Optimal. Leaf size=98 \[ \frac {2}{5} x (c-a c x)^{3/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (-\frac {5}{2},\frac {n-3}{2};-\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 98, 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 = {6176, 6181, 132} \[ \frac {2}{5} x (c-a c x)^{3/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (-\frac {5}{2},\frac {n-3}{2};-\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]
Antiderivative was successfully verified.
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Rule 132
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac {(c-a c x)^{3/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {2}{5} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-3+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{3/2} \, _2F_1\left (-\frac {5}{2},\frac {1}{2} (-3+n);-\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 101, normalized size = 1.03 \[ -\frac {2 c (a x+1)^2 \sqrt {c-a c x} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left (\frac {a x-1}{a x+1}\right )^{\frac {n-1}{2}} \, _2F_1\left (-\frac {5}{2},\frac {n-3}{2};-\frac {3}{2};\frac {2}{a x+1}\right )}{5 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, 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 = 0.38, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{3/2} \,d x \]
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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