Optimal. Leaf size=149 \[ -\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+\frac {1}{3} x^3 \left (C \left (2 a c+b^2\right )+2 A b c\right )+x \left (A \left (2 a c+b^2\right )+2 a b C\right )-\frac {a (a C+2 A b)}{x}+\frac {1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac {1}{5} c x^5 (A c+2 b C)+\frac {1}{2} b B c x^4+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7 \]
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Rubi [A] time = 0.14, antiderivative size = 149, 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 = {1628} \[ -\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+\frac {1}{3} x^3 \left (C \left (2 a c+b^2\right )+2 A b c\right )+x \left (A \left (2 a c+b^2\right )+2 a b C\right )-\frac {a (a C+2 A b)}{x}+\frac {1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac {1}{5} c x^5 (A c+2 b C)+\frac {1}{2} b B c x^4+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7 \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2}{x^4} \, dx &=\int \left (A b^2 \left (1+\frac {2 a (A c+b C)}{A b^2}\right )+\frac {a^2 A}{x^4}+\frac {a^2 B}{x^3}+\frac {a (2 A b+a C)}{x^2}+\frac {2 a b B}{x}+B \left (b^2+2 a c\right ) x+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^2+2 b B c x^3+c (A c+2 b C) x^4+B c^2 x^5+c^2 C x^6\right ) \, dx\\ &=-\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}-\frac {a (2 A b+a C)}{x}+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x+\frac {1}{2} B \left (b^2+2 a c\right ) x^2+\frac {1}{3} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^3+\frac {1}{2} b B c x^4+\frac {1}{5} c (A c+2 b C) x^5+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7+2 a b B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 151, normalized size = 1.01 \[ \frac {a^2 (-C)-2 a A b}{x}-\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+\frac {1}{3} x^3 \left (2 a c C+2 A b c+b^2 C\right )+x \left (2 a A c+2 a b C+A b^2\right )+\frac {1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac {1}{5} c x^5 (A c+2 b C)+\frac {1}{2} b B c x^4+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 145, normalized size = 0.97 \[ \frac {30 \, C c^{2} x^{10} + 35 \, B c^{2} x^{9} + 105 \, B b c x^{7} + 42 \, {\left (2 \, C b c + A c^{2}\right )} x^{8} + 70 \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{6} + 420 \, B a b x^{3} \log \relax (x) + 105 \, {\left (B b^{2} + 2 \, B a c\right )} x^{5} + 210 \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 105 \, B a^{2} x - 70 \, A a^{2} - 210 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{210 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 146, normalized size = 0.98 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, C b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, B b c x^{4} + \frac {1}{3} \, C b^{2} x^{3} + \frac {2}{3} \, C a c x^{3} + \frac {2}{3} \, A b c x^{3} + \frac {1}{2} \, B b^{2} x^{2} + B a c x^{2} + 2 \, C a b x + A b^{2} x + 2 \, A a c x + 2 \, B a b \log \left ({\left | x \right |}\right ) - \frac {3 \, B a^{2} x + 2 \, A a^{2} + 6 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 146, normalized size = 0.98 \[ \frac {C \,c^{2} x^{7}}{7}+\frac {B \,c^{2} x^{6}}{6}+\frac {A \,c^{2} x^{5}}{5}+\frac {2 C b c \,x^{5}}{5}+\frac {B b c \,x^{4}}{2}+\frac {2 A b c \,x^{3}}{3}+\frac {2 C a c \,x^{3}}{3}+\frac {C \,b^{2} x^{3}}{3}+B a c \,x^{2}+\frac {B \,b^{2} x^{2}}{2}+2 A a c x +A \,b^{2} x +2 B a b \ln \relax (x )+2 C a b x -\frac {2 A a b}{x}-\frac {C \,a^{2}}{x}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 140, normalized size = 0.94 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{2} \, B b c x^{4} + \frac {1}{5} \, {\left (2 \, C b c + A c^{2}\right )} x^{5} + \frac {1}{3} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{3} + 2 \, B a b \log \relax (x) + \frac {1}{2} \, {\left (B b^{2} + 2 \, B a c\right )} x^{2} + {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x - \frac {3 \, B a^{2} x + 2 \, A a^{2} + 6 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 137, normalized size = 0.92 \[ x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,C\,b\,c}{5}\right )-\frac {x^2\,\left (C\,a^2+2\,A\,b\,a\right )+\frac {A\,a^2}{3}+\frac {B\,a^2\,x}{2}}{x^3}+x\,\left (A\,b^2+2\,C\,a\,b+2\,A\,a\,c\right )+x^3\,\left (\frac {C\,b^2}{3}+\frac {2\,A\,c\,b}{3}+\frac {2\,C\,a\,c}{3}\right )+\frac {B\,c^2\,x^6}{6}+\frac {C\,c^2\,x^7}{7}+\frac {B\,x^2\,\left (b^2+2\,a\,c\right )}{2}+\frac {B\,b\,c\,x^4}{2}+2\,B\,a\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 160, normalized size = 1.07 \[ 2 B a b \log {\relax (x )} + \frac {B b c x^{4}}{2} + \frac {B c^{2} x^{6}}{6} + \frac {C c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 C b c}{5}\right ) + x^{3} \left (\frac {2 A b c}{3} + \frac {2 C a c}{3} + \frac {C b^{2}}{3}\right ) + x^{2} \left (B a c + \frac {B b^{2}}{2}\right ) + x \left (2 A a c + A b^{2} + 2 C a b\right ) + \frac {- 2 A a^{2} - 3 B a^{2} x + x^{2} \left (- 12 A a b - 6 C a^{2}\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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