Integrand size = 16, antiderivative size = 143 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p}{1+p}+\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} \sqrt {1+\frac {1}{a x}} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,1-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96, 134} \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {\frac {1}{a x}+1} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,1-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (p+1) \sqrt {1-\frac {1}{a x}}}+\frac {x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} (c-a c x)^p}{p+1} \]
[In]
[Out]
Rule 96
Rule 134
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p x^p \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \text {Subst}\left (\int x^{-2-p} \left (1-\frac {x}{a}\right )^{-\frac {1}{2}+p} \sqrt {1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p}{1+p}-\frac {\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \text {Subst}\left (\int \frac {x^{-1-p} \left (1-\frac {x}{a}\right )^{-\frac {1}{2}+p}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a (1+p)} \\ & = \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p}{1+p}+\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} \sqrt {1+\frac {1}{a x}} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,1-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.92 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {1+\frac {1}{a x}} \left (\frac {-1+a x}{1+a x}\right )^{-p} (c-a c x)^p \left (p (-1+a x) \left (\frac {-1+a x}{1+a x}\right )^p+\sqrt {\frac {-1+a x}{1+a x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,1-p,\frac {2}{1+a x}\right )\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
\[\int \frac {\left (-a c x +c \right )^{p}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \frac {{\left (c-a\,c\,x\right )}^p}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
[In]
[Out]