Optimal. Leaf size=100 \[ -\frac{(d+e x)^{m+1} (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,m+2 p+2;m+p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(m+p+1) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.104822, antiderivative size = 123, normalized size of antiderivative = 1.23, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {679, 677, 70, 69} \[ \frac{(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-m-p} \, _2F_1\left (-m-p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]
Antiderivative was successfully verified.
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Rule 679
Rule 677
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \left (1+\frac{e x}{d}\right )^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\\ &=\left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^m \left (1+\frac{e x}{d}\right )^{-m-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (1+\frac{e x}{d}\right )^{m+p} \, dx\\ &=\left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^m \left (\frac{c d^2 \left (1+\frac{e x}{d}\right )}{c d^2-a e^2}\right )^{-m-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (\frac{c d^2}{c d^2-a e^2}+\frac{c d e x}{c d^2-a e^2}\right )^{m+p} \, dx\\ &=\frac{(a e+c d x) (d+e x)^m \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-m-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (-m-p,1+p;2+p;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0539297, size = 107, normalized size = 1.07 \[ \frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{p+1} \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-m-p} \, _2F_1\left (-m-p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{c d (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.319, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \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 d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} 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 d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} 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 d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} 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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