Optimal. Leaf size=56 \[ \frac {2 x (1-a x)^{3/2} F_1\left (\frac {5}{2};\frac {n+3}{2},-\frac {n}{2};\frac {7}{2};a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6134, 6129, 133} \[ \frac {2 x (1-a x)^{3/2} F_1\left (\frac {5}{2};\frac {n+3}{2},-\frac {n}{2};\frac {7}{2};a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\frac {(1-a x)^{3/2} \int \frac {e^{n \tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {(1-a x)^{3/2} \int x^{3/2} (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{n/2} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {2 x (1-a x)^{3/2} F_1\left (\frac {5}{2};\frac {3+n}{2},-\frac {n}{2};\frac {7}{2};a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a c x - c}{a x}}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________