Optimal. Leaf size=57 \[ -\frac{d^2 2^{p+1} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-1,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]
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Rubi [A] time = 0.0156348, antiderivative size = 56, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {641, 245} \[ d x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+1}}{2 e (p+1)} \]
Antiderivative was successfully verified.
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Rule 641
Rule 245
Rubi steps
\begin{align*} \int (d+e x) \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx &=-\frac{d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{1+p}}{2 e (1+p)}+d \int \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx\\ &=-\frac{d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{1+p}}{2 e (1+p)}+d x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0282315, size = 56, normalized size = 0.98 \[ d x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+1}}{2 e (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.354, size = 47, normalized size = 0.8 \begin{align*}{\frac{e{x}^{2}}{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+dx{\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 )}{\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 x + d\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 [A] time = 3.25252, size = 78, normalized size = 1.37 \begin{align*} d 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 \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 ) \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 )}{\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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