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 = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 69
Rule 676
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*}
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Mathematica [A] time = 0.08, 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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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (-\frac {e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\left (-\frac {e^{2} x^{2}}{d^{2}}+1\right )^{p}}{\left (e x +d \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (1-\frac {e^2\,x^2}{d^2}\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (-1 + \frac {e x}{d}\right ) \left (1 + \frac {e x}{d}\right )\right )^{p}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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