Optimal. Leaf size=143 \[ \frac{e^2 (3-p) \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (1-p)}+\frac{2 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 x}-\frac{\left (d^2-e^2 x^2\right )^{p-1}}{2 x^2} \]
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Rubi [A] time = 0.16156, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {852, 1807, 764, 365, 364, 266, 65} \[ \frac{e^2 (3-p) \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (1-p)}+\frac{2 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 x}-\frac{\left (d^2-e^2 x^2\right )^{p-1}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1807
Rule 764
Rule 365
Rule 364
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^p}{x^3 (d+e x)^2} \, dx &=\int \frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p}}{x^3} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{-1+p}}{2 x^2}-\frac{\int \frac{\left (4 d^3 e-2 d^2 e^2 (3-p) x\right ) \left (d^2-e^2 x^2\right )^{-2+p}}{x^2} \, dx}{2 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{-1+p}}{2 x^2}-(2 d e) \int \frac{\left (d^2-e^2 x^2\right )^{-2+p}}{x^2} \, dx+\left (e^2 (3-p)\right ) \int \frac{\left (d^2-e^2 x^2\right )^{-2+p}}{x} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{-1+p}}{2 x^2}+\frac{1}{2} \left (e^2 (3-p)\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-2+p}}{x} \, dx,x,x^2\right )-\frac{\left (2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int \frac{\left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p}}{x^2} \, dx}{d^3}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{-1+p}}{2 x^2}+\frac{2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 x}+\frac{e^2 (3-p) \left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (1,-1+p;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (1-p)}\\ \end{align*}
Mathematica [A] time = 0.538843, size = 283, normalized size = 1.98 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{2 d^3 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{6 d e^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+\frac{8 d^2 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{3 e^2 2^{p+1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{e^2 2^p (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{4 d^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{3} \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}}{{\left (e x + d\right )}^{2} x^{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 (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{2} x^{5} + 2 \, d e x^{4} + d^{2} x^{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 (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{3} \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}}{{\left (e x + d\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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