Optimal. Leaf size=69 \[ \frac{c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
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Rubi [A] time = 0.0435253, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {626, 12, 43} \[ \frac{c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx &=\int c^2 x^2 (d+e x)^{2+m} \, dx\\ &=c^2 \int x^2 (d+e x)^{2+m} \, dx\\ &=c^2 \int \left (\frac{d^2 (d+e x)^{2+m}}{e^2}-\frac{2 d (d+e x)^{3+m}}{e^2}+\frac{(d+e x)^{4+m}}{e^2}\right ) \, dx\\ &=\frac{c^2 d^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac{2 c^2 d (d+e x)^{4+m}}{e^3 (4+m)}+\frac{c^2 (d+e x)^{5+m}}{e^3 (5+m)}\\ \end{align*}
Mathematica [A] time = 0.030712, size = 60, normalized size = 0.87 \[ \frac{c^2 (d+e x)^{m+3} \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )}{e^3 (m+3) (m+4) (m+5)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 76, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3+m} \left ({e}^{2}{m}^{2}{x}^{2}+7\,{e}^{2}m{x}^{2}-2\,demx+12\,{x}^{2}{e}^{2}-6\,dxe+2\,{d}^{2} \right ){c}^{2}}{{e}^{3} \left ({m}^{3}+12\,{m}^{2}+47\,m+60 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.30397, size = 436, normalized size = 6.32 \begin{align*} \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^{2} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac{2 \,{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} +{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \,{\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )}{\left (e x + d\right )}^{m} c^{2} d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac{{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} +{\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \,{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \,{\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )}{\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09147, size = 394, normalized size = 5.71 \begin{align*} -\frac{{\left (2 \, c^{2} d^{4} e m x - 2 \, c^{2} d^{5} -{\left (c^{2} e^{5} m^{2} + 7 \, c^{2} e^{5} m + 12 \, c^{2} e^{5}\right )} x^{5} -{\left (3 \, c^{2} d e^{4} m^{2} + 19 \, c^{2} d e^{4} m + 30 \, c^{2} d e^{4}\right )} x^{4} -{\left (3 \, c^{2} d^{2} e^{3} m^{2} + 15 \, c^{2} d^{2} e^{3} m + 20 \, c^{2} d^{2} e^{3}\right )} x^{3} -{\left (c^{2} d^{3} e^{2} m^{2} + c^{2} d^{3} e^{2} m\right )} x^{2}\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.87346, size = 1047, normalized size = 15.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22437, size = 394, normalized size = 5.71 \begin{align*} \frac{{\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + 3 \,{\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} + 3 \,{\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} +{\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 7 \,{\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 19 \,{\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} + 15 \,{\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} +{\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 2 \,{\left (x e + d\right )}^{m} c^{2} d^{4} m x e + 12 \,{\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 30 \,{\left (x e + d\right )}^{m} c^{2} d x^{4} e^{4} + 20 \,{\left (x e + d\right )}^{m} c^{2} d^{2} x^{3} e^{3} + 2 \,{\left (x e + d\right )}^{m} c^{2} d^{5}}{m^{3} e^{3} + 12 \, m^{2} e^{3} + 47 \, m e^{3} + 60 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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