Optimal. Leaf size=28 \[ \frac {1}{2} x^2 (A c+b B)+A b x+\frac {1}{3} B c x^3 \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} \frac {1}{2} x^2 (A c+b B)+A b x+\frac {1}{3} B c x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x} \, dx &=\int \left (A b+(b B+A c) x+B c x^2\right ) \, dx\\ &=A b x+\frac {1}{2} (b B+A c) x^2+\frac {1}{3} B c 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 c+b B)+A b x+\frac {1}{3} B c x^3 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 32, normalized size = 1.14 \begin {gather*} A b x+\frac {1}{2} A c x^2+\frac {1}{2} b B x^2+\frac {1}{3} B c x^3 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, B c x^{3} + A b x + \frac {1}{2} \, {\left (B b + A c\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, B c x^{3} + \frac {1}{2} \, B b x^{2} + \frac {1}{2} \, A c x^{2} + A b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.89 \begin {gather*} \frac {B c \,x^{3}}{3}+A b x +\frac {\left (A c +b B \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, B c x^{3} + A b x + \frac {1}{2} \, {\left (B b + A c\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} \frac {B\,c\,x^3}{3}+\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x^2+A\,b\,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 b x + \frac {B c x^{3}}{3} + x^{2} \left (\frac {A c}{2} + \frac {B b}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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