Optimal. Leaf size=159 \[ 2 d e^2 (1-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{3 e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac{e \left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)}-\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{x} \]
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Rubi [A] time = 0.184813, antiderivative size = 159, 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 = {1807, 1652, 446, 80, 65, 12, 246, 245} \[ 2 d e^2 (1-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{3 e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac{e \left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)}-\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{x} \]
Antiderivative was successfully verified.
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Rule 1807
Rule 1652
Rule 446
Rule 80
Rule 65
Rule 12
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^p \left (-3 d^4 e-2 d^3 e^2 (1-p) x-d^2 e^3 x^2\right )}{x} \, dx}{d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac{\int -2 d^3 e^2 (1-p) \left (d^2-e^2 x^2\right )^p \, dx}{d^2}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^p \left (-3 d^4 e-d^2 e^3 x^2\right )}{x} \, dx}{d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac{\operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^p \left (-3 d^4 e-d^2 e^3 x\right )}{x} \, dx,x,x^2\right )}{2 d^2}+\left (2 d e^2 (1-p)\right ) \int \left (d^2-e^2 x^2\right )^p \, dx\\ &=-\frac{e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{x}+\frac{1}{2} \left (3 d^2 e\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (2 d e^2 (1-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 )^p \, dx\\ &=-\frac{e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{x}+2 d e^2 (1-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},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{3 e \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac{e^2 x^2}{d^2}\right )}{2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0970604, size = 158, normalized size = 0.99 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (e x \left (6 d e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\left (d^2-e^2 x^2\right ) \left (1-\frac{e^2 x^2}{d^2}\right )^p \left (3 \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )+1\right )\right )-2 d^3 (p+1) \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )\right )}{2 (p+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.61, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{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 x + d\right )}^{3}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}\,{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^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.36551, size = 177, normalized size = 1.11 \begin{align*} - \frac{d^{3} d^{2 p}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - p \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac{3 d^{2} e e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ 1 - p \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + 3 d d^{2 p} e^{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{3} \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) \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 x + d\right )}^{3}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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