Optimal. Leaf size=21 \[ \frac{c^2 (d+e x)^{m+5}}{e (m+5)} \]
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Rubi [A] time = 0.0093606, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ \frac{c^2 (d+e x)^{m+5}}{e (m+5)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
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 )^2 \, dx &=\int c^2 (d+e x)^{4+m} \, dx\\ &=c^2 \int (d+e x)^{4+m} \, dx\\ &=\frac{c^2 (d+e x)^{5+m}}{e (5+m)}\\ \end{align*}
Mathematica [A] time = 0.0168074, size = 22, normalized size = 1.05 \[ \frac{c^2 (d+e x)^{m+5}}{e m+5 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 40, normalized size = 1.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m}{c}^{2} \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) ^{2}}{e \left ( 5+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44608, size = 169, normalized size = 8.05 \begin{align*} \frac{{\left (c^{2} e^{5} x^{5} + 5 \, c^{2} d e^{4} x^{4} + 10 \, c^{2} d^{2} e^{3} x^{3} + 10 \, c^{2} d^{3} e^{2} x^{2} + 5 \, c^{2} d^{4} e x + c^{2} d^{5}\right )}{\left (e x + d\right )}^{m}}{e m + 5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.45693, size = 185, normalized size = 8.81 \begin{align*} \begin{cases} \frac{c^{2} x}{d} & \text{for}\: e = 0 \wedge m = -5 \\c^{2} d^{4} d^{m} x & \text{for}\: e = 0 \\\frac{c^{2} \log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: m = -5 \\\frac{c^{2} d^{5} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{5 c^{2} d^{4} e x \left (d + e x\right )^{m}}{e m + 5 e} + \frac{10 c^{2} d^{3} e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{10 c^{2} d^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{5 c^{2} d e^{4} x^{4} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{c^{2} e^{5} x^{5} \left (d + e x\right )^{m}}{e m + 5 e} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33514, size = 169, normalized size = 8.05 \begin{align*} \frac{{\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 5 \,{\left (x e + d\right )}^{m} c^{2} d x^{4} e^{4} + 10 \,{\left (x e + d\right )}^{m} c^{2} d^{2} x^{3} e^{3} + 10 \,{\left (x e + d\right )}^{m} c^{2} d^{3} x^{2} e^{2} + 5 \,{\left (x e + d\right )}^{m} c^{2} d^{4} x e +{\left (x e + d\right )}^{m} c^{2} d^{5}}{m e + 5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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