Optimal. Leaf size=90 \[ -\frac {a^2 A}{3 x^3}-\frac {A \left (2 a c+b^2\right )+2 a b B}{x}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac {a (a B+2 A b)}{2 x^2}+c x (A c+2 b B)+\frac {1}{2} B c^2 x^2 \]
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Rubi [A] time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} -\frac {a^2 A}{3 x^3}-\frac {A \left (2 a c+b^2\right )+2 a b B}{x}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac {a (a B+2 A b)}{2 x^2}+c x (A c+2 b B)+\frac {1}{2} B c^2 x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^4} \, dx &=\int \left (c (2 b B+A c)+\frac {a^2 A}{x^4}+\frac {a (2 A b+a B)}{x^3}+\frac {2 a b B+A \left (b^2+2 a c\right )}{x^2}+\frac {b^2 B+2 A b c+2 a B c}{x}+B c^2 x\right ) \, dx\\ &=-\frac {a^2 A}{3 x^3}-\frac {a (2 A b+a B)}{2 x^2}-\frac {2 a b B+A \left (b^2+2 a c\right )}{x}+c (2 b B+A c) x+\frac {1}{2} B c^2 x^2+\left (b^2 B+2 A b c+2 a B c\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 1.00 \begin {gather*} -\frac {a^2 (2 A+3 B x)}{6 x^3}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac {a (A b+2 A c x+2 b B x)}{x^2}-\frac {A b^2}{x}+A c^2 x+2 b B c x+\frac {1}{2} B c^2 x^2 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 95, normalized size = 1.06 \begin {gather*} \frac {3 \, B c^{2} x^{5} + 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} \log \relax (x) - 2 \, A a^{2} - 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 89, normalized size = 0.99 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x + {\left (B b^{2} + 2 \, B a c + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 95, normalized size = 1.06 \begin {gather*} \frac {B \,c^{2} x^{2}}{2}+2 A b c \ln \relax (x )+A \,c^{2} x +2 B a c \ln \relax (x )+B \,b^{2} \ln \relax (x )+2 B b c x -\frac {2 A a c}{x}-\frac {A \,b^{2}}{x}-\frac {2 B a b}{x}-\frac {A a b}{x^{2}}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 89, normalized size = 0.99 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} + {\left (2 \, B b c + A c^{2}\right )} x + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} \log \relax (x) - \frac {2 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 87, normalized size = 0.97 \begin {gather*} x\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {\frac {A\,a^2}{3}+x^2\,\left (A\,b^2+2\,B\,a\,b+2\,A\,a\,c\right )+x\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )}{x^3}+\ln \relax (x)\,\left (B\,b^2+2\,A\,c\,b+2\,B\,a\,c\right )+\frac {B\,c^2\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 99, normalized size = 1.10 \begin {gather*} \frac {B c^{2} x^{2}}{2} + x \left (A c^{2} + 2 B b c\right ) + \left (2 A b c + 2 B a c + B b^{2}\right ) \log {\relax (x )} + \frac {- 2 A a^{2} + x^{2} \left (- 12 A a c - 6 A b^{2} - 12 B a b\right ) + x \left (- 6 A a b - 3 B a^{2}\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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