Optimal. Leaf size=43 \[ \frac{(d+e x)^{m+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (m+2 p+1)} \]
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Rubi [A] time = 0.0177609, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {644, 32} \[ \frac{(d+e x)^{m+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (m+2 p+1)} \]
Antiderivative was successfully verified.
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Rule 644
Rule 32
Rubi steps
\begin{align*} \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\left ((d+e x)^{-2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^p\right ) \int (d+e x)^{m+2 p} \, dx\\ &=\frac{(d+e x)^{1+m} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (1+m+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0169445, size = 32, normalized size = 0.74 \[ \frac{(d+e x)^{m+1} \left (c (d+e x)^2\right )^p}{e m+2 e p+e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 44, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{e \left ( 1+m+2\,p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09992, size = 58, normalized size = 1.35 \begin{align*} \frac{{\left (c^{p} e x + c^{p} d\right )} e^{\left (m \log \left (e x + d\right ) + 2 \, p \log \left (e x + d\right )\right )}}{e{\left (m + 2 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43167, size = 99, normalized size = 2.3 \begin{align*} \frac{{\left (e x + d\right )}{\left (e x + d\right )}^{m} e^{\left (2 \, p \log \left (e x + d\right ) + p \log \left (c\right )\right )}}{e m + 2 \, e p + e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24254, size = 93, normalized size = 2.16 \begin{align*} \frac{{\left (x e + d\right )}^{m} x e^{\left (2 \, p \log \left (x e + d\right ) + p \log \left (c\right ) + 1\right )} +{\left (x e + d\right )}^{m} d e^{\left (2 \, p \log \left (x e + d\right ) + p \log \left (c\right )\right )}}{m e + 2 \, p e + e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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