Optimal. Leaf size=254 \[ \frac{1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac{1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac{1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac{1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac{1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac{2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.325565, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1657} \[ \frac{1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac{1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac{1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac{1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac{1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac{2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1657
Rubi steps
\begin{align*} \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx &=\int \left (a^4 A+4 a^3 A b x+a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^2+4 a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^3+\left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^4+4 b \left (b^2+3 a c\right ) (A c+a C) x^5+\left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^6+4 b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^7+c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^8+4 b c^3 C x^9+c^4 C x^{10}\right ) \, dx\\ &=a^4 A x+2 a^3 A b x^2+\frac{1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^4+\frac{1}{5} \left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^5+\frac{2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac{1}{7} \left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^7+\frac{1}{2} b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^8+\frac{1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11}\\ \end{align*}
Mathematica [A] time = 0.0970406, size = 256, normalized size = 1.01 \[ \frac{1}{7} x^7 \left (6 a^2 c^2 C+4 a A c^3+12 a b^2 c C+6 A b^2 c^2+b^4 C\right )+\frac{1}{5} x^5 \left (6 a^2 A c^2+6 a^2 b^2 C+4 a^3 c C+12 a A b^2 c+A b^4\right )+a b x^4 \left (a^2 C+3 a A c+A b^2\right )+\frac{1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac{1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac{1}{2} b c x^8 \left (3 a c C+A c^2+b^2 C\right )+\frac{2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 343, normalized size = 1.4 \begin{align*}{\frac{{c}^{4}C{x}^{11}}{11}}+{\frac{2\,b{c}^{3}C{x}^{10}}{5}}+{\frac{ \left ( \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) C+{c}^{4}A \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) C+4\,b{c}^{3}A \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) C+ \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) A \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) C+ \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) A \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) C+ \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{3}bC+ \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) A \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{4}C+ \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) A \right ){x}^{3}}{3}}+2\,{a}^{3}Ab{x}^{2}+{a}^{4}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00883, size = 355, normalized size = 1.4 \begin{align*} \frac{1}{11} \, C c^{4} x^{11} + \frac{2}{5} \, C b c^{3} x^{10} + \frac{1}{9} \,{\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac{1}{2} \,{\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac{1}{7} \,{\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \,{\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac{2}{3} \,{\left (C a b^{3} + 3 \, A a b c^{2} +{\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac{1}{5} \,{\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \,{\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} +{\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac{1}{3} \,{\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.22246, size = 718, normalized size = 2.83 \begin{align*} \frac{1}{11} x^{11} c^{4} C + \frac{2}{5} x^{10} c^{3} b C + \frac{2}{3} x^{9} c^{2} b^{2} C + \frac{4}{9} x^{9} c^{3} a C + \frac{1}{9} x^{9} c^{4} A + \frac{1}{2} x^{8} c b^{3} C + \frac{3}{2} x^{8} c^{2} b a C + \frac{1}{2} x^{8} c^{3} b A + \frac{1}{7} x^{7} b^{4} C + \frac{12}{7} x^{7} c b^{2} a C + \frac{6}{7} x^{7} c^{2} a^{2} C + \frac{6}{7} x^{7} c^{2} b^{2} A + \frac{4}{7} x^{7} c^{3} a A + \frac{2}{3} x^{6} b^{3} a C + 2 x^{6} c b a^{2} C + \frac{2}{3} x^{6} c b^{3} A + 2 x^{6} c^{2} b a A + \frac{6}{5} x^{5} b^{2} a^{2} C + \frac{4}{5} x^{5} c a^{3} C + \frac{1}{5} x^{5} b^{4} A + \frac{12}{5} x^{5} c b^{2} a A + \frac{6}{5} x^{5} c^{2} a^{2} A + x^{4} b a^{3} C + x^{4} b^{3} a A + 3 x^{4} c b a^{2} A + \frac{1}{3} x^{3} a^{4} C + 2 x^{3} b^{2} a^{2} A + \frac{4}{3} x^{3} c a^{3} A + 2 x^{2} b a^{3} A + x a^{4} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.112548, size = 320, normalized size = 1.26 \begin{align*} A a^{4} x + 2 A a^{3} b x^{2} + \frac{2 C b c^{3} x^{10}}{5} + \frac{C c^{4} x^{11}}{11} + x^{9} \left (\frac{A c^{4}}{9} + \frac{4 C a c^{3}}{9} + \frac{2 C b^{2} c^{2}}{3}\right ) + x^{8} \left (\frac{A b c^{3}}{2} + \frac{3 C a b c^{2}}{2} + \frac{C b^{3} c}{2}\right ) + x^{7} \left (\frac{4 A a c^{3}}{7} + \frac{6 A b^{2} c^{2}}{7} + \frac{6 C a^{2} c^{2}}{7} + \frac{12 C a b^{2} c}{7} + \frac{C b^{4}}{7}\right ) + x^{6} \left (2 A a b c^{2} + \frac{2 A b^{3} c}{3} + 2 C a^{2} b c + \frac{2 C a b^{3}}{3}\right ) + x^{5} \left (\frac{6 A a^{2} c^{2}}{5} + \frac{12 A a b^{2} c}{5} + \frac{A b^{4}}{5} + \frac{4 C a^{3} c}{5} + \frac{6 C a^{2} b^{2}}{5}\right ) + x^{4} \left (3 A a^{2} b c + A a b^{3} + C a^{3} b\right ) + x^{3} \left (\frac{4 A a^{3} c}{3} + 2 A a^{2} b^{2} + \frac{C a^{4}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1739, size = 416, normalized size = 1.64 \begin{align*} \frac{1}{11} \, C c^{4} x^{11} + \frac{2}{5} \, C b c^{3} x^{10} + \frac{2}{3} \, C b^{2} c^{2} x^{9} + \frac{4}{9} \, C a c^{3} x^{9} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, C b^{3} c x^{8} + \frac{3}{2} \, C a b c^{2} x^{8} + \frac{1}{2} \, A b c^{3} x^{8} + \frac{1}{7} \, C b^{4} x^{7} + \frac{12}{7} \, C a b^{2} c x^{7} + \frac{6}{7} \, C a^{2} c^{2} x^{7} + \frac{6}{7} \, A b^{2} c^{2} x^{7} + \frac{4}{7} \, A a c^{3} x^{7} + \frac{2}{3} \, C a b^{3} x^{6} + 2 \, C a^{2} b c x^{6} + \frac{2}{3} \, A b^{3} c x^{6} + 2 \, A a b c^{2} x^{6} + \frac{6}{5} \, C a^{2} b^{2} x^{5} + \frac{1}{5} \, A b^{4} x^{5} + \frac{4}{5} \, C a^{3} c x^{5} + \frac{12}{5} \, A a b^{2} c x^{5} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + C a^{3} b x^{4} + A a b^{3} x^{4} + 3 \, A a^{2} b c x^{4} + \frac{1}{3} \, C a^{4} x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac{4}{3} \, A a^{3} c x^{3} + 2 \, A a^{3} b x^{2} + A a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]