Optimal. Leaf size=33 \[ \frac {1}{2} a A x^2+\frac {1}{3} (A b+a B) x^3+\frac {1}{4} b B x^4 \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {77}
\begin {gather*} \frac {1}{3} x^3 (a B+A b)+\frac {1}{2} a A x^2+\frac {1}{4} b B x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x (a+b x) (A+B x) \, dx &=\int \left (a A x+(A b+a B) x^2+b B x^3\right ) \, dx\\ &=\frac {1}{2} a A x^2+\frac {1}{3} (A b+a B) x^3+\frac {1}{4} b B x^4\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{12} x^2 (b x (4 A+3 B x)+a (6 A+4 B x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 28, normalized size = 0.85
method | result | size |
default | \(\frac {a A \,x^{2}}{2}+\frac {\left (A b +B a \right ) x^{3}}{3}+\frac {b B \,x^{4}}{4}\) | \(28\) |
norman | \(\frac {b B \,x^{4}}{4}+\left (\frac {A b}{3}+\frac {B a}{3}\right ) x^{3}+\frac {a A \,x^{2}}{2}\) | \(29\) |
gosper | \(\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} B a +\frac {1}{2} a A \,x^{2}\) | \(30\) |
risch | \(\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} B a +\frac {1}{2} a A \,x^{2}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, B b x^{4} + \frac {1}{2} \, A a x^{2} + \frac {1}{3} \, {\left (B a + A b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.95, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{4} x^{4} b B + \frac {1}{3} x^{3} a B + \frac {1}{3} x^{3} b A + \frac {1}{2} x^{2} a A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 29, normalized size = 0.88 \begin {gather*} \frac {A a x^{2}}{2} + \frac {B b x^{4}}{4} + x^{3} \left (\frac {A b}{3} + \frac {B a}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, B b x^{4} + \frac {1}{3} \, B a x^{3} + \frac {1}{3} \, A b x^{3} + \frac {1}{2} \, A a x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,b\,x^4}{4}+\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )\,x^3+\frac {A\,a\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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