Optimal. Leaf size=62 \[ -\frac{(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac{d (d+e x)^5 (c d-b e)}{5 e^3}+\frac{c (d+e x)^7}{7 e^3} \]
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Rubi [A] time = 0.0667449, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {698} \[ -\frac{(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac{d (d+e x)^5 (c d-b e)}{5 e^3}+\frac{c (d+e x)^7}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^4 \left (b x+c x^2\right ) \, dx &=\int \left (\frac{d (c d-b e) (d+e x)^4}{e^2}+\frac{(-2 c d+b e) (d+e x)^5}{e^2}+\frac{c (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac{d (c d-b e) (d+e x)^5}{5 e^3}-\frac{(2 c d-b e) (d+e x)^6}{6 e^3}+\frac{c (d+e x)^7}{7 e^3}\\ \end{align*}
Mathematica [A] time = 0.0153295, size = 99, normalized size = 1.6 \[ \frac{1}{2} d^2 e x^4 (3 b e+2 c d)+\frac{1}{3} d^3 x^3 (4 b e+c d)+\frac{1}{6} e^3 x^6 (b e+4 c d)+\frac{2}{5} d e^2 x^5 (2 b e+3 c d)+\frac{1}{2} b d^4 x^2+\frac{1}{7} c e^4 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 100, normalized size = 1.6 \begin{align*}{\frac{{e}^{4}c{x}^{7}}{7}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,d{e}^{3}b+6\,{d}^{2}{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}b+4\,{d}^{3}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,{d}^{3}eb+{d}^{4}c \right ){x}^{3}}{3}}+{\frac{b{d}^{4}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12101, size = 134, normalized size = 2.16 \begin{align*} \frac{1}{7} \, c e^{4} x^{7} + \frac{1}{2} \, b d^{4} x^{2} + \frac{1}{6} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + \frac{2}{5} \,{\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{4} + 4 \, b d^{3} e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41085, size = 231, normalized size = 3.73 \begin{align*} \frac{1}{7} x^{7} e^{4} c + \frac{2}{3} x^{6} e^{3} d c + \frac{1}{6} x^{6} e^{4} b + \frac{6}{5} x^{5} e^{2} d^{2} c + \frac{4}{5} x^{5} e^{3} d b + x^{4} e d^{3} c + \frac{3}{2} x^{4} e^{2} d^{2} b + \frac{1}{3} x^{3} d^{4} c + \frac{4}{3} x^{3} e d^{3} b + \frac{1}{2} x^{2} d^{4} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.278048, size = 107, normalized size = 1.73 \begin{align*} \frac{b d^{4} x^{2}}{2} + \frac{c e^{4} x^{7}}{7} + x^{6} \left (\frac{b e^{4}}{6} + \frac{2 c d e^{3}}{3}\right ) + x^{5} \left (\frac{4 b d e^{3}}{5} + \frac{6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac{3 b d^{2} e^{2}}{2} + c d^{3} e\right ) + x^{3} \left (\frac{4 b d^{3} e}{3} + \frac{c d^{4}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26549, size = 130, normalized size = 2.1 \begin{align*} \frac{1}{7} \, c x^{7} e^{4} + \frac{2}{3} \, c d x^{6} e^{3} + \frac{6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac{1}{3} \, c d^{4} x^{3} + \frac{1}{6} \, b x^{6} e^{4} + \frac{4}{5} \, b d x^{5} e^{3} + \frac{3}{2} \, b d^{2} x^{4} e^{2} + \frac{4}{3} \, b d^{3} x^{3} e + \frac{1}{2} \, b d^{4} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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