Optimal. Leaf size=42 \[ a d x+\frac {1}{2} (b d+a e) x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4 \]
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Rubi [A]
time = 0.02, 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 = {645}
\begin {gather*} \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 \end {gather*}
Antiderivative was successfully verified.
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Rule 645
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 \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 37, normalized size = 0.88
method | result | size |
default | \(a d x +\frac {\left (a e +b d \right ) x^{2}}{2}+\frac {\left (b e +c d \right ) x^{3}}{3}+\frac {c e \,x^{4}}{4}\) | \(37\) |
norman | \(\frac {c e \,x^{4}}{4}+\left (\frac {b e}{3}+\frac {c d}{3}\right ) x^{3}+\left (\frac {a e}{2}+\frac {b d}{2}\right ) x^{2}+a d x\) | \(39\) |
gosper | \(\frac {1}{4} c e \,x^{4}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+\frac {1}{2} b d \,x^{2}+a d x\) | \(41\) |
risch | \(\frac {1}{4} c e \,x^{4}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+\frac {1}{2} b d \,x^{2}+a d x\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 39, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, c x^{4} e + \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.56, size = 42, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, c d x^{3} + \frac {1}{2} \, b d x^{2} + a d x + \frac {1}{12} \, {\left (3 \, c x^{4} + 4 \, b x^{3} + 6 \, a x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 39, normalized size = 0.93 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 43, normalized size = 1.02 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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