Optimal. Leaf size=65 \[ -\frac {A b^3}{2 x^2}-\frac {b^2 (3 A c+b B)}{x}+c^2 x (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac {1}{2} B c^3 x^2 \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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} \[ -\frac {b^2 (3 A c+b B)}{x}-\frac {A b^3}{2 x^2}+c^2 x (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac {1}{2} B c^3 x^2 \]
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 )^3}{x^6} \, dx &=\int \left (c^2 (3 b B+A c)+\frac {A b^3}{x^3}+\frac {b^2 (b B+3 A c)}{x^2}+\frac {3 b c (b B+A c)}{x}+B c^3 x\right ) \, dx\\ &=-\frac {A b^3}{2 x^2}-\frac {b^2 (b B+3 A c)}{x}+c^2 (3 b B+A c) x+\frac {1}{2} B c^3 x^2+3 b c (b B+A c) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 1.09 \[ -\frac {A b^3}{2 x^2}+3 \log (x) \left (A b c^2+b^2 B c\right )+\frac {b^3 (-B)-3 A b^2 c}{x}+c^2 x (A c+3 b B)+\frac {1}{2} B c^3 x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 74, normalized size = 1.14 \[ \frac {B c^{3} x^{4} - A b^{3} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} \log \relax (x) - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 69, normalized size = 1.06 \[ \frac {1}{2} \, B c^{3} x^{2} + 3 \, B b c^{2} x + A c^{3} x + 3 \, {\left (B b^{2} c + A b c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A b^{3} + 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 71, normalized size = 1.09 \[ \frac {B \,c^{3} x^{2}}{2}+3 A b \,c^{2} \ln \relax (x )+A \,c^{3} x +3 B \,b^{2} c \ln \relax (x )+3 B b \,c^{2} x -\frac {3 A \,b^{2} c}{x}-\frac {B \,b^{3}}{x}-\frac {A \,b^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 69, normalized size = 1.06 \[ \frac {1}{2} \, B c^{3} x^{2} + {\left (3 \, B b c^{2} + A c^{3}\right )} x + 3 \, {\left (B b^{2} c + A b c^{2}\right )} \log \relax (x) - \frac {A b^{3} + 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 70, normalized size = 1.08 \[ \ln \relax (x)\,\left (3\,B\,b^2\,c+3\,A\,b\,c^2\right )-\frac {x\,\left (B\,b^3+3\,A\,c\,b^2\right )+\frac {A\,b^3}{2}}{x^2}+x\,\left (A\,c^3+3\,B\,b\,c^2\right )+\frac {B\,c^3\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 68, normalized size = 1.05 \[ \frac {B c^{3} x^{2}}{2} + 3 b c \left (A c + B b\right ) \log {\relax (x )} + x \left (A c^{3} + 3 B b c^{2}\right ) + \frac {- A b^{3} + x \left (- 6 A b^{2} c - 2 B b^{3}\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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