Optimal. Leaf size=128 \[ -2 e^2 p x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{d (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.115745, antiderivative size = 128, 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, 266, 65, 246, 245} \[ -2 e^2 p x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{d (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1807
Rule 764
Rule 266
Rule 65
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac{\int \frac{\left (-2 d^3 e+2 d^2 e^2 p x\right ) \left (d^2-e^2 x^2\right )^p}{x} \, dx}{d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{x}+(2 d e) \int \frac{\left (d^2-e^2 x^2\right )^p}{x} \, dx-\left (2 e^2 p\right ) \int \left (d^2-e^2 x^2\right )^p \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{x}+(d e) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )-\left (2 e^2 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{\left (d^2-e^2 x^2\right )^{1+p}}{x}-2 e^2 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{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 )}{d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0771194, size = 153, normalized size = 1.2 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (e x \left (d e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\left (d^2-e^2 x^2\right ) \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )\right )-d^3 (p+1) \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )\right )}{d (p+1) x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.587, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 6.27673, size = 116, normalized size = 0.91 \begin{align*} - \frac{d^{2} 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{d 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 )}}{\Gamma \left (1 - p\right )} + 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]