Optimal. Leaf size=103 \[ \frac{\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac{e x}{d}\right )^{-p} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1)} \]
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Rubi [A] time = 0.0491622, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {759, 133} \[ \frac{\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac{e x}{d}\right )^{-p} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 759
Rule 133
Rubi steps
\begin{align*} \int (d+e x)^m \left (b x+c x^2\right )^p \, dx &=\frac{\left (\left (b x+c x^2\right )^p \left (1-\frac{d+e x}{d}\right )^{-p} \left (1-\frac{d+e x}{d-\frac{b e}{c}}\right )^{-p}\right ) \operatorname{Subst}\left (\int x^m \left (1-\frac{x}{d}\right )^p \left (1-\frac{c x}{c d-b e}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac{\left (-\frac{e x}{d}\right )^{-p} (d+e x)^{1+m} \left (b x+c x^2\right )^p \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (1+m;-p,-p;2+m;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.101776, size = 76, normalized size = 0.74 \[ \frac{x \left (\frac{b+c x}{b}\right )^{-p} (x (b+c x))^p (d+e x)^m \left (\frac{d+e x}{d}\right )^{-m} F_1\left (p+1;-p,-m;p+2;-\frac{c x}{b},-\frac{e x}{d}\right )}{p+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.664, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{p}\, 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 x^{2} + b x\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 x^{2} + b x\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 x^{2} + b x\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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