Optimal. Leaf size=57 \[ -\frac{d^3 2^{p+2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0345681, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {676, 69} \[ -\frac{d^3 2^{p+2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 676
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx &=\left (d \left (\frac{d-e x}{d}\right )^{1+p} \left (\frac{1}{d}-\frac{e x}{d^2}\right )^{-1-p}\right ) \int \left (\frac{1}{d}-\frac{e x}{d^2}\right )^p \left (1+\frac{e x}{d}\right )^{2+p} \, dx\\ &=-\frac{2^{2+p} d^3 \left (\frac{d-e x}{d}\right )^{1+p} \, _2F_1\left (-2-p,1+p;2+p;\frac{d-e x}{2 d}\right )}{e (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0656816, size = 86, normalized size = 1.51 \[ d^2 x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+\frac{1}{3} e^2 x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+1}}{e (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.495, size = 75, normalized size = 1.3 \begin{align*}{\frac{{e}^{2}{x}^{3}}{3}{\mbox{$_2$F$_1$}({\frac{3}{2}},-p;\,{\frac{5}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+ed{x}^{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}+{d}^{2}x{\mbox{$_2$F$_1$}({\frac{1}{2}},-p;\,{\frac{3}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})} \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{\left (e x + d\right )}^{2}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{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 ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \left (-\frac{e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 3.89185, size = 116, normalized size = 2.04 \begin{align*} d^{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 )} + 2 d e \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: e^{2} = 0 \\- \frac{d^{2} \left (\begin{cases} \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )} & \text{otherwise} \end{cases}\right )}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \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{\left (e x + d\right )}^{2}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]