Optimal. Leaf size=275 \[ -\frac{2 e (-2 m-3 p+2) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},3-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 g^2 (m+2) (-m-2 p+2)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p-2}}{g^2 (-m-2 p+2)}-\frac{2 (2 m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},3-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g (m+1) (-m-2 p+3)}+\frac{3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-2}}{g (-m-2 p+3)} \]
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Rubi [A] time = 0.441517, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {852, 1809, 808, 365, 364} \[ -\frac{2 e (-2 m-3 p+2) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},3-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 g^2 (m+2) (-m-2 p+2)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p-2}}{g^2 (-m-2 p+2)}-\frac{2 (2 m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},3-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g (m+1) (-m-2 p+3)}+\frac{3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-2}}{g (-m-2 p+3)} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1809
Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx &=\int (g x)^m (d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p} \, dx\\ &=-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}+\frac{\int (g x)^m \left (d^2-e^2 x^2\right )^{-3+p} \left (d^3 e^2 (2-m-2 p)-2 d^2 e^3 (2-2 m-3 p) x+3 d e^4 (2-m-2 p) x^2\right ) \, dx}{e^2 (2-m-2 p)}\\ &=\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}+\frac{\int (g x)^m \left (-2 d^3 e^4 (2-m-2 p) (2 m+p)-2 d^2 e^5 (2-2 m-3 p) (3-m-2 p) x\right ) \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{e^4 (2-m-2 p) (3-m-2 p)}\\ &=\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac{\left (2 d^2 e (2-2 m-3 p)\right ) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{g (2-m-2 p)}-\frac{\left (2 d^3 (2 m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{3-m-2 p}\\ &=\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac{\left (2 e (2-2 m-3 p) \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac{e^2 x^2}{d^2}\right )^{-3+p} \, dx}{d^4 g (2-m-2 p)}-\frac{\left (2 (2 m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{-3+p} \, dx}{d^3 (3-m-2 p)}\\ &=\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g (3-m-2 p)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{-2+p}}{g^2 (2-m-2 p)}-\frac{2 (2 m+p) (g x)^{1+m} \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+m}{2},3-p;\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g (1+m) (3-m-2 p)}-\frac{2 e (2-2 m-3 p) (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{2+m}{2},3-p;\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 g^2 (2+m) (2-m-2 p)}\\ \end{align*}
Mathematica [A] time = 0.203333, size = 206, normalized size = 0.75 \[ \frac{x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (\frac{d^3 \, _2F_1\left (\frac{m+1}{2},3-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}+e x \left (e x \left (\frac{3 d \, _2F_1\left (\frac{m+3}{2},3-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}-\frac{e x \, _2F_1\left (\frac{m+4}{2},3-p;\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right )}{m+4}\right )-\frac{3 d^2 \, _2F_1\left (\frac{m+2}{2},3-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{m+2}\right )\right )}{d^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.714, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx \right ) ^{m} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ \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 (g x\right )^{m}}{{\left (e x + d\right )}^{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} \left (g x\right )^{m}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{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 (g x\right )^{m} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\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 (g x\right )^{m}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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