Optimal. Leaf size=149 \[ -\frac {a^2 A}{6 x^6}-\frac {a^2 B}{5 x^5}-\frac {a (2 A b+a C)}{4 x^4}-\frac {2 a b B}{3 x^3}-\frac {A \left (b^2+2 a c\right )+2 a b C}{2 x^2}-\frac {B \left (b^2+2 a c\right )}{x}+2 b B c x+\frac {1}{2} c (A c+2 b C) x^2+\frac {1}{3} B c^2 x^3+\frac {1}{4} c^2 C x^4+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) \log (x) \]
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Rubi [A]
time = 0.09, 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 = {1642}
\begin {gather*} -\frac {a^2 A}{6 x^6}-\frac {a^2 B}{5 x^5}-\frac {A \left (2 a c+b^2\right )+2 a b C}{2 x^2}+\log (x) \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac {a (a C+2 A b)}{4 x^4}-\frac {B \left (2 a c+b^2\right )}{x}-\frac {2 a b B}{3 x^3}+\frac {1}{2} c x^2 (A c+2 b C)+2 b B c x+\frac {1}{3} B c^2 x^3+\frac {1}{4} c^2 C x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
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^7} \, dx &=\int \left (2 b B c+\frac {a^2 A}{x^7}+\frac {a^2 B}{x^6}+\frac {a (2 A b+a C)}{x^5}+\frac {2 a b B}{x^4}+\frac {A \left (b^2+2 a c\right )+2 a b C}{x^3}+\frac {B \left (b^2+2 a c\right )}{x^2}+\frac {2 A b c+\left (b^2+2 a c\right ) C}{x}+c (A c+2 b C) x+B c^2 x^2+c^2 C x^3\right ) \, dx\\ &=-\frac {a^2 A}{6 x^6}-\frac {a^2 B}{5 x^5}-\frac {a (2 A b+a C)}{4 x^4}-\frac {2 a b B}{3 x^3}-\frac {A \left (b^2+2 a c\right )+2 a b C}{2 x^2}-\frac {B \left (b^2+2 a c\right )}{x}+2 b B c x+\frac {1}{2} c (A c+2 b C) x^2+\frac {1}{3} B c^2 x^3+\frac {1}{4} c^2 C x^4+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 144, normalized size = 0.97 \begin {gather*} -\frac {b^2 B}{x}+b c x (2 B+C x)+\frac {1}{12} c^2 x^3 (4 B+3 C x)+\frac {A \left (-b^2+c^2 x^4\right )}{2 x^2}-\frac {a^2 (10 A+3 x (4 B+5 C x))}{60 x^6}-\frac {a \left (3 A \left (b+2 c x^2\right )+2 x \left (2 b B+3 b C x+6 B c x^2\right )\right )}{6 x^4}+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 136, normalized size = 0.91
method | result | size |
default | \(\frac {c^{2} C \,x^{4}}{4}+\frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+C b c \,x^{2}+2 b B c x -\frac {a^{2} B}{5 x^{5}}-\frac {a \left (2 A b +a C \right )}{4 x^{4}}-\frac {2 a c A +A \,b^{2}+2 a b C}{2 x^{2}}-\frac {B \left (2 a c +b^{2}\right )}{x}-\frac {2 a b B}{3 x^{3}}+\left (2 b c A +2 a c C +C \,b^{2}\right ) \ln \left (x \right )-\frac {a^{2} A}{6 x^{6}}\) | \(136\) |
norman | \(\frac {\left (\frac {1}{2} A \,c^{2}+b c C \right ) x^{8}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} C \right ) x^{2}+\left (-a c A -\frac {1}{2} A \,b^{2}-a b C \right ) x^{4}+\left (-2 a c B -b^{2} B \right ) x^{5}-\frac {a^{2} A}{6}+\frac {B \,c^{2} x^{9}}{3}-\frac {a^{2} B x}{5}+\frac {c^{2} C \,x^{10}}{4}-\frac {2 a b B \,x^{3}}{3}+2 b B c \,x^{7}}{x^{6}}+\left (2 b c A +2 a c C +C \,b^{2}\right ) \ln \left (x \right )\) | \(141\) |
risch | \(\frac {c^{2} C \,x^{4}}{4}+\frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+C b c \,x^{2}+2 b B c x +\frac {\left (-2 a c B -b^{2} B \right ) x^{5}+\left (-a c A -\frac {1}{2} A \,b^{2}-a b C \right ) x^{4}-\frac {2 a b B \,x^{3}}{3}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} C \right ) x^{2}-\frac {a^{2} B x}{5}-\frac {a^{2} A}{6}}{x^{6}}+2 A \ln \left (x \right ) b c +2 C \ln \left (x \right ) a c +C \ln \left (x \right ) b^{2}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 140, normalized size = 0.94 \begin {gather*} \frac {1}{4} \, C c^{2} x^{4} + \frac {1}{3} \, B c^{2} x^{3} + 2 \, B b c x + \frac {1}{2} \, {\left (2 \, C b c + A c^{2}\right )} x^{2} + {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} \log \left (x\right ) - \frac {40 \, B a b x^{3} + 60 \, {\left (B b^{2} + 2 \, B a c\right )} x^{5} + 30 \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 12 \, B a^{2} x + 10 \, A a^{2} + 15 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 145, normalized size = 0.97 \begin {gather*} \frac {15 \, C c^{2} x^{10} + 20 \, B c^{2} x^{9} + 120 \, B b c x^{7} + 30 \, {\left (2 \, C b c + A c^{2}\right )} x^{8} + 60 \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{6} \log \left (x\right ) - 40 \, B a b x^{3} - 60 \, {\left (B b^{2} + 2 \, B a c\right )} x^{5} - 30 \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 12 \, B a^{2} x - 10 \, A a^{2} - 15 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 29.62, size = 158, normalized size = 1.06 \begin {gather*} 2 B b c x + \frac {B c^{2} x^{3}}{3} + \frac {C c^{2} x^{4}}{4} + x^{2} \left (\frac {A c^{2}}{2} + C b c\right ) + \left (2 A b c + 2 C a c + C b^{2}\right ) \log {\left (x \right )} + \frac {- 10 A a^{2} - 12 B a^{2} x - 40 B a b x^{3} + x^{5} \left (- 120 B a c - 60 B b^{2}\right ) + x^{4} \left (- 60 A a c - 30 A b^{2} - 60 C a b\right ) + x^{2} \left (- 30 A a b - 15 C a^{2}\right )}{60 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.53, size = 141, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, C c^{2} x^{4} + \frac {1}{3} \, B c^{2} x^{3} + C b c x^{2} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B b c x + {\left (C b^{2} + 2 \, C a c + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) - \frac {40 \, B a b x^{3} + 60 \, {\left (B b^{2} + 2 \, B a c\right )} x^{5} + 30 \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 12 \, B a^{2} x + 10 \, A a^{2} + 15 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 136, normalized size = 0.91 \begin {gather*} x^2\,\left (\frac {A\,c^2}{2}+C\,b\,c\right )-\frac {x^2\,\left (\frac {C\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^5\,\left (B\,b^2+2\,B\,a\,c\right )+\frac {A\,a^2}{6}+x^4\,\left (\frac {A\,b^2}{2}+C\,a\,b+A\,a\,c\right )+\frac {B\,a^2\,x}{5}+\frac {2\,B\,a\,b\,x^3}{3}}{x^6}+\ln \left (x\right )\,\left (C\,b^2+2\,A\,c\,b+2\,C\,a\,c\right )+\frac {B\,c^2\,x^3}{3}+\frac {C\,c^2\,x^4}{4}+2\,B\,b\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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