Optimal. Leaf size=42 \[ \frac {1}{2} x^2 (a e+b d)+a d x+\frac {1}{3} x^3 (b e+c d)+\frac {1}{4} c e x^4 \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {631} \[ \frac {1}{2} x^2 (a e+b d)+a d x+\frac {1}{3} x^3 (b e+c d)+\frac {1}{4} c e x^4 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx\\ &=a d x+\frac {1}{2} (b d+a e) x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \[ \frac {1}{2} x^2 (a e+b d)+a d x+\frac {1}{3} x^3 (b e+c d)+\frac {1}{4} c e x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 40, normalized size = 0.95 \[ \frac {1}{4} x^{4} e c + \frac {1}{3} x^{3} d c + \frac {1}{3} x^{3} e b + \frac {1}{2} x^{2} d b + \frac {1}{2} x^{2} e a + x d a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 43, normalized size = 1.02 \[ \frac {1}{4} \, c x^{4} e + \frac {1}{3} \, c d x^{3} + \frac {1}{3} \, b x^{3} e + \frac {1}{2} \, b d x^{2} + \frac {1}{2} \, a x^{2} e + a d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.88 \[ \frac {c e \,x^{4}}{4}+a d x +\frac {\left (b e +c d \right ) x^{3}}{3}+\frac {\left (a e +b d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 36, normalized size = 0.86 \[ \frac {1}{4} \, c e x^{4} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} + a d x + \frac {1}{2} \, {\left (b d + a e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.90 \[ \frac {c\,e\,x^4}{4}+\left (\frac {b\,e}{3}+\frac {c\,d}{3}\right )\,x^3+\left (\frac {a\,e}{2}+\frac {b\,d}{2}\right )\,x^2+a\,d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 39, normalized size = 0.93 \[ a d x + \frac {c e x^{4}}{4} + x^{3} \left (\frac {b e}{3} + \frac {c d}{3}\right ) + x^{2} \left (\frac {a e}{2} + \frac {b d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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