Optimal. Leaf size=19 \[ \frac {c (d+e x)^{m+3}}{e (m+3)} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 32} \begin {gather*} \frac {c (d+e x)^{m+3}}{e (m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
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 ) \, dx &=\int c (d+e x)^{2+m} \, dx\\ &=c \int (d+e x)^{2+m} \, dx\\ &=\frac {c (d+e x)^{3+m}}{e (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.05 \begin {gather*} \frac {c (d+e x)^{m+3}}{e m+3 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 49, normalized size = 2.58 \begin {gather*} \frac {{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} {\left (e x + d\right )}^{m}}{e m + 3 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 75, normalized size = 3.95 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \, {\left (x e + d\right )}^{m} c d x^{2} e^{2} + 3 \, {\left (x e + d\right )}^{m} c d^{2} x e + {\left (x e + d\right )}^{m} c d^{3}}{m e + 3 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.89 \begin {gather*} \frac {\left (e^{2} x^{2}+2 d x e +d^{2}\right ) c \left (e x +d \right )^{m +1}}{\left (m +3\right ) e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 137, normalized size = 7.21 \begin {gather*} \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} e} + \frac {{\left (e x + d\right )}^{m + 1} c d^{2}}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 60, normalized size = 3.16 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {3\,c\,d^2\,x}{m+3}+\frac {c\,d^3}{e\,\left (m+3\right )}+\frac {c\,e^2\,x^3}{m+3}+\frac {3\,c\,d\,e\,x^2}{m+3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 116, normalized size = 6.11 \begin {gather*} \begin {cases} \frac {c x}{d} & \text {for}\: e = 0 \wedge m = -3 \\c d^{2} d^{m} x & \text {for}\: e = 0 \\\frac {c \log {\left (\frac {d}{e} + x \right )}}{e} & \text {for}\: m = -3 \\\frac {c d^{3} \left (d + e x\right )^{m}}{e m + 3 e} + \frac {3 c d^{2} e x \left (d + e x\right )^{m}}{e m + 3 e} + \frac {3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 3 e} + \frac {c e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 3 e} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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