Optimal. Leaf size=78 \[ \frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \, _2F_1\left (\frac {1}{2} (-1-n),-\frac {n}{2};\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{a c (1+n) \sqrt {c-a c x}} \]
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Rubi [A]
time = 0.05, antiderivative size = 78, 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} (1-a x)^{-n/2} \, _2F_1\left (\frac {1}{2} (-n-1),-\frac {n}{2};\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{a c (n+1) \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 71
Rule 6265
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\int \frac {(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} \, _2F_1\left (\frac {1}{2} (-1-n),-\frac {n}{2};\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{a c (1+n) \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 78, normalized size = 1.00 \begin {gather*} \frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \, _2F_1\left (-\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {1}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a c (1+n) \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (-c x a +c \right )^{\frac {3}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {{\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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