Optimal. Leaf size=55 \[ \frac{1}{2} a^2 A x^2+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0265814, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {76} \[ \frac{1}{2} a^2 A x^2+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rubi steps
\begin{align*} \int x (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A x+a (2 A b+a B) x^2+b (A b+2 a B) x^3+b^2 B x^4\right ) \, dx\\ &=\frac{1}{2} a^2 A x^2+\frac{1}{3} a (2 A b+a B) x^3+\frac{1}{4} b (A b+2 a B) x^4+\frac{1}{5} b^2 B x^5\\ \end{align*}
Mathematica [A] time = 0.0092655, size = 50, normalized size = 0.91 \[ \frac{1}{60} x^2 \left (10 a^2 (3 A+2 B x)+10 a b x (4 A+3 B x)+3 b^2 x^2 (5 A+4 B x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{5}}{5}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{3}}{3}}+{\frac{{a}^{2}A{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.02775, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{2} \, A a^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74787, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{5} x^{5} b^{2} B + \frac{1}{2} x^{4} b a B + \frac{1}{4} x^{4} b^{2} A + \frac{1}{3} x^{3} a^{2} B + \frac{2}{3} x^{3} b a A + \frac{1}{2} x^{2} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.075575, size = 54, normalized size = 0.98 \begin{align*} \frac{A a^{2} x^{2}}{2} + \frac{B b^{2} x^{5}}{5} + x^{4} \left (\frac{A b^{2}}{4} + \frac{B a b}{2}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{B a^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20159, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{2} \, B a b x^{4} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + \frac{1}{2} \, A a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]