Optimal. Leaf size=78 \[ \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (\frac {1}{2} (-n-1),-\frac {n}{2};\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{a c (n+1) \sqrt {c-a c x}} \]
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Rubi [A] time = 0.07, 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 = {6130, 23, 69} \[ \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (\frac {1}{2} (-n-1),-\frac {n}{2};\frac {1-n}{2};\frac {1}{2} (1-a x)\right )}{a c (n+1) \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
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 = 0.03, size = 78, normalized size = 1.00 \[ \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},-\frac {n}{2};\frac {1}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a c (n+1) \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a c x + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (-a c x +c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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