Optimal. Leaf size=57 \[ -\frac{2^{p-3} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e (p+1)} \]
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Rubi [A] time = 0.0365362, 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} \[ -\frac{2^{p-3} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e (p+1)} \]
Antiderivative was successfully verified.
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Rule 676
Rule 69
Rubi steps
\begin{align*} \int \frac{\left (1-\frac{e^2 x^2}{d^2}\right )^p}{(d+e x)^3} \, dx &=\frac{\left (\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}{d^4}\\ &=-\frac{2^{-3+p} \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 )}{d^2 e (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0768739, size = 76, normalized size = 1.33 \[ -\frac{2^{p-3} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^3 e (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.61, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{3}} \left ( 1-{\frac{{e}^{2}{x}^{2}}{{d}^{2}}} \right ) ^{p}}\, 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 (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{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 (-\frac{e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (-1 + \frac{e x}{d}\right ) \left (1 + \frac{e x}{d}\right )\right )^{p}}{\left (d + e x\right )^{3}}\, dx \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 (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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