Optimal. Leaf size=96 \[ \frac {1}{3} x^3 \left (a^2 C+A \left (2 a c+b^2\right )\right )+a^2 A x+\frac {1}{5} x^5 \left (C \left (2 a c+b^2\right )+A c^2\right )+\frac {1}{2} b x^4 (a C+A c)+a A b x^2+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1657} \[ \frac {1}{3} x^3 \left (a^2 C+A \left (2 a c+b^2\right )\right )+a^2 A x+\frac {1}{5} x^5 \left (C \left (2 a c+b^2\right )+A c^2\right )+\frac {1}{2} b x^4 (a C+A c)+a A b x^2+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1657
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^2 \left (A+C x^2\right ) \, dx &=\int \left (a^2 A+2 a A b x+\left (A \left (b^2+2 a c\right )+a^2 C\right ) x^2+2 b (A c+a C) x^3+\left (A c^2+\left (b^2+2 a c\right ) C\right ) x^4+2 b c C x^5+c^2 C x^6\right ) \, dx\\ &=a^2 A x+a A b x^2+\frac {1}{3} \left (A \left (b^2+2 a c\right )+a^2 C\right ) x^3+\frac {1}{2} b (A c+a C) x^4+\frac {1}{5} \left (A c^2+\left (b^2+2 a c\right ) C\right ) x^5+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 96, normalized size = 1.00 \[ \frac {1}{3} x^3 \left (a^2 C+2 a A c+A b^2\right )+a^2 A x+\frac {1}{5} x^5 \left (2 a c C+A c^2+b^2 C\right )+\frac {1}{2} b x^4 (a C+A c)+a A b x^2+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 99, normalized size = 1.03 \[ \frac {1}{7} x^{7} c^{2} C + \frac {1}{3} x^{6} c b C + \frac {1}{5} x^{5} b^{2} C + \frac {2}{5} x^{5} c a C + \frac {1}{5} x^{5} c^{2} A + \frac {1}{2} x^{4} b a C + \frac {1}{2} x^{4} c b A + \frac {1}{3} x^{3} a^{2} C + \frac {1}{3} x^{3} b^{2} A + \frac {2}{3} x^{3} c a A + x^{2} b a A + x a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 99, normalized size = 1.03 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{3} \, C b c x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {2}{5} \, C a c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, C a b x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {1}{3} \, C a^{2} x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {2}{3} \, A a c x^{3} + A a b x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 90, normalized size = 0.94 \[ \frac {C \,c^{2} x^{7}}{7}+\frac {C b c \,x^{6}}{3}+A a b \,x^{2}+\frac {\left (A \,c^{2}+\left (2 a c +b^{2}\right ) C \right ) x^{5}}{5}+A \,a^{2} x +\frac {\left (2 b c A +2 a b C \right ) x^{4}}{4}+\frac {\left (C \,a^{2}+\left (2 a c +b^{2}\right ) A \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 87, normalized size = 0.91 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{3} \, C b c x^{6} + \frac {1}{5} \, {\left (C b^{2} + 2 \, C a c + A c^{2}\right )} x^{5} + A a b x^{2} + \frac {1}{2} \, {\left (C a b + A b c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + A b^{2} + 2 \, A a c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.09, size = 88, normalized size = 0.92 \[ x^3\,\left (\frac {C\,a^2}{3}+\frac {2\,A\,c\,a}{3}+\frac {A\,b^2}{3}\right )+x^5\,\left (\frac {C\,b^2}{5}+\frac {A\,c^2}{5}+\frac {2\,C\,a\,c}{5}\right )+\frac {C\,c^2\,x^7}{7}+A\,a^2\,x+\frac {b\,x^4\,\left (A\,c+C\,a\right )}{2}+A\,a\,b\,x^2+\frac {C\,b\,c\,x^6}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 102, normalized size = 1.06 \[ A a^{2} x + A a b x^{2} + \frac {C b c x^{6}}{3} + \frac {C c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 C a c}{5} + \frac {C b^{2}}{5}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {C a b}{2}\right ) + x^{3} \left (\frac {2 A a c}{3} + \frac {A b^{2}}{3} + \frac {C a^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________