Optimal. Leaf size=39 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+2}}{2 c^2 e (p+2)} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {643, 629} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+2}}{2 c^2 e (p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rule 643
Rubi steps
\begin {align*} \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\frac {\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{1+p} \, dx}{c}\\ &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{2+p}}{2 c^2 e (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.77 \begin {gather*} \frac {(d+e x)^4 \left (c (d+e x)^2\right )^p}{2 e (p+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^3 \left (c d^2+2 c d e 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 = 0.42, size = 71, normalized size = 1.82 \begin {gather*} \frac {{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, {\left (e p + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 160, normalized size = 4.10 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x^{4} e^{4} + 4 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d x^{3} e^{3} + 6 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{2} x^{2} e^{2} + 4 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{3} x e + {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{4}}{2 \, {\left (p e + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.03 \begin {gather*} \frac {\left (e x +d \right )^{4} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{2 \left (p +2\right ) e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.69, size = 313, normalized size = 8.03 \begin {gather*} \frac {{\left (c^{p} e x + c^{p} d\right )} {\left (e x + d\right )}^{2 \, p} d^{3}}{e {\left (2 \, p + 1\right )}} + \frac {3 \, {\left (c^{p} e^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, c^{p} d e p x - c^{p} d^{2}\right )} {\left (e x + d\right )}^{2 \, p} d^{2}}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} e} + \frac {3 \, {\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} c^{p} e^{3} x^{3} + {\left (2 \, p^{2} + p\right )} c^{p} d e^{2} x^{2} - 2 \, c^{p} d^{2} e p x + c^{p} d^{3}\right )} {\left (e x + d\right )}^{2 \, p} d}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} e} + \frac {{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} c^{p} e^{4} x^{4} + 2 \, {\left (2 \, p^{3} + 3 \, p^{2} + p\right )} c^{p} d e^{3} x^{3} - 3 \, {\left (2 \, p^{2} + p\right )} c^{p} d^{2} e^{2} x^{2} + 6 \, c^{p} d^{3} e p x - 3 \, c^{p} d^{4}\right )} {\left (e x + d\right )}^{2 \, p}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 90, normalized size = 2.31 \begin {gather*} {\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p\,\left (\frac {d^4}{2\,e\,\left (p+2\right )}+\frac {e^3\,x^4}{2\,\left (p+2\right )}+\frac {2\,d^3\,x}{p+2}+\frac {3\,d^2\,e\,x^2}{p+2}+\frac {2\,d\,e^2\,x^3}{p+2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.14, size = 233, normalized size = 5.97 \begin {gather*} \begin {cases} \frac {x}{c^{2} d} & \text {for}\: e = 0 \wedge p = -2 \\d^{3} x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c^{2} e} & \text {for}\: p = -2 \\\frac {d^{4} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 4 e} + \frac {4 d^{3} e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 4 e} + \frac {6 d^{2} e^{2} x^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 4 e} + \frac {4 d e^{3} x^{3} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 4 e} + \frac {e^{4} x^{4} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 4 e} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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