Optimal. Leaf size=64 \[ \frac{3 a^2 (a+b x)^7}{7 b^4}-\frac{a^3 (a+b x)^6}{6 b^4}+\frac{(a+b x)^9}{9 b^4}-\frac{3 a (a+b x)^8}{8 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0264511, antiderivative size = 64, 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} \[ \frac{3 a^2 (a+b x)^7}{7 b^4}-\frac{a^3 (a+b x)^6}{6 b^4}+\frac{(a+b x)^9}{9 b^4}-\frac{3 a (a+b x)^8}{8 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int x^3 (a+b x)^5 \, dx &=\int \left (-\frac{a^3 (a+b x)^5}{b^3}+\frac{3 a^2 (a+b x)^6}{b^3}-\frac{3 a (a+b x)^7}{b^3}+\frac{(a+b x)^8}{b^3}\right ) \, dx\\ &=-\frac{a^3 (a+b x)^6}{6 b^4}+\frac{3 a^2 (a+b x)^7}{7 b^4}-\frac{3 a (a+b x)^8}{8 b^4}+\frac{(a+b x)^9}{9 b^4}\\ \end{align*}
Mathematica [A] time = 0.0025793, size = 66, normalized size = 1.03 \[ \frac{10}{7} a^2 b^3 x^7+\frac{5}{3} a^3 b^2 x^6+a^4 b x^5+\frac{a^5 x^4}{4}+\frac{5}{8} a b^4 x^8+\frac{b^5 x^9}{9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 57, normalized size = 0.9 \begin{align*}{\frac{{b}^{5}{x}^{9}}{9}}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{7}}{7}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{6}}{3}}+{a}^{4}b{x}^{5}+{\frac{{a}^{5}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07099, size = 76, normalized size = 1.19 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{8} \, a b^{4} x^{8} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac{1}{4} \, a^{5} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.35036, size = 124, normalized size = 1.94 \begin{align*} \frac{1}{9} x^{9} b^{5} + \frac{5}{8} x^{8} b^{4} a + \frac{10}{7} x^{7} b^{3} a^{2} + \frac{5}{3} x^{6} b^{2} a^{3} + x^{5} b a^{4} + \frac{1}{4} x^{4} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.076684, size = 63, normalized size = 0.98 \begin{align*} \frac{a^{5} x^{4}}{4} + a^{4} b x^{5} + \frac{5 a^{3} b^{2} x^{6}}{3} + \frac{10 a^{2} b^{3} x^{7}}{7} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15571, size = 76, normalized size = 1.19 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{8} \, a b^{4} x^{8} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac{1}{4} \, a^{5} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]