Integrand size = 25, antiderivative size = 159 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=-\frac {e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}+2 d e^2 (1-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )-\frac {3 e \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )}{2 (1+p)} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1821, 1666, 457, 81, 67, 12, 252, 251} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=2 d e^2 (1-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )-\frac {3 e \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,1-\frac {e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac {e \left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)}-\frac {d \left (d^2-e^2 x^2\right )^{p+1}}{x} \]
[In]
[Out]
Rule 12
Rule 67
Rule 81
Rule 251
Rule 252
Rule 457
Rule 1666
Rule 1821
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^p \left (-3 d^4 e-2 d^3 e^2 (1-p) x-d^2 e^3 x^2\right )}{x} \, dx}{d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac {\int -2 d^3 e^2 (1-p) \left (d^2-e^2 x^2\right )^p \, dx}{d^2}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^p \left (-3 d^4 e-d^2 e^3 x^2\right )}{x} \, dx}{d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac {\text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p \left (-3 d^4 e-d^2 e^3 x\right )}{x} \, dx,x,x^2\right )}{2 d^2}+\left (2 d e^2 (1-p)\right ) \int \left (d^2-e^2 x^2\right )^p \, dx \\ & = -\frac {e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}+\frac {1}{2} \left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (2 d e^2 (1-p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx \\ & = -\frac {e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}+2 d e^2 (1-p) x \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 {3}{2};\frac {e^2 x^2}{d^2}\right )-\frac {3 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 (1+p)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\frac {\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (-2 d^3 (1+p) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )+e x \left (6 d e (1+p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )-\left (d^2-e^2 x^2\right ) \left (1-\frac {e^2 x^2}{d^2}\right )^p \left (1+3 \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {e^2 x^2}{d^2}\right )\right )\right )\right )}{2 (1+p) x} \]
[In]
[Out]
\[\int \frac {\left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{2}}d x\]
[In]
[Out]
\[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
[In]
[Out]
Time = 2.49 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=- \frac {d^{3} d^{2 p} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac {3 d^{2} e e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + 3 d d^{2 p} e^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{3} \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
\[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3}{x^2} \,d x \]
[In]
[Out]