Optimal. Leaf size=101 \[ \frac {1}{3} a^2 A x^3+\frac {1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{7} c x^7 (A c+2 b B)+\frac {1}{8} B c^2 x^8 \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} \frac {1}{3} a^2 A x^3+\frac {1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{7} c x^7 (A c+2 b B)+\frac {1}{8} B c^2 x^8 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A x^2+a (2 A b+a B) x^3+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^4+\left (b^2 B+2 A b c+2 a B c\right ) x^5+c (2 b B+A c) x^6+B c^2 x^7\right ) \, dx\\ &=\frac {1}{3} a^2 A x^3+\frac {1}{4} a (2 A b+a B) x^4+\frac {1}{5} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^5+\frac {1}{6} \left (b^2 B+2 A b c+2 a B c\right ) x^6+\frac {1}{7} c (2 b B+A c) x^7+\frac {1}{8} B c^2 x^8\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 101, normalized size = 1.00 \begin {gather*} \frac {1}{3} a^2 A x^3+\frac {1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{5} x^5 \left (2 a A c+2 a b B+A b^2\right )+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{7} c x^7 (A c+2 b B)+\frac {1}{8} B c^2 x^8 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.37, size = 103, normalized size = 1.02 \begin {gather*} \frac {1}{8} x^{8} c^{2} B + \frac {2}{7} x^{7} c b B + \frac {1}{7} x^{7} c^{2} A + \frac {1}{6} x^{6} b^{2} B + \frac {1}{3} x^{6} c a B + \frac {1}{3} x^{6} c b A + \frac {2}{5} x^{5} b a B + \frac {1}{5} x^{5} b^{2} A + \frac {2}{5} x^{5} c a A + \frac {1}{4} x^{4} a^{2} B + \frac {1}{2} x^{4} b a A + \frac {1}{3} x^{3} a^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 103, normalized size = 1.02 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {2}{7} \, B b c x^{7} + \frac {1}{7} \, A c^{2} x^{7} + \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, B a c x^{6} + \frac {1}{3} \, A b c x^{6} + \frac {2}{5} \, B a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {2}{5} \, A a c x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, A a^{2} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 94, normalized size = 0.93 \begin {gather*} \frac {B \,c^{2} x^{8}}{8}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{7}}{7}+\frac {A \,a^{2} x^{3}}{3}+\frac {\left (2 A b c +\left (2 a c +b^{2}\right ) B \right ) x^{6}}{6}+\frac {\left (2 B a b +\left (2 a c +b^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (2 A a b +B \,a^{2}\right ) x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 93, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.16, size = 93, normalized size = 0.92 \begin {gather*} x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^7\,\left (\frac {A\,c^2}{7}+\frac {2\,B\,b\,c}{7}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^6\,\left (\frac {B\,b^2}{6}+\frac {A\,c\,b}{3}+\frac {B\,a\,c}{3}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,c^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 105, normalized size = 1.04 \begin {gather*} \frac {A a^{2} x^{3}}{3} + \frac {B c^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2}}{7} + \frac {2 B b c}{7}\right ) + x^{6} \left (\frac {A b c}{3} + \frac {B a c}{3} + \frac {B b^{2}}{6}\right ) + x^{5} \left (\frac {2 A a c}{5} + \frac {A b^{2}}{5} + \frac {2 B a b}{5}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________