Optimal. Leaf size=44 \[ -\frac {A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac {1}{2} B c^2 x^2 \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} -\frac {A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+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 (b x+c x^2\right )^2}{x^4} \, dx &=\int \left (c (2 b B+A c)+\frac {A b^2}{x^2}+\frac {b (b B+2 A c)}{x}+B c^2 x\right ) \, dx\\ &=-\frac {A b^2}{x}+c (2 b B+A c) x+\frac {1}{2} B c^2 x^2+b (b B+2 A c) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.98 \begin {gather*} A \left (c^2 x-\frac {b^2}{x}\right )+b \log (x) (2 A c+b B)+\frac {1}{2} B c x (4 b+c x) \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 (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 = 52, normalized size = 1.18 \begin {gather*} \frac {B c^{2} x^{3} - 2 \, A b^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x \log \relax (x)}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x - \frac {A b^{2}}{x} + {\left (B b^{2} + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 1.05 \begin {gather*} \frac {B \,c^{2} x^{2}}{2}+2 A b c \ln \relax (x )+A \,c^{2} x +B \,b^{2} \ln \relax (x )+2 B b c x -\frac {A \,b^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} - \frac {A b^{2}}{x} + {\left (2 \, B b c + A c^{2}\right )} x + {\left (B b^{2} + 2 \, A b c\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 46, normalized size = 1.05 \begin {gather*} x\,\left (A\,c^2+2\,B\,b\,c\right )+\ln \relax (x)\,\left (B\,b^2+2\,A\,c\,b\right )-\frac {A\,b^2}{x}+\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 = 0.19, size = 42, normalized size = 0.95 \begin {gather*} - \frac {A b^{2}}{x} + \frac {B c^{2} x^{2}}{2} + b \left (2 A c + B b\right ) \log {\relax (x )} + x \left (A c^{2} + 2 B b c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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