Integrand size = 23, antiderivative size = 110 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=-\frac {e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (1+p)} \]
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Time = 0.04 (sec) , antiderivative size = 110, 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, 272, 67, 372, 371} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=-\frac {e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (p+1)} \]
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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^3} \, dx+e \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x^2} \, dx,x,x^2\right )+\left (e \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 {e \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^2 \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (1+p)} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\frac {1}{2} e \left (d^2-e^2 x^2\right )^p \left (-\frac {2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}+\frac {e \left (-d^2+e^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{d^3 (1+p)}\right ) \]
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\[\int \frac {\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{3}}d x\]
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\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.87 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=- \frac {d e^{2 p} x^{2 p - 2} e^{i \pi p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (2 - p\right )} - \frac {d^{2 p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} \]
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\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}} \,d x } \]
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\[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right )}{x^3} \,d x \]
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