Optimal. Leaf size=166 \[ \frac {1}{4} a^3 A x^4+\frac {1}{6} a^2 x^6 (a B+3 A b)+\frac {3}{14} c x^{14} \left (a B c+A b c+b^2 B\right )+\frac {3}{8} a x^8 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{12} x^{12} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{10} x^{10} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{16} c^2 x^{16} (A c+3 b B)+\frac {1}{18} B c^3 x^{18} \]
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Rubi [A] time = 0.39, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1251, 765} \[ \frac {1}{6} a^2 x^6 (a B+3 A b)+\frac {1}{4} a^3 A x^4+\frac {1}{12} x^{12} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {3}{14} c x^{14} \left (a B c+A b c+b^2 B\right )+\frac {1}{10} x^{10} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {3}{8} a x^8 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{16} c^2 x^{16} (A c+3 b B)+\frac {1}{18} B c^3 x^{18} \]
Antiderivative was successfully verified.
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Rule 765
Rule 1251
Rubi steps
\begin {align*} \int x^3 \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (A+B x) \left (a+b x+c x^2\right )^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^3 A x+a^2 (3 A b+a B) x^2+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^5+3 c \left (b^2 B+A b c+a B c\right ) x^6+c^2 (3 b B+A c) x^7+B c^3 x^8\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4} a^3 A x^4+\frac {1}{6} a^2 (3 A b+a B) x^6+\frac {3}{8} a \left (a b B+A \left (b^2+a c\right )\right ) x^8+\frac {1}{10} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{10}+\frac {1}{12} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{12}+\frac {3}{14} c \left (b^2 B+A b c+a B c\right ) x^{14}+\frac {1}{16} c^2 (3 b B+A c) x^{16}+\frac {1}{18} B c^3 x^{18}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 166, normalized size = 1.00 \[ \frac {1}{4} a^3 A x^4+\frac {1}{6} a^2 x^6 (a B+3 A b)+\frac {3}{14} c x^{14} \left (a B c+A b c+b^2 B\right )+\frac {3}{8} a x^8 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{12} x^{12} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{10} x^{10} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{16} c^2 x^{16} (A c+3 b B)+\frac {1}{18} B c^3 x^{18} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 193, normalized size = 1.16 \[ \frac {1}{18} x^{18} c^{3} B + \frac {3}{16} x^{16} c^{2} b B + \frac {1}{16} x^{16} c^{3} A + \frac {3}{14} x^{14} c b^{2} B + \frac {3}{14} x^{14} c^{2} a B + \frac {3}{14} x^{14} c^{2} b A + \frac {1}{12} x^{12} b^{3} B + \frac {1}{2} x^{12} c b a B + \frac {1}{4} x^{12} c b^{2} A + \frac {1}{4} x^{12} c^{2} a A + \frac {3}{10} x^{10} b^{2} a B + \frac {3}{10} x^{10} c a^{2} B + \frac {1}{10} x^{10} b^{3} A + \frac {3}{5} x^{10} c b a A + \frac {3}{8} x^{8} b a^{2} B + \frac {3}{8} x^{8} b^{2} a A + \frac {3}{8} x^{8} c a^{2} A + \frac {1}{6} x^{6} a^{3} B + \frac {1}{2} x^{6} b a^{2} A + \frac {1}{4} x^{4} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 193, normalized size = 1.16 \[ \frac {1}{18} \, B c^{3} x^{18} + \frac {3}{16} \, B b c^{2} x^{16} + \frac {1}{16} \, A c^{3} x^{16} + \frac {3}{14} \, B b^{2} c x^{14} + \frac {3}{14} \, B a c^{2} x^{14} + \frac {3}{14} \, A b c^{2} x^{14} + \frac {1}{12} \, B b^{3} x^{12} + \frac {1}{2} \, B a b c x^{12} + \frac {1}{4} \, A b^{2} c x^{12} + \frac {1}{4} \, A a c^{2} x^{12} + \frac {3}{10} \, B a b^{2} x^{10} + \frac {1}{10} \, A b^{3} x^{10} + \frac {3}{10} \, B a^{2} c x^{10} + \frac {3}{5} \, A a b c x^{10} + \frac {3}{8} \, B a^{2} b x^{8} + \frac {3}{8} \, A a b^{2} x^{8} + \frac {3}{8} \, A a^{2} c x^{8} + \frac {1}{6} \, B a^{3} x^{6} + \frac {1}{2} \, A a^{2} b x^{6} + \frac {1}{4} \, A a^{3} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 226, normalized size = 1.36 \[ \frac {B \,c^{3} x^{18}}{18}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{16}}{16}+\frac {\left (3 A b \,c^{2}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) B \right ) x^{14}}{14}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) A +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) B \right ) x^{12}}{12}+\frac {\left (\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) A +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) B \right ) x^{10}}{10}+\frac {A \,a^{3} x^{4}}{4}+\frac {\left (3 B \,a^{2} b +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) A \right ) x^{8}}{8}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 166, normalized size = 1.00 \[ \frac {1}{18} \, B c^{3} x^{18} + \frac {1}{16} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{16} + \frac {3}{14} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{12} + \frac {1}{10} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{10} + \frac {3}{8} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{8} + \frac {1}{4} \, A a^{3} x^{4} + \frac {1}{6} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 169, normalized size = 1.02 \[ x^{10}\,\left (\frac {3\,B\,c\,a^2}{10}+\frac {3\,B\,a\,b^2}{10}+\frac {3\,A\,c\,a\,b}{5}+\frac {A\,b^3}{10}\right )+x^{12}\,\left (\frac {B\,b^3}{12}+\frac {A\,b^2\,c}{4}+\frac {B\,a\,b\,c}{2}+\frac {A\,a\,c^2}{4}\right )+x^6\,\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )+x^{16}\,\left (\frac {A\,c^3}{16}+\frac {3\,B\,b\,c^2}{16}\right )+x^8\,\left (\frac {3\,B\,a^2\,b}{8}+\frac {3\,A\,c\,a^2}{8}+\frac {3\,A\,a\,b^2}{8}\right )+x^{14}\,\left (\frac {3\,B\,b^2\,c}{14}+\frac {3\,A\,b\,c^2}{14}+\frac {3\,B\,a\,c^2}{14}\right )+\frac {A\,a^3\,x^4}{4}+\frac {B\,c^3\,x^{18}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 202, normalized size = 1.22 \[ \frac {A a^{3} x^{4}}{4} + \frac {B c^{3} x^{18}}{18} + x^{16} \left (\frac {A c^{3}}{16} + \frac {3 B b c^{2}}{16}\right ) + x^{14} \left (\frac {3 A b c^{2}}{14} + \frac {3 B a c^{2}}{14} + \frac {3 B b^{2} c}{14}\right ) + x^{12} \left (\frac {A a c^{2}}{4} + \frac {A b^{2} c}{4} + \frac {B a b c}{2} + \frac {B b^{3}}{12}\right ) + x^{10} \left (\frac {3 A a b c}{5} + \frac {A b^{3}}{10} + \frac {3 B a^{2} c}{10} + \frac {3 B a b^{2}}{10}\right ) + x^{8} \left (\frac {3 A a^{2} c}{8} + \frac {3 A a b^{2}}{8} + \frac {3 B a^{2} b}{8}\right ) + x^{6} \left (\frac {A a^{2} b}{2} + \frac {B a^{3}}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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