Optimal. Leaf size=81 \[ -\frac {2^{\frac {n}{2}+1} (c-a c x)^{5/2} (1-a x)^{-n/2} \, _2F_1\left (\frac {5-n}{2},-\frac {n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a c (5-n)} \]
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Rubi [A] time = 0.07, 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 = {6130, 23, 69} \[ -\frac {2^{\frac {n}{2}+1} (c-a c x)^{5/2} (1-a x)^{-n/2} \, _2F_1\left (\frac {5-n}{2},-\frac {n}{2};\frac {7-n}{2};\frac {1}{2} (1-a x)\right )}{a c (5-n)} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
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 \[ \frac {c 2^{\frac {n}{2}+1} \sqrt {c-a c x} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (\frac {5}{2}-\frac {n}{2},-\frac {n}{2};\frac {7}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a (n-5)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c x - c\right )} \sqrt {-a c x + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a c x +c \right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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