Optimal. Leaf size=159 \[ \frac {1}{3} a^2 A x^3+\frac {1}{4} a^2 B x^4+\frac {1}{5} a (2 A b+a C) x^5+\frac {1}{3} a b B x^6+\frac {1}{7} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^7+\frac {1}{8} B \left (b^2+2 a c\right ) x^8+\frac {1}{9} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^9+\frac {1}{5} b B c x^{10}+\frac {1}{11} c (A c+2 b C) x^{11}+\frac {1}{12} B c^2 x^{12}+\frac {1}{13} c^2 C x^{13} \]
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Rubi [A]
time = 0.15, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1642}
\begin {gather*} \frac {1}{3} a^2 A x^3+\frac {1}{4} a^2 B x^4+\frac {1}{9} x^9 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{7} x^7 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac {1}{5} a x^5 (a C+2 A b)+\frac {1}{8} B x^8 \left (2 a c+b^2\right )+\frac {1}{3} a b B x^6+\frac {1}{11} c x^{11} (A c+2 b C)+\frac {1}{5} b B c x^{10}+\frac {1}{12} B c^2 x^{12}+\frac {1}{13} c^2 C x^{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int x^2 \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 A x^2+a^2 B x^3+a (2 A b+a C) x^4+2 a b B x^5+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^6+B \left (b^2+2 a c\right ) x^7+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^8+2 b B c x^9+c (A c+2 b C) x^{10}+B c^2 x^{11}+c^2 C x^{12}\right ) \, dx\\ &=\frac {1}{3} a^2 A x^3+\frac {1}{4} a^2 B x^4+\frac {1}{5} a (2 A b+a C) x^5+\frac {1}{3} a b B x^6+\frac {1}{7} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^7+\frac {1}{8} B \left (b^2+2 a c\right ) x^8+\frac {1}{9} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^9+\frac {1}{5} b B c x^{10}+\frac {1}{11} c (A c+2 b C) x^{11}+\frac {1}{12} B c^2 x^{12}+\frac {1}{13} c^2 C x^{13}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 159, normalized size = 1.00 \begin {gather*} \frac {1}{3} a^2 A x^3+\frac {1}{4} a^2 B x^4+\frac {1}{5} a (2 A b+a C) x^5+\frac {1}{3} a b B x^6+\frac {1}{7} \left (A b^2+2 a A c+2 a b C\right ) x^7+\frac {1}{8} B \left (b^2+2 a c\right ) x^8+\frac {1}{9} \left (2 A b c+b^2 C+2 a c C\right ) x^9+\frac {1}{5} b B c x^{10}+\frac {1}{11} c (A c+2 b C) x^{11}+\frac {1}{12} B c^2 x^{12}+\frac {1}{13} c^2 C x^{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 142, normalized size = 0.89
method | result | size |
default | \(\frac {c^{2} C \,x^{13}}{13}+\frac {B \,c^{2} x^{12}}{12}+\frac {\left (A \,c^{2}+2 b c C \right ) x^{11}}{11}+\frac {b B c \,x^{10}}{5}+\frac {\left (2 b c A +\left (2 a c +b^{2}\right ) C \right ) x^{9}}{9}+\frac {B \left (2 a c +b^{2}\right ) x^{8}}{8}+\frac {\left (A \left (2 a c +b^{2}\right )+2 a b C \right ) x^{7}}{7}+\frac {a b B \,x^{6}}{3}+\frac {\left (2 a b A +a^{2} C \right ) x^{5}}{5}+\frac {a^{2} B \,x^{4}}{4}+\frac {a^{2} A \,x^{3}}{3}\) | \(142\) |
norman | \(\frac {c^{2} C \,x^{13}}{13}+\frac {B \,c^{2} x^{12}}{12}+\left (\frac {1}{11} A \,c^{2}+\frac {2}{11} b c C \right ) x^{11}+\frac {b B c \,x^{10}}{5}+\left (\frac {2}{9} b c A +\frac {2}{9} a c C +\frac {1}{9} C \,b^{2}\right ) x^{9}+\left (\frac {1}{4} a c B +\frac {1}{8} b^{2} B \right ) x^{8}+\left (\frac {2}{7} a c A +\frac {1}{7} A \,b^{2}+\frac {2}{7} a b C \right ) x^{7}+\frac {a b B \,x^{6}}{3}+\left (\frac {2}{5} a b A +\frac {1}{5} a^{2} C \right ) x^{5}+\frac {a^{2} B \,x^{4}}{4}+\frac {a^{2} A \,x^{3}}{3}\) | \(144\) |
gosper | \(\frac {1}{13} c^{2} C \,x^{13}+\frac {1}{12} B \,c^{2} x^{12}+\frac {1}{11} x^{11} A \,c^{2}+\frac {2}{11} x^{11} b c C +\frac {1}{5} b B c \,x^{10}+\frac {2}{9} x^{9} b c A +\frac {2}{9} x^{9} a c C +\frac {1}{9} x^{9} C \,b^{2}+\frac {1}{4} x^{8} a c B +\frac {1}{8} x^{8} b^{2} B +\frac {2}{7} x^{7} a c A +\frac {1}{7} x^{7} A \,b^{2}+\frac {2}{7} x^{7} a b C +\frac {1}{3} a b B \,x^{6}+\frac {2}{5} x^{5} a b A +\frac {1}{5} x^{5} a^{2} C +\frac {1}{4} a^{2} B \,x^{4}+\frac {1}{3} a^{2} A \,x^{3}\) | \(155\) |
