Optimal. Leaf size=162 \[ -\frac {a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac {3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac {3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{10} c^2 x^{10} (A c+3 b B)+\frac {1}{12} B c^3 x^{12} \]
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Rubi [A] time = 0.23, antiderivative size = 162, 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} \[ a^2 \log (x) (a B+3 A b)-\frac {a^3 A}{2 x^2}+\frac {1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{10} c^2 x^{10} (A c+3 b B)+\frac {1}{12} B c^3 x^{12} \]
Antiderivative was successfully verified.
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Rule 765
Rule 1251
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (3 a \left (a b B+A \left (b^2+a c\right )\right )+\frac {a^3 A}{x^2}+\frac {a^2 (3 A b+a B)}{x}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^2+3 c \left (b^2 B+A b c+a B c\right ) x^3+c^2 (3 b B+A c) x^4+B c^3 x^5\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3 A}{2 x^2}+\frac {3}{2} a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\frac {1}{4} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\frac {1}{6} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^6+\frac {3}{8} c \left (b^2 B+A b c+a B c\right ) x^8+\frac {1}{10} c^2 (3 b B+A c) x^{10}+\frac {1}{12} B c^3 x^{12}+a^2 (3 A b+a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 162, normalized size = 1.00 \[ -\frac {a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac {3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac {3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{10} c^2 x^{10} (A c+3 b B)+\frac {1}{12} B c^3 x^{12} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 170, normalized size = 1.05 \[ \frac {10 \, B c^{3} x^{14} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + 45 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{10} + 20 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 30 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 180 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} - 60 \, A a^{3} + 120 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \log \relax (x)}{120 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 212, normalized size = 1.31 \[ \frac {1}{12} \, B c^{3} x^{12} + \frac {3}{10} \, B b c^{2} x^{10} + \frac {1}{10} \, A c^{3} x^{10} + \frac {3}{8} \, B b^{2} c x^{8} + \frac {3}{8} \, B a c^{2} x^{8} + \frac {3}{8} \, A b c^{2} x^{8} + \frac {1}{6} \, B b^{3} x^{6} + B a b c x^{6} + \frac {1}{2} \, A b^{2} c x^{6} + \frac {1}{2} \, A a c^{2} x^{6} + \frac {3}{4} \, B a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + \frac {3}{4} \, B a^{2} c x^{4} + \frac {3}{2} \, A a b c x^{4} + \frac {3}{2} \, B a^{2} b x^{2} + \frac {3}{2} \, A a b^{2} x^{2} + \frac {3}{2} \, A a^{2} c x^{2} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x^{2}\right ) - \frac {B a^{3} x^{2} + 3 \, A a^{2} b x^{2} + A a^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 190, normalized size = 1.17 \[ \frac {B \,c^{3} x^{12}}{12}+\frac {A \,c^{3} x^{10}}{10}+\frac {3 B b \,c^{2} x^{10}}{10}+\frac {3 A b \,c^{2} x^{8}}{8}+\frac {3 B a \,c^{2} x^{8}}{8}+\frac {3 B \,b^{2} c \,x^{8}}{8}+\frac {A a \,c^{2} x^{6}}{2}+\frac {A \,b^{2} c \,x^{6}}{2}+B a b c \,x^{6}+\frac {B \,b^{3} x^{6}}{6}+\frac {3 A a b c \,x^{4}}{2}+\frac {A \,b^{3} x^{4}}{4}+\frac {3 B \,a^{2} c \,x^{4}}{4}+\frac {3 B a \,b^{2} x^{4}}{4}+\frac {3 A \,a^{2} c \,x^{2}}{2}+\frac {3 A a \,b^{2} x^{2}}{2}+\frac {3 B \,a^{2} b \,x^{2}}{2}+3 A \,a^{2} b \ln \relax (x )+B \,a^{3} \ln \relax (x )-\frac {A \,a^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 167, normalized size = 1.03 \[ \frac {1}{12} \, B c^{3} x^{12} + \frac {1}{10} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac {3}{8} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{8} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{6} + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + \frac {3}{2} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - \frac {A a^{3}}{2 \, x^{2}} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 166, normalized size = 1.02 \[ x^4\,\left (\frac {3\,B\,c\,a^2}{4}+\frac {3\,B\,a\,b^2}{4}+\frac {3\,A\,c\,a\,b}{2}+\frac {A\,b^3}{4}\right )+x^6\,\left (\frac {B\,b^3}{6}+\frac {A\,b^2\,c}{2}+B\,a\,b\,c+\frac {A\,a\,c^2}{2}\right )+x^{10}\,\left (\frac {A\,c^3}{10}+\frac {3\,B\,b\,c^2}{10}\right )+\ln \relax (x)\,\left (B\,a^3+3\,A\,b\,a^2\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{2}+\frac {3\,A\,c\,a^2}{2}+\frac {3\,A\,a\,b^2}{2}\right )+x^8\,\left (\frac {3\,B\,b^2\,c}{8}+\frac {3\,A\,b\,c^2}{8}+\frac {3\,B\,a\,c^2}{8}\right )-\frac {A\,a^3}{2\,x^2}+\frac {B\,c^3\,x^{12}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 197, normalized size = 1.22 \[ - \frac {A a^{3}}{2 x^{2}} + \frac {B c^{3} x^{12}}{12} + a^{2} \left (3 A b + B a\right ) \log {\relax (x )} + x^{10} \left (\frac {A c^{3}}{10} + \frac {3 B b c^{2}}{10}\right ) + x^{8} \left (\frac {3 A b c^{2}}{8} + \frac {3 B a c^{2}}{8} + \frac {3 B b^{2} c}{8}\right ) + x^{6} \left (\frac {A a c^{2}}{2} + \frac {A b^{2} c}{2} + B a b c + \frac {B b^{3}}{6}\right ) + x^{4} \left (\frac {3 A a b c}{2} + \frac {A b^{3}}{4} + \frac {3 B a^{2} c}{4} + \frac {3 B a b^{2}}{4}\right ) + x^{2} \left (\frac {3 A a^{2} c}{2} + \frac {3 A a b^{2}}{2} + \frac {3 B a^{2} b}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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