Optimal. Leaf size=93 \[ \frac{3}{5} a^2 A c x^5+\frac{1}{3} a^3 A x^3+\frac{1}{2} a^2 B c x^6+\frac{1}{4} a^3 B x^4+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.088014, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{3}{5} a^2 A c x^5+\frac{1}{3} a^3 A x^3+\frac{1}{2} a^2 B c x^6+\frac{1}{4} a^3 B x^4+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 766
Rubi steps
\begin{align*} \int x^2 (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A x^2+a^3 B x^3+3 a^2 A c x^4+3 a^2 B c x^5+3 a A c^2 x^6+3 a B c^2 x^7+A c^3 x^8+B c^3 x^9\right ) \, dx\\ &=\frac{1}{3} a^3 A x^3+\frac{1}{4} a^3 B x^4+\frac{3}{5} a^2 A c x^5+\frac{1}{2} a^2 B c x^6+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10}\\ \end{align*}
Mathematica [A] time = 0.003031, size = 93, normalized size = 1. \[ \frac{3}{5} a^2 A c x^5+\frac{1}{3} a^3 A x^3+\frac{1}{2} a^2 B c x^6+\frac{1}{4} a^3 B x^4+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 78, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}A{x}^{3}}{3}}+{\frac{{a}^{3}B{x}^{4}}{4}}+{\frac{3\,{a}^{2}Ac{x}^{5}}{5}}+{\frac{{a}^{2}Bc{x}^{6}}{2}}+{\frac{3\,aA{c}^{2}{x}^{7}}{7}}+{\frac{3\,aB{c}^{2}{x}^{8}}{8}}+{\frac{A{c}^{3}{x}^{9}}{9}}+{\frac{B{c}^{3}{x}^{10}}{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04009, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{10} \, B c^{3} x^{10} + \frac{1}{9} \, A c^{3} x^{9} + \frac{3}{8} \, B a c^{2} x^{8} + \frac{3}{7} \, A a c^{2} x^{7} + \frac{1}{2} \, B a^{2} c x^{6} + \frac{3}{5} \, A a^{2} c x^{5} + \frac{1}{4} \, B a^{3} x^{4} + \frac{1}{3} \, A a^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54742, size = 185, normalized size = 1.99 \begin{align*} \frac{1}{10} x^{10} c^{3} B + \frac{1}{9} x^{9} c^{3} A + \frac{3}{8} x^{8} c^{2} a B + \frac{3}{7} x^{7} c^{2} a A + \frac{1}{2} x^{6} c a^{2} B + \frac{3}{5} x^{5} c a^{2} A + \frac{1}{4} x^{4} a^{3} B + \frac{1}{3} x^{3} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.083413, size = 92, normalized size = 0.99 \begin{align*} \frac{A a^{3} x^{3}}{3} + \frac{3 A a^{2} c x^{5}}{5} + \frac{3 A a c^{2} x^{7}}{7} + \frac{A c^{3} x^{9}}{9} + \frac{B a^{3} x^{4}}{4} + \frac{B a^{2} c x^{6}}{2} + \frac{3 B a c^{2} x^{8}}{8} + \frac{B c^{3} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13507, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{10} \, B c^{3} x^{10} + \frac{1}{9} \, A c^{3} x^{9} + \frac{3}{8} \, B a c^{2} x^{8} + \frac{3}{7} \, A a c^{2} x^{7} + \frac{1}{2} \, B a^{2} c x^{6} + \frac{3}{5} \, A a^{2} c x^{5} + \frac{1}{4} \, B a^{3} x^{4} + \frac{1}{3} \, A a^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]