Optimal. Leaf size=38 \[ \frac{(a+b x)^3 (A b-a B)}{3 b^2}+\frac{B (a+b x)^4}{4 b^2} \]
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Rubi [A] time = 0.0282879, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {27, 43} \[ \frac{(a+b x)^3 (A b-a B)}{3 b^2}+\frac{B (a+b x)^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (A+B x) \, dx\\ &=\int \left (\frac{(A b-a B) (a+b x)^2}{b}+\frac{B (a+b x)^3}{b}\right ) \, dx\\ &=\frac{(A b-a B) (a+b x)^3}{3 b^2}+\frac{B (a+b x)^4}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.0091377, size = 46, normalized size = 1.21 \[ \frac{1}{12} x \left (6 a^2 (2 A+B x)+4 a b x (3 A+2 B x)+b^2 x^2 (4 A+3 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 49, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}B{x}^{4}}{4}}+{\frac{ \left ( A{b}^{2}+2\,abB \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Aab+B{a}^{2} \right ){x}^{2}}{2}}+{a}^{2}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26164, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, B b^{2} x^{4} + A a^{2} x + \frac{1}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34649, size = 115, normalized size = 3.03 \begin{align*} \frac{1}{4} x^{4} b^{2} B + \frac{2}{3} x^{3} b a B + \frac{1}{3} x^{3} b^{2} A + \frac{1}{2} x^{2} a^{2} B + x^{2} b a A + x a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072378, size = 49, normalized size = 1.29 \begin{align*} A a^{2} x + \frac{B b^{2} x^{4}}{4} + x^{3} \left (\frac{A b^{2}}{3} + \frac{2 B a b}{3}\right ) + x^{2} \left (A a b + \frac{B a^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14464, size = 66, normalized size = 1.74 \begin{align*} \frac{1}{4} \, B b^{2} x^{4} + \frac{2}{3} \, B a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a b x^{2} + A a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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