Optimal. Leaf size=57 \[ -\frac{d^4 2^{p+3} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-3,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]
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Rubi [A] time = 0.0386394, 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} \[ -\frac{d^4 2^{p+3} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-3,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]
Antiderivative was successfully verified.
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Rule 676
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^3 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx &=\left (d^2 \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 )^{3+p} \, dx\\ &=-\frac{2^{3+p} d^4 \left (\frac{d-e x}{d}\right )^{1+p} \, _2F_1\left (-3-p,1+p;2+p;\frac{d-e x}{2 d}\right )}{e (1+p)}\\ \end{align*}
Mathematica [B] time = 0.112035, size = 116, normalized size = 2.04 \[ d^3 x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+d e^2 x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{2 d^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+1}}{e (p+1)}+\frac{d^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+2}}{2 e (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.616, size = 104, normalized size = 1.8 \begin{align*}{\frac{{e}^{3}{x}^{4}}{4}{\mbox{$_2$F$_1$}(2,-p;\,3;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+d{e}^{2}{x}^{3}{\mbox{$_2$F$_1$}({\frac{3}{2}},-p;\,{\frac{5}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}+{\frac{3\,{d}^{2}e{x}^{2}}{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+{d}^{3}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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{3}{\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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\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.
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Sympy [B] time = 6.26666, size = 479, normalized size = 8.4 \begin{align*} d^{3} 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 )} + 3 d^{2} 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 ) + d 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 )} + e^{3} \left (\begin{cases} \frac{x^{4}}{4} & \text{for}\: e = 0 \\- \frac{d^{6} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{6} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{6}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{d^{4} e^{2} x^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{d^{4} e^{2} x^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text{for}\: p = -2 \\- \frac{d^{4} \log{\left (- \frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{4} \log{\left (\frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{2} x^{2}}{2 e^{2}} & \text{for}\: p = -1 \\- \frac{d^{4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac{d^{2} e^{2} p x^{2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} p x^{4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} x^{4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text{otherwise} \end{cases}\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{\left (e x + d\right )}^{3}{\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.
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