Optimal. Leaf size=80 \[ \frac {2 x \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{n/2} (c-a c x)^{\frac {n-2}{2}} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 80, 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 = {6176, 6181, 131} \[ \frac {2 x \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{n/2} (c-a c x)^{\frac {n-2}{2}} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 131
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx &=\left (\left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} x^{1-\frac {n}{2}} (c-a c x)^{-1+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{-1+\frac {n}{2}} x^{-1+\frac {n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{x}\right )^{-1+\frac {n}{2}} (c-a c x)^{-1+\frac {n}{2}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x (c-a c x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 78, normalized size = 0.98 \[ -\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} (c-a c x)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {2}{a x+1}\right )}{a c n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{\frac {1}{2} \, n - 1} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 1} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{-1+\frac {n}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 1} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (a x - 1\right )\right )^{\frac {n}{2} - 1} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________