Optimal. Leaf size=38 \[ \frac{(a+b x)^4 (A b-a B)}{4 b^2}+\frac{B (a+b x)^5}{5 b^2} \]
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Rubi [A] time = 0.014005, antiderivative size = 38, normalized size of antiderivative = 1., 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 = {43} \[ \frac{(a+b x)^4 (A b-a B)}{4 b^2}+\frac{B (a+b x)^5}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 (A+B x) \, dx &=\int \left (\frac{(A b-a B) (a+b x)^3}{b}+\frac{B (a+b x)^4}{b}\right ) \, dx\\ &=\frac{(A b-a B) (a+b x)^4}{4 b^2}+\frac{B (a+b x)^5}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0092553, size = 67, normalized size = 1.76 \[ \frac{1}{2} a^2 x^2 (a B+3 A b)+a^3 A x+\frac{1}{4} b^2 x^4 (3 a B+A b)+a b x^3 (a B+A b)+\frac{1}{5} b^3 B x^5 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 73, normalized size = 1.9 \begin{align*}{\frac{{b}^{3}B{x}^{5}}{5}}+{\frac{ \left ({b}^{3}A+3\,a{b}^{2}B \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{2}bA+{a}^{3}B \right ){x}^{2}}{2}}+{a}^{3}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01911, size = 93, normalized size = 2.45 \begin{align*} \frac{1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} +{\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47611, size = 163, normalized size = 4.29 \begin{align*} \frac{1}{5} x^{5} b^{3} B + \frac{3}{4} x^{4} b^{2} a B + \frac{1}{4} x^{4} b^{3} A + x^{3} b a^{2} B + x^{3} b^{2} a A + \frac{1}{2} x^{2} a^{3} B + \frac{3}{2} x^{2} b a^{2} A + x a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.080009, size = 73, normalized size = 1.92 \begin{align*} A a^{3} x + \frac{B b^{3} x^{5}}{5} + x^{4} \left (\frac{A b^{3}}{4} + \frac{3 B a b^{2}}{4}\right ) + x^{3} \left (A a b^{2} + B a^{2} b\right ) + x^{2} \left (\frac{3 A a^{2} b}{2} + \frac{B a^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19566, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{5} \, B b^{3} x^{5} + \frac{3}{4} \, B a b^{2} x^{4} + \frac{1}{4} \, A b^{3} x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + \frac{1}{2} \, B a^{3} x^{2} + \frac{3}{2} \, A a^{2} b x^{2} + A a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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