Optimal. Leaf size=64 \[ -\frac {A b^3}{3 x^3}-\frac {b^2 (3 A c+b B)}{2 x^2}+c^2 \log (x) (A c+3 b B)-\frac {3 b c (A c+b B)}{x}+B c^3 x \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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} \[ -\frac {b^2 (3 A c+b B)}{2 x^2}-\frac {A b^3}{3 x^3}+c^2 \log (x) (A c+3 b B)-\frac {3 b c (A c+b B)}{x}+B c^3 x \]
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 )^3}{x^7} \, dx &=\int \left (B c^3+\frac {A b^3}{x^4}+\frac {b^2 (b B+3 A c)}{x^3}+\frac {3 b c (b B+A c)}{x^2}+\frac {c^2 (3 b B+A c)}{x}\right ) \, dx\\ &=-\frac {A b^3}{3 x^3}-\frac {b^2 (b B+3 A c)}{2 x^2}-\frac {3 b c (b B+A c)}{x}+B c^3 x+c^2 (3 b B+A c) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.14 \[ -\frac {A b^3}{3 x^3}-\frac {3 \left (A b c^2+b^2 B c\right )}{x}+\frac {b^3 (-B)-3 A b^2 c}{2 x^2}+\log (x) \left (A c^3+3 b B c^2\right )+B c^3 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 75, normalized size = 1.17 \[ \frac {6 \, B c^{3} x^{4} + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} \log \relax (x) - 2 \, A b^{3} - 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} - 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 70, normalized size = 1.09 \[ B c^{3} x + {\left (3 \, B b c^{2} + A c^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A b^{3} + 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 72, normalized size = 1.12 \[ A \,c^{3} \ln \relax (x )+3 B b \,c^{2} \ln \relax (x )+B \,c^{3} x -\frac {3 A b \,c^{2}}{x}-\frac {3 B \,b^{2} c}{x}-\frac {3 A \,b^{2} c}{2 x^{2}}-\frac {B \,b^{3}}{2 x^{2}}-\frac {A \,b^{3}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 69, normalized size = 1.08 \[ B c^{3} x + {\left (3 \, B b c^{2} + A c^{3}\right )} \log \relax (x) - \frac {2 \, A b^{3} + 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 70, normalized size = 1.09 \[ \ln \relax (x)\,\left (A\,c^3+3\,B\,b\,c^2\right )-\frac {x^2\,\left (3\,B\,b^2\,c+3\,A\,b\,c^2\right )+x\,\left (\frac {B\,b^3}{2}+\frac {3\,A\,c\,b^2}{2}\right )+\frac {A\,b^3}{3}}{x^3}+B\,c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 73, normalized size = 1.14 \[ B c^{3} x + c^{2} \left (A c + 3 B b\right ) \log {\relax (x )} + \frac {- 2 A b^{3} + x^{2} \left (- 18 A b c^{2} - 18 B b^{2} c\right ) + x \left (- 9 A b^{2} c - 3 B b^{3}\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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