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 = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.70, size = 49, normalized size = 1.29 \[ \frac {1}{4} x^{4} b^{2} B + \frac {2}{3} x^{3} b a B + \frac {1}{3} x^{3} b^{2} A + \frac {1}{2} x^{2} a^{2} B + x^{2} b a A + x a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 49, normalized size = 1.29 \[ \frac {1}{4} \, B b^{2} x^{4} + \frac {2}{3} \, B a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a b x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 1.29 \[ \frac {B \,b^{2} x^{4}}{4}+A \,a^{2} x +\frac {\left (A \,b^{2}+2 a b B \right ) x^{3}}{3}+\frac {\left (2 A a b +a^{2} B \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 48, normalized size = 1.26 \[ \frac {1}{4} \, B b^{2} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 47, normalized size = 1.24 \[ x^2\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+x^3\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}\right )+\frac {B\,b^2\,x^4}{4}+A\,a^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 49, normalized size = 1.29 \[ A a^{2} x + \frac {B b^{2} x^{4}}{4} + x^{3} \left (\frac {A b^{2}}{3} + \frac {2 B a b}{3}\right ) + x^{2} \left (A a b + \frac {B a^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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