Optimal. Leaf size=47 \[ \frac{a^2 (a+b x)^6}{6 b^3}+\frac{(a+b x)^8}{8 b^3}-\frac{2 a (a+b x)^7}{7 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0205076, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{a^2 (a+b x)^6}{6 b^3}+\frac{(a+b x)^8}{8 b^3}-\frac{2 a (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int x^2 (a+b x)^5 \, dx &=\int \left (\frac{a^2 (a+b x)^5}{b^2}-\frac{2 a (a+b x)^6}{b^2}+\frac{(a+b x)^7}{b^2}\right ) \, dx\\ &=\frac{a^2 (a+b x)^6}{6 b^3}-\frac{2 a (a+b x)^7}{7 b^3}+\frac{(a+b x)^8}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.0048094, size = 67, normalized size = 1.43 \[ \frac{5}{3} a^2 b^3 x^6+2 a^3 b^2 x^5+\frac{5}{4} a^4 b x^4+\frac{a^5 x^3}{3}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^8}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 58, normalized size = 1.2 \begin{align*}{\frac{{b}^{5}{x}^{8}}{8}}+{\frac{5\,a{b}^{4}{x}^{7}}{7}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+2\,{a}^{3}{b}^{2}{x}^{5}+{\frac{5\,{a}^{4}b{x}^{4}}{4}}+{\frac{{a}^{5}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0072, size = 77, normalized size = 1.64 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{4} \, a^{4} b x^{4} + \frac{1}{3} \, a^{5} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.39859, size = 126, normalized size = 2.68 \begin{align*} \frac{1}{8} x^{8} b^{5} + \frac{5}{7} x^{7} b^{4} a + \frac{5}{3} x^{6} b^{3} a^{2} + 2 x^{5} b^{2} a^{3} + \frac{5}{4} x^{4} b a^{4} + \frac{1}{3} x^{3} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.081626, size = 65, normalized size = 1.38 \begin{align*} \frac{a^{5} x^{3}}{3} + \frac{5 a^{4} b x^{4}}{4} + 2 a^{3} b^{2} x^{5} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{7}}{7} + \frac{b^{5} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17811, size = 77, normalized size = 1.64 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{4} \, a^{4} b x^{4} + \frac{1}{3} \, a^{5} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]