Optimal. Leaf size=42 \[ \frac{(d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{p+1}}{e} \]
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Rubi [A] time = 0.0534318, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 61, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.016, Rules used = {786} \[ \frac{(d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{p+1}}{e} \]
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin{align*} \int (d+e x)^m (c d m-b e (1+m+p)-c e (2+m+2 p) x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^p \, dx &=\frac{(d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{1+p}}{e}\\ \end{align*}
Mathematica [A] time = 0.151669, size = 34, normalized size = 0.81 \[ \frac{(d+e x)^m ((d+e x) (c (d-e x)-b e))^{p+1}}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 56, normalized size = 1.3 \begin{align*} -{\frac{ \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{p} \left ( ex+d \right ) ^{1+m} \left ( cex+be-cd \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.35576, size = 86, normalized size = 2.05 \begin{align*} -\frac{{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )} e^{\left (p \log \left (-c e x + c d - b e\right ) + m \log \left (e x + d\right ) + p \log \left (e x + d\right )\right )}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4081, size = 128, normalized size = 3.05 \begin{align*} -\frac{{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\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}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.68928, size = 173, normalized size = 4.12 \begin{align*} \begin{cases} - b d \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p} - b e x \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p} + \frac{c d^{2} \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p}}{e} - c e x^{2} \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p} & \text{for}\: e \neq 0 \\c d d^{m} m x \left (c d^{2}\right )^{p} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26095, size = 234, normalized size = 5.57 \begin{align*} -{\left ({\left (x e + d\right )}^{m} c x^{2} e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right ) + 2\right )} -{\left (x e + d\right )}^{m} c d^{2} e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right )\right )} +{\left (x e + d\right )}^{m} b x e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right ) + 2\right )} +{\left (x e + d\right )}^{m} b d e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right ) + 1\right )}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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