Optimal. Leaf size=175 \[ \frac{\left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d (1-p)}-\frac{2 e (4-3 p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 (3-2 p)}-\frac{e x \left (d^2-e^2 x^2\right )^{p-2}}{3-2 p}+\frac{2 d \left (d^2-e^2 x^2\right )^{p-2}}{2-p} \]
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Rubi [A] time = 0.155502, antiderivative size = 175, 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, 388, 246, 245} \[ \frac{\left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d (1-p)}-\frac{2 e (4-3 p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 (3-2 p)}-\frac{e x \left (d^2-e^2 x^2\right )^{p-2}}{3-2 p}+\frac{2 d \left (d^2-e^2 x^2\right )^{p-2}}{2-p} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1652
Rule 446
Rule 79
Rule 65
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^p}{x (d+e x)^3} \, dx &=\int \frac{(d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p}}{x} \, dx\\ &=\int \frac{\left (d^2-e^2 x^2\right )^{-3+p} \left (d^3+3 d e^2 x^2\right )}{x} \, dx+\int \left (d^2-e^2 x^2\right )^{-3+p} \left (-3 d^2 e-e^3 x^2\right ) \, dx\\ &=-\frac{e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-3+p} \left (d^3+3 d e^2 x\right )}{x} \, dx,x,x^2\right )-\frac{\left (2 d^2 e (4-3 p)\right ) \int \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{3-2 p}\\ &=\frac{2 d \left (d^2-e^2 x^2\right )^{-2+p}}{2-p}-\frac{e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-2+p}}{x} \, dx,x,x^2\right )-\frac{\left (2 e (4-3 p) \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 )^{-3+p} \, dx}{d^4 (3-2 p)}\\ &=\frac{2 d \left (d^2-e^2 x^2\right )^{-2+p}}{2-p}-\frac{e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}-\frac{2 e (4-3 p) 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},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 (3-2 p)}+\frac{\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 (1-p)}\\ \end{align*}
Mathematica [C] time = 0.0971274, size = 82, normalized size = 0.47 \[ \frac{\left (1-\frac{d^2}{e^2 x^2}\right )^{-p} (d-e x)^p (d+e x)^p F_1\left (3-2 p;-p,3-p;4-2 p;\frac{d}{e x},-\frac{d}{e x}\right )}{e^3 (2 p-3) x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.65, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{x \left ( ex+d \right ) ^{3}}}\, 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 )}^{3} 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^{3} x^{4} + 3 \, d e^{2} x^{3} + 3 \, d^{2} e x^{2} + d^{3} 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 )^{3}}\, 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 )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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