Optimal. Leaf size=49 \[ -\frac {A b^2}{3 x^3}-\frac {b (2 A c+b B)}{2 x^2}-\frac {c (A c+2 b B)}{x}+B c^2 \log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 49, 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} \[ -\frac {A b^2}{3 x^3}-\frac {b (2 A c+b B)}{2 x^2}-\frac {c (A c+2 b B)}{x}+B c^2 \log (x) \]
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^6} \, dx &=\int \left (\frac {A b^2}{x^4}+\frac {b (b B+2 A c)}{x^3}+\frac {c (2 b B+A c)}{x^2}+\frac {B c^2}{x}\right ) \, dx\\ &=-\frac {A b^2}{3 x^3}-\frac {b (b B+2 A c)}{2 x^2}-\frac {c (2 b B+A c)}{x}+B c^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 0.96 \[ B c^2 \log (x)-\frac {2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 53, normalized size = 1.08 \[ \frac {6 \, B c^{2} x^{3} \log \relax (x) - 2 \, A b^{2} - 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \, {\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 51, normalized size = 1.04 \[ B c^{2} \log \left ({\left | x \right |}\right ) - \frac {2 \, A b^{2} + 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 1.06 \[ B \,c^{2} \ln \relax (x )-\frac {A \,c^{2}}{x}-\frac {2 B b c}{x}-\frac {A b c}{x^{2}}-\frac {B \,b^{2}}{2 x^{2}}-\frac {A \,b^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 50, normalized size = 1.02 \[ B c^{2} \log \relax (x) - \frac {2 \, A b^{2} + 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 48, normalized size = 0.98 \[ B\,c^2\,\ln \relax (x)-\frac {x^2\,\left (A\,c^2+2\,B\,b\,c\right )+\frac {A\,b^2}{3}+x\,\left (\frac {B\,b^2}{2}+A\,c\,b\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 54, normalized size = 1.10 \[ B c^{2} \log {\relax (x )} + \frac {- 2 A b^{2} + x^{2} \left (- 6 A c^{2} - 12 B b c\right ) + x \left (- 6 A b c - 3 B b^{2}\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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