Optimal. Leaf size=139 \[ -\frac{2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]
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Rubi [A] time = 0.124664, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1807, 764, 365, 364, 266, 65} \[ -\frac{2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 1807
Rule 764
Rule 365
Rule 364
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac{\int \frac{\left (-4 d^3 e-2 d^2 e^2 (1-p) x\right ) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx}{2 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}+(2 d e) \int \frac{\left (d^2-e^2 x^2\right )^p}{x^2} \, dx+\left (e^2 (1-p)\right ) \int \frac{\left (d^2-e^2 x^2\right )^p}{x} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}+\frac{1}{2} \left (e^2 (1-p)\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (2 d 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{\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac{2 d 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 (1-p) \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*}
Mathematica [A] time = 0.087817, size = 131, normalized size = 0.94 \[ \frac{e \left (d^2-e^2 x^2\right )^p \left (\frac{e \left (e^2 x^2-d^2\right ) \left (\, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )+\, _2F_1\left (2,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )\right )}{p+1}-\frac{4 d^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.632, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{2}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.07993, size = 139, normalized size = 1. \begin{align*} - \frac{d^{2} e^{2 p} x^{2 p} 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 x^{2} \Gamma \left (2 - p\right )} - \frac{2 d 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} - \frac{e^{2} 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{2}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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