Optimal. Leaf size=77 \[ -\frac{c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac{(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{c^2 d^2 (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.0882099, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {610, 43} \[ -\frac{c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac{(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{c^2 d^2 (d+e x)^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 610
Rule 43
Rubi steps
\begin{align*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx &=\frac{\int \left (c d^2+c d e x\right )^2 \left (a e^2+c d e x\right )^2 \, dx}{c^2 d^2 e^2}\\ &=\frac{\int \left (\left (c d^2-a e^2\right )^2 \left (c d^2+c d e x\right )^2-2 \left (c d^2-a e^2\right ) \left (c d^2+c d e x\right )^3+\left (c d^2+c d e x\right )^4\right ) \, dx}{c^2 d^2 e^2}\\ &=\frac{\left (c d^2-a e^2\right )^2 (d+e x)^3}{3 e^3}-\frac{c d \left (c d^2-a e^2\right ) (d+e x)^4}{2 e^3}+\frac{c^2 d^2 (d+e x)^5}{5 e^3}\\ \end{align*}
Mathematica [A] time = 0.0242541, size = 87, normalized size = 1.13 \[ \frac{1}{30} x \left (10 a^2 e^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a c d e x \left (6 d^2+8 d e x+3 e^2 x^2\right )+c^2 d^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 93, normalized size = 1.2 \begin{align*}{\frac{{d}^{2}{e}^{2}{c}^{2}{x}^{5}}{5}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ) dec{x}^{4}}{2}}+{\frac{ \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ){x}^{3}}{3}}+ade \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}+{a}^{2}{d}^{2}{e}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0831, size = 126, normalized size = 1.64 \begin{align*} \frac{1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} c d e x^{4} + a^{2} d^{2} e^{2} x + \frac{1}{3} \,{\left (c d^{2} + a e^{2}\right )}^{2} x^{3} + \frac{1}{3} \,{\left (2 \, c d e x^{3} + 3 \,{\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a d e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37704, size = 225, normalized size = 2.92 \begin{align*} \frac{1}{5} x^{5} e^{2} d^{2} c^{2} + \frac{1}{2} x^{4} e d^{3} c^{2} + \frac{1}{2} x^{4} e^{3} d c a + \frac{1}{3} x^{3} d^{4} c^{2} + \frac{4}{3} x^{3} e^{2} d^{2} c a + \frac{1}{3} x^{3} e^{4} a^{2} + x^{2} e d^{3} c a + x^{2} e^{3} d a^{2} + x e^{2} d^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.119938, size = 104, normalized size = 1.35 \begin{align*} a^{2} d^{2} e^{2} x + \frac{c^{2} d^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac{a c d e^{3}}{2} + \frac{c^{2} d^{3} e}{2}\right ) + x^{3} \left (\frac{a^{2} e^{4}}{3} + \frac{4 a c d^{2} e^{2}}{3} + \frac{c^{2} d^{4}}{3}\right ) + x^{2} \left (a^{2} d e^{3} + a c d^{3} e\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18371, size = 136, normalized size = 1.77 \begin{align*} \frac{1}{5} \, c^{2} d^{2} x^{5} e^{2} + \frac{1}{2} \, c^{2} d^{3} x^{4} e + \frac{1}{3} \, c^{2} d^{4} x^{3} + \frac{1}{2} \, a c d x^{4} e^{3} + \frac{4}{3} \, a c d^{2} x^{3} e^{2} + a c d^{3} x^{2} e + \frac{1}{3} \, a^{2} x^{3} e^{4} + a^{2} d x^{2} e^{3} + a^{2} d^{2} x e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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