Integrand size = 23, antiderivative size = 108 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=-\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {778, 372, 371, 272, 67} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=-\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)} \]
[In]
[Out]
Rule 67
Rule 272
Rule 371
Rule 372
Rule 778
Rubi steps \begin{align*} \text {integral}& = d \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2} \, dx+e \int \frac {\left (d^2-e^2 x^2\right )^p}{x} \, dx \\ & = \frac {1}{2} e \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int \frac {\left (1-\frac {e^2 x^2}{d^2}\right )^p}{x^2} \, dx \\ & = -\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=-\frac {d \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)} \]
[In]
[Out]
\[\int \frac {\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{2}}d x\]
[In]
[Out]
\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=- \frac {d d^{2 p} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac {e e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \]
[In]
[Out]
\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right )}{x^2} \,d x \]
[In]
[Out]