Optimal. Leaf size=28 \[ \frac {1}{2} x^2 (a d+b c)+a c x+\frac {1}{3} b d x^3 \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {1}{2} x^2 (a d+b c)+a c x+\frac {1}{3} b d x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x) (c+d x) \, dx &=\int \left (a c+(b c+a d) x+b d x^2\right ) \, dx\\ &=a c x+\frac {1}{2} (b c+a d) x^2+\frac {1}{3} b d x^3\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (a d+b c)+a c x+\frac {1}{3} b d x^3 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (c+d x) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.56, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{3} x^{3} d b + \frac {1}{2} x^{2} c b + \frac {1}{2} x^{2} d a + x c a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, b d x^{3} + \frac {1}{2} \, b c x^{2} + \frac {1}{2} \, a d x^{2} + a c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.89 \begin {gather*} \frac {b d \,x^{3}}{3}+a c x +\frac {\left (a d +b c \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, b d x^{3} + a c x + \frac {1}{2} \, {\left (b c + a d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.89 \begin {gather*} \frac {b\,d\,x^3}{3}+\left (\frac {a\,d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 26, normalized size = 0.93 \begin {gather*} a c x + \frac {b d x^{3}}{3} + x^{2} \left (\frac {a d}{2} + \frac {b c}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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