Optimal. Leaf size=154 \[ a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac {1}{3} a x^3 (a C+2 A b)+\frac {1}{6} B x^6 \left (2 a c+b^2\right )+\frac {1}{2} a b B x^4+\frac {1}{9} c x^9 (A c+2 b C)+\frac {1}{4} b B c x^8+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11} \]
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Rubi [A] time = 0.11, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1657} \begin {gather*} a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac {1}{3} a x^3 (a C+2 A b)+\frac {1}{6} B x^6 \left (2 a c+b^2\right )+\frac {1}{2} a b B x^4+\frac {1}{9} c x^9 (A c+2 b C)+\frac {1}{4} b B c x^8+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin {align*} \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 A+a^2 B x+a (2 A b+a C) x^2+2 a b B x^3+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^4+B \left (b^2+2 a c\right ) x^5+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^6+2 b B c x^7+c (A c+2 b C) x^8+B c^2 x^9+c^2 C x^{10}\right ) \, dx\\ &=a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{3} a (2 A b+a C) x^3+\frac {1}{2} a b B x^4+\frac {1}{5} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^5+\frac {1}{6} B \left (b^2+2 a c\right ) x^6+\frac {1}{7} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^7+\frac {1}{4} b B c x^8+\frac {1}{9} c (A c+2 b C) x^9+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 154, normalized size = 1.00 \begin {gather*} a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{7} x^7 \left (2 a c C+2 A b c+b^2 C\right )+\frac {1}{5} x^5 \left (2 a A c+2 a b C+A b^2\right )+\frac {1}{3} a x^3 (a C+2 A b)+\frac {1}{6} B x^6 \left (2 a c+b^2\right )+\frac {1}{2} a b B x^4+\frac {1}{9} c x^9 (A c+2 b C)+\frac {1}{4} b B c x^8+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.98, size = 151, normalized size = 0.98 \begin {gather*} \frac {1}{11} x^{11} c^{2} C + \frac {1}{10} x^{10} c^{2} B + \frac {2}{9} x^{9} c b C + \frac {1}{9} x^{9} c^{2} A + \frac {1}{4} x^{8} c b B + \frac {1}{7} x^{7} b^{2} C + \frac {2}{7} x^{7} c a C + \frac {2}{7} x^{7} c b A + \frac {1}{6} x^{6} b^{2} B + \frac {1}{3} x^{6} c a B + \frac {2}{5} x^{5} b a C + \frac {1}{5} x^{5} b^{2} A + \frac {2}{5} x^{5} c a A + \frac {1}{2} x^{4} b a B + \frac {1}{3} x^{3} a^{2} C + \frac {2}{3} x^{3} b a A + \frac {1}{2} x^{2} a^{2} B + x a^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 151, normalized size = 0.98 \begin {gather*} \frac {1}{11} \, C c^{2} x^{11} + \frac {1}{10} \, B c^{2} x^{10} + \frac {2}{9} \, C b c x^{9} + \frac {1}{9} \, A c^{2} x^{9} + \frac {1}{4} \, B b c x^{8} + \frac {1}{7} \, C b^{2} x^{7} + \frac {2}{7} \, C a c x^{7} + \frac {2}{7} \, A b c x^{7} + \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, B a c x^{6} + \frac {2}{5} \, C a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {2}{5} \, A a c x^{5} + \frac {1}{2} \, B a b x^{4} + \frac {1}{3} \, C a^{2} x^{3} + \frac {2}{3} \, A a b x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 139, normalized size = 0.90 \begin {gather*} \frac {C \,c^{2} x^{11}}{11}+\frac {B \,c^{2} x^{10}}{10}+\frac {B b c \,x^{8}}{4}+\frac {\left (A \,c^{2}+2 C b c \right ) x^{9}}{9}+\frac {B a b \,x^{4}}{2}+\frac {\left (2 a c +b^{2}\right ) B \,x^{6}}{6}+\frac {\left (2 A b c +\left (2 a c +b^{2}\right ) C \right ) x^{7}}{7}+\frac {B \,a^{2} x^{2}}{2}+\frac {\left (2 C a b +\left (2 a c +b^{2}\right ) A \right ) x^{5}}{5}+A \,a^{2} x +\frac {\left (2 A a b +C \,a^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 140, normalized size = 0.91 \begin {gather*} \frac {1}{11} \, C c^{2} x^{11} + \frac {1}{10} \, B c^{2} x^{10} + \frac {1}{4} \, B b c x^{8} + \frac {1}{9} \, {\left (2 \, C b c + A c^{2}\right )} x^{9} + \frac {1}{7} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{7} + \frac {1}{2} \, B a b x^{4} + \frac {1}{6} \, {\left (B b^{2} + 2 \, B a c\right )} x^{6} + \frac {1}{5} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 138, normalized size = 0.90 \begin {gather*} x^3\,\left (\frac {C\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+x^9\,\left (\frac {A\,c^2}{9}+\frac {2\,C\,b\,c}{9}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,C\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^7\,\left (\frac {C\,b^2}{7}+\frac {2\,A\,c\,b}{7}+\frac {2\,C\,a\,c}{7}\right )+\frac {B\,a^2\,x^2}{2}+\frac {B\,c^2\,x^{10}}{10}+\frac {C\,c^2\,x^{11}}{11}+\frac {B\,x^6\,\left (b^2+2\,a\,c\right )}{6}+A\,a^2\,x+\frac {B\,a\,b\,x^4}{2}+\frac {B\,b\,c\,x^8}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 165, normalized size = 1.07 \begin {gather*} A a^{2} x + \frac {B a^{2} x^{2}}{2} + \frac {B a b x^{4}}{2} + \frac {B b c x^{8}}{4} + \frac {B c^{2} x^{10}}{10} + \frac {C c^{2} x^{11}}{11} + x^{9} \left (\frac {A c^{2}}{9} + \frac {2 C b c}{9}\right ) + x^{7} \left (\frac {2 A b c}{7} + \frac {2 C a c}{7} + \frac {C b^{2}}{7}\right ) + x^{6} \left (\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 C a b}{5}\right ) + x^{3} \left (\frac {2 A a b}{3} + \frac {C a^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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