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 = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{p+1}}{e} \end {gather*}
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*}
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Mathematica [A] time = 0.14, size = 34, normalized size = 0.81 \begin {gather*} \frac {(d+e x)^m ((d+e x) (c (d-e x)-b e))^{p+1}}{e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 66, normalized size = 1.57 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 173, normalized size = 4.12 \begin {gather*} -{\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 56, normalized size = 1.33 \begin {gather*} -\frac {\left (c e x +b e -c d \right ) \left (e x +d \right )^{m +1} \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{p}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 64, normalized size = 1.52 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.58, size = 79, normalized size = 1.88 \begin {gather*} -\left (b\,e\,x\,{\left (d+e\,x\right )}^m-\frac {\left (c\,d^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^m}{e}+c\,e\,x^2\,{\left (d+e\,x\right )}^m\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^p \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.86, size = 173, normalized size = 4.12 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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