Optimal. Leaf size=30 \[ \frac {a^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {b^2 x^4}{4} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45}
\begin {gather*} \frac {a^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {b^2 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x (a+b x)^2 \, dx &=\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx\\ &=\frac {a^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {b^2 x^4}{4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {b^2 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.65, size = 24, normalized size = 0.80 \begin {gather*} \frac {x^2 \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 25, normalized size = 0.83
method | result | size |
gosper | \(\frac {1}{2} a^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} b^{2} x^{4}\) | \(25\) |
default | \(\frac {1}{2} a^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} b^{2} x^{4}\) | \(25\) |
norman | \(\frac {1}{2} a^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} b^{2} x^{4}\) | \(25\) |
risch | \(\frac {1}{2} a^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} b^{2} x^{4}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{2}}{2} + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^{4} b^{2}+\frac {2}{3} x^{3} b a+\frac {1}{2} x^{2} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {2\,a\,b\,x^3}{3}+\frac {b^2\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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