {n}{2},-\frac {n}{2};\frac {7}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a (-5+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-c x a +c \right )^{\frac {3}{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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