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 = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 69
Rule 678
Rule 793
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*}
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Mathematica [A] time = 0.09, 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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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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