risch | \(\frac {1}{13} c^{2} C \,x^{13}+\frac {1}{12} B \,c^{2} x^{12}+\frac {1}{11} x^{11} A \,c^{2}+\frac {2}{11} x^{11} b c C +\frac {1}{5} b B c \,x^{10}+\frac {2}{9} x^{9} b c A +\frac {2}{9} x^{9} a c C +\frac {1}{9} x^{9} C \,b^{2}+\frac {1}{4} x^{8} a c B +\frac {1}{8} x^{8} b^{2} B +\frac {2}{7} x^{7} a c A +\frac {1}{7} x^{7} A \,b^{2}+\frac {2}{7} x^{7} a b C +\frac {1}{3} a b B \,x^{6}+\frac {2}{5} x^{5} a b A +\frac {1}{5} x^{5} a^{2} C +\frac {1}{4} a^{2} B \,x^{4}+\frac {1}{3} a^{2} A \,x^{3}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 143, normalized size = 0.90 \begin {gather*} \frac {1}{13} \, C c^{2} x^{13} + \frac {1}{12} \, B c^{2} x^{12} + \frac {1}{5} \, B b c x^{10} + \frac {1}{11} \, {\left (2 \, C b c + A c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{9} + \frac {1}{3} \, B a b x^{6} + \frac {1}{8} \, {\left (B b^{2} + 2 \, B a c\right )} x^{8} + \frac {1}{7} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{7} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (C a^{2} + 2 \, A a b\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 143, normalized size = 0.90 \begin {gather*} \frac {1}{13} \, C c^{2} x^{13} + \frac {1}{12} \, B c^{2} x^{12} + \frac {1}{5} \, B b c x^{10} + \frac {1}{11} \, {\left (2 \, C b c + A c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{9} + \frac {1}{3} \, B a b x^{6} + \frac {1}{8} \, {\left (B b^{2} + 2 \, B a c\right )} x^{8} + \frac {1}{7} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{7} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (C a^{2} + 2 \, A a b\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 168, normalized size = 1.06 \begin {gather*} \frac {A a^{2} x^{3}}{3} + \frac {B a^{2} x^{4}}{4} + \frac {B a b x^{6}}{3} + \frac {B b c x^{10}}{5} + \frac {B c^{2} x^{12}}{12} + \frac {C c^{2} x^{13}}{13} + x^{11} \left (\frac {A c^{2}}{11} + \frac {2 C b c}{11}\right ) + x^{9} \cdot \left (\frac {2 A b c}{9} + \frac {2 C a c}{9} + \frac {C b^{2}}{9}\right ) + x^{8} \left (\frac {B a c}{4} + \frac {B b^{2}}{8}\right ) + x^{7} \cdot \left (\frac {2 A a c}{7} + \frac {A b^{2}}{7} + \frac {2 C a b}{7}\right ) + x^{5} \cdot \left (\frac {2 A a b}{5} + \frac {C a^{2}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.40, size = 154, normalized size = 0.97 \begin {gather*} \frac {1}{13} \, C c^{2} x^{13} + \frac {1}{12} \, B c^{2} x^{12} + \frac {2}{11} \, C b c x^{11} + \frac {1}{11} \, A c^{2} x^{11} + \frac {1}{5} \, B b c x^{10} + \frac {1}{9} \, C b^{2} x^{9} + \frac {2}{9} \, C a c x^{9} + \frac {2}{9} \, A b c x^{9} + \frac {1}{8} \, B b^{2} x^{8} + \frac {1}{4} \, B a c x^{8} + \frac {2}{7} \, C a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {2}{7} \, A a c x^{7} + \frac {1}{3} \, B a b x^{6} + \frac {1}{5} \, C a^{2} x^{5} + \frac {2}{5} \, A a b x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{3} \, A a^{2} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 141, normalized size = 0.89 \begin {gather*} x^5\,\left (\frac {C\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+x^{11}\,\left (\frac {A\,c^2}{11}+\frac {2\,C\,b\,c}{11}\right )+x^7\,\left (\frac {A\,b^2}{7}+\frac {2\,C\,a\,b}{7}+\frac {2\,A\,a\,c}{7}\right )+x^9\,\left (\frac {C\,b^2}{9}+\frac {2\,A\,c\,b}{9}+\frac {2\,C\,a\,c}{9}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,a^2\,x^4}{4}+\frac {B\,c^2\,x^{12}}{12}+\frac {C\,c^2\,x^{13}}{13}+\frac {B\,x^8\,\left (b^2+2\,a\,c\right )}{8}+\frac {B\,a\,b\,x^6}{3}+\frac {B\,b\,c\,x^{10}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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