Optimal. Leaf size=128 \[ -\frac{2 e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3}-\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}+\frac{\left (d^2-e^2 x^2\right )^{p-1}}{1-p} \]
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Rubi [A] time = 0.124908, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {852, 1652, 446, 79, 65, 12, 246, 245} \[ -\frac{2 e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3}-\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}+\frac{\left (d^2-e^2 x^2\right )^{p-1}}{1-p} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1652
Rule 446
Rule 79
Rule 65
Rule 12
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^p}{x (d+e x)^2} \, dx &=\int \frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p}}{x} \, dx\\ &=\int -2 d e \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int \frac{\left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right )}{x} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-2+p} \left (d^2+e^2 x\right )}{x} \, dx,x,x^2\right )-(2 d e) \int \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac{\left (d^2-e^2 x^2\right )^{-1+p}}{1-p}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-1+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 \left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3}\\ &=\frac{\left (d^2-e^2 x^2\right )^{-1+p}}{1-p}-\frac{2 e x \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{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3}-\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;1+p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}\\ \end{align*}
Mathematica [A] time = 0.167914, size = 201, normalized size = 1.57 \[ \frac{2^{p-2} \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (2 p (d-e x) \left (1-\frac{d^2}{e^2 x^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+p (d-e x) \left (1-\frac{d^2}{e^2 x^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )+2 d (p+1) \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )\right )}{d^3 p (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.645, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{x \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}\,{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^{3} + 2 \, d e x^{2} + d^{2} x}, 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 \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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