Optimal. Leaf size=62 \[ -\frac {(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac {d (d+e x)^4 (c d-b e)}{4 e^3}+\frac {c (d+e x)^6}{6 e^3} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, 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} \[ -\frac {(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac {d (d+e x)^4 (c d-b e)}{4 e^3}+\frac {c (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^3 \left (b x+c x^2\right ) \, dx &=\int \left (\frac {d (c d-b e) (d+e x)^3}{e^2}+\frac {(-2 c d+b e) (d+e x)^4}{e^2}+\frac {c (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {d (c d-b e) (d+e x)^4}{4 e^3}-\frac {(2 c d-b e) (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 1.08 \[ \frac {1}{60} x^2 \left (20 d^2 x (3 b e+c d)+12 e^2 x^3 (b e+3 c d)+45 d e x^2 (b e+c d)+30 b d^3+10 c e^3 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 76, normalized size = 1.23 \[ \frac {1}{6} x^{6} e^{3} c + \frac {3}{5} x^{5} e^{2} d c + \frac {1}{5} x^{5} e^{3} b + \frac {3}{4} x^{4} e d^{2} c + \frac {3}{4} x^{4} e^{2} d b + \frac {1}{3} x^{3} d^{3} c + x^{3} e d^{2} b + \frac {1}{2} x^{2} d^{3} b \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 74, normalized size = 1.19 \[ \frac {1}{6} \, c x^{6} e^{3} + \frac {3}{5} \, c d x^{5} e^{2} + \frac {3}{4} \, c d^{2} x^{4} e + \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{5} \, b x^{5} e^{3} + \frac {3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac {1}{2} \, b d^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.23 \[ \frac {c \,e^{3} x^{6}}{6}+\frac {b \,d^{3} x^{2}}{2}+\frac {\left (e^{3} b +3 d \,e^{2} c \right ) x^{5}}{5}+\frac {\left (3 d \,e^{2} b +3 d^{2} e c \right ) x^{4}}{4}+\frac {\left (3 d^{2} e b +d^{3} c \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 73, normalized size = 1.18 \[ \frac {1}{6} \, c e^{3} x^{6} + \frac {1}{2} \, b d^{3} x^{2} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + \frac {3}{4} \, {\left (c d^{2} e + b d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 68, normalized size = 1.10 \[ x^3\,\left (\frac {c\,d^3}{3}+b\,e\,d^2\right )+x^5\,\left (\frac {b\,e^3}{5}+\frac {3\,c\,d\,e^2}{5}\right )+\frac {b\,d^3\,x^2}{2}+\frac {c\,e^3\,x^6}{6}+\frac {3\,d\,e\,x^4\,\left (b\,e+c\,d\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 80, normalized size = 1.29 \[ \frac {b d^{3} x^{2}}{2} + \frac {c e^{3} x^{6}}{6} + x^{5} \left (\frac {b e^{3}}{5} + \frac {3 c d e^{2}}{5}\right ) + x^{4} \left (\frac {3 b d e^{2}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (b d^{2} e + \frac {c d^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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