Optimal. Leaf size=31 \[ a A x+\frac {B \left (a+c x^2\right )^2}{4 c}+\frac {1}{3} A c x^3 \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {641} \begin {gather*} a A x+\frac {B \left (a+c x^2\right )^2}{4 c}+\frac {1}{3} A c x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 641
Rubi steps
\begin {align*} \int (A+B x) \left (a+c x^2\right ) \, dx &=\frac {B \left (a+c x^2\right )^2}{4 c}+A \int \left (a+c x^2\right ) \, dx\\ &=a A x+\frac {1}{3} A c x^3+\frac {B \left (a+c x^2\right )^2}{4 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.03 \begin {gather*} a A x+\frac {1}{2} a B x^2+\frac {1}{3} A c x^3+\frac {1}{4} B c x^4 \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) \left (a+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{4} x^{4} c B + \frac {1}{3} x^{3} c A + \frac {1}{2} x^{2} a B + x a A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, A c x^{3} + \frac {1}{2} \, B a x^{2} + A a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{4} B c \,x^{4}+\frac {1}{3} A c \,x^{3}+\frac {1}{2} B a \,x^{2}+A a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, A c x^{3} + \frac {1}{2} \, B a x^{2} + A a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.84 \begin {gather*} \frac {B\,c\,x^4}{4}+\frac {A\,c\,x^3}{3}+\frac {B\,a\,x^2}{2}+A\,a\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 29, normalized size = 0.94 \begin {gather*} A a x + \frac {A c x^{3}}{3} + \frac {B a x^{2}}{2} + \frac {B c x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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