Optimal. Leaf size=69 \[ \frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac {(d+e x)^4 (2 c d-b e)}{4 e^3}+\frac {c (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \[ \frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac {(d+e x)^4 (2 c d-b e)}{4 e^3}+\frac {c (d+e x)^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^2}{e^2}+\frac {(-2 c d+b e) (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {(2 c d-b e) (d+e x)^4}{4 e^3}+\frac {c (d+e x)^5}{5 e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 1.06 \[ \frac {1}{3} x^3 \left (a e^2+2 b d e+c d^2\right )+\frac {1}{2} d x^2 (2 a e+b d)+a d^2 x+\frac {1}{4} e x^4 (b e+2 c d)+\frac {1}{5} c e^2 x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 75, normalized size = 1.09 \[ \frac {1}{5} x^{5} e^{2} c + \frac {1}{2} x^{4} e d c + \frac {1}{4} x^{4} e^{2} b + \frac {1}{3} x^{3} d^{2} c + \frac {2}{3} x^{3} e d b + \frac {1}{3} x^{3} e^{2} a + \frac {1}{2} x^{2} d^{2} b + x^{2} e d a + x d^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 75, normalized size = 1.09 \[ \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{4} \, b x^{4} e^{2} + \frac {2}{3} \, b d x^{3} e + \frac {1}{2} \, b d^{2} x^{2} + \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 1.01 \[ \frac {c \,e^{2} x^{5}}{5}+a \,d^{2} x +\frac {\left (b \,e^{2}+2 c d e \right ) x^{4}}{4}+\frac {\left (a \,e^{2}+2 b d e +c \,d^{2}\right ) x^{3}}{3}+\frac {\left (2 a d e +d^{2} b \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 69, normalized size = 1.00 \[ \frac {1}{5} \, c e^{2} x^{5} + \frac {1}{4} \, {\left (2 \, c d e + b e^{2}\right )} x^{4} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{2} + 2 \, a d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 69, normalized size = 1.00 \[ x^3\,\left (\frac {c\,d^2}{3}+\frac {2\,b\,d\,e}{3}+\frac {a\,e^2}{3}\right )+x^2\,\left (\frac {b\,d^2}{2}+a\,e\,d\right )+x^4\,\left (\frac {b\,e^2}{4}+\frac {c\,d\,e}{2}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 73, normalized size = 1.06 \[ a d^{2} x + \frac {c e^{2} x^{5}}{5} + x^{4} \left (\frac {b e^{2}}{4} + \frac {c d e}{2}\right ) + x^{3} \left (\frac {a e^{2}}{3} + \frac {2 b d e}{3} + \frac {c d^{2}}{3}\right ) + x^{2} \left (a d e + \frac {b d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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