Optimal. Leaf size=115 \[ \frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}-\frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^2 \left (1-p^2\right )} \]
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Rubi [A] time = 0.0539433, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {793, 678, 69} \[ \frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}-\frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^2 \left (1-p^2\right )} \]
Antiderivative was successfully verified.
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Rule 793
Rule 678
Rule 69
Rubi steps
\begin{align*} \int \frac{x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx &=\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^2 (1-p) (d+e x)^2}+\frac{\int \frac{\left (d^2-e^2 x^2\right )^p}{d+e x} \, dx}{e (1-p)}\\ &=\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^2 (1-p) (d+e x)^2}+\frac{\left ((d-e x)^{-1-p} \left (1+\frac{e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p}\right ) \int (d-e x)^p \left (1+\frac{e x}{d}\right )^{-1+p} \, dx}{d^2 e (1-p)}\\ &=\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^2 (1-p) (d+e x)^2}-\frac{2^{-1+p} \left (1+\frac{e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (1-p,1+p;2+p;\frac{d-e x}{2 d}\right )}{d^2 e^2 \left (1-p^2\right )}\\ \end{align*}
Mathematica [A] time = 0.096461, size = 102, normalized size = 0.89 \[ \frac{2^{p-2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (\, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-2 \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )\right )}{d e^2 (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.683, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, 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 (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}}\,{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 (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e^{2} x^{2} + 2 \, d e x + d^{2}}, 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{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, 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 (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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