Optimal. Leaf size=222 \[ \frac{g (d+e x)^{m-1} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{p+2}}{c e^2 (m+2 p+3)}-\frac{(d+e x)^m (-b e+c d-c e x)^2 \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^p \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-p} (b e g (m+p+1)+c (d g (1-m)-e f (m+2 p+3))) \, _2F_1\left (-m-p,p+2;p+3;\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (p+2) (m+2 p+3)} \]
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Rubi [A] time = 0.40739, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 70, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {1632, 794, 679, 677, 70, 69} \[ \frac{g (d+e x)^{m-1} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{p+2}}{c e^2 (m+2 p+3)}-\frac{(d+e x)^m (-b e+c d-c e x)^2 \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^p \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-p} (b e g (m+p+1)+c (d g (1-m)-e f (m+2 p+3))) \, _2F_1\left (-m-p,p+2;p+3;\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (p+2) (m+2 p+3)} \]
Antiderivative was successfully verified.
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Rule 1632
Rule 794
Rule 679
Rule 677
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^m \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p \left (-(c d-b e) f+(c e f-c d g+b e g) x+c e g x^2\right ) \, dx &=(d e) \int (d+e x)^{-1+m} \left (\frac{f}{d e}+\frac{g x}{d e}\right ) \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^{1+p} \, dx\\ &=\frac{g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac{\left (d \left (\left (\frac{c e^2 f}{d}+\frac{\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac{2 c e f}{d}-\frac{b e g}{d}\right ) (2+p)\right )\right ) \int (d+e x)^{-1+m} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^{1+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac{g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac{\left (\left (\left (\frac{c e^2 f}{d}+\frac{\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac{2 c e f}{d}-\frac{b e g}{d}\right ) (2+p)\right ) (d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \left (1+\frac{e x}{d}\right )^{-1+m} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^{1+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac{g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac{\left (\left (\left (\frac{c e^2 f}{d}+\frac{\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac{2 c e f}{d}-\frac{b e g}{d}\right ) (2+p)\right ) (d+e x)^m \left (1+\frac{e x}{d}\right )^{-m-p} \left (-c d^2+b d e+c d e x\right )^{-p} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p\right ) \int \left (1+\frac{e x}{d}\right )^{m+p} \left (-c d^2+b d e+c d e x\right )^{1+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac{g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac{\left (\left (\left (\frac{c e^2 f}{d}+\frac{\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac{2 c e f}{d}-\frac{b e g}{d}\right ) (2+p)\right ) (d+e x)^m \left (\frac{c d e \left (1+\frac{e x}{d}\right )}{c d e-\frac{e \left (-c d^2+b d e\right )}{d}}\right )^{-m-p} \left (-c d^2+b d e+c d e x\right )^{-p} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p\right ) \int \left (-c d^2+b d e+c d e x\right )^{1+p} \left (\frac{c d}{2 c d-b e}+\frac{c e x}{2 c d-b e}\right )^{m+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac{g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}-\frac{(c d g (1-m)+b e g (1+m+p)-c e f (3+m+2 p)) (d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-p} (c d-b e-c e x)^2 \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^p \, _2F_1\left (-m-p,2+p;3+p;\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (2+p) (3+m+2 p)}\\ \end{align*}
Mathematica [A] time = 0.361065, size = 165, normalized size = 0.74 \[ \frac{(d+e x)^m (b e-c d+c e x)^2 (-(d+e x) (c (d-e x)-b e))^p \left (\frac{e \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-p} (-b e g (m+p+1)+c d g (m-1)+c e f (m+2 p+3)) \, _2F_1\left (-m-p,p+2;p+3;\frac{-c d+b e+c e x}{b e-2 c d}\right )}{p+2}+c e g (d+e x)\right )}{c^2 e^3 (m+2 p+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.615, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( c{e}^{2}{x}^{2}+xb{e}^{2}+bde-c{d}^{2} \right ) ^{p} \left ( - \left ( -be+cd \right ) f+ \left ( beg-cdg+cef \right ) x+ceg{x}^{2} \right ) \, 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{\left (c e g x^{2} -{\left (c d - b e\right )} f +{\left (c e f - c d g + b e g\right )} x\right )}{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )}^{p}{\left (e x + d\right )}^{m}\,{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 ({\left (c e g x^{2} -{\left (c d - b e\right )} f +{\left (c e f -{\left (c d - b e\right )} g\right )} x\right )}{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )}^{p}{\left (e x + d\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (c e g x^{2} -{\left (c d - b e\right )} f +{\left (c e f - c d g + b e g\right )} x\right )}{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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