Optimal. Leaf size=109 \[ -\frac {3 c \log (x) (b B-2 A c)}{b^5}+\frac {3 c (b B-2 A c) \log (b+c x)}{b^5}-\frac {b B-3 A c}{b^4 x}-\frac {c (2 b B-3 A c)}{b^4 (b+c x)}-\frac {c (b B-A c)}{2 b^3 (b+c x)^2}-\frac {A}{2 b^3 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \begin {gather*} -\frac {b B-3 A c}{b^4 x}-\frac {c (2 b B-3 A c)}{b^4 (b+c x)}-\frac {c (b B-A c)}{2 b^3 (b+c x)^2}-\frac {3 c \log (x) (b B-2 A c)}{b^5}+\frac {3 c (b B-2 A c) \log (b+c x)}{b^5}-\frac {A}{2 b^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {A}{b^3 x^3}+\frac {b B-3 A c}{b^4 x^2}-\frac {3 c (b B-2 A c)}{b^5 x}+\frac {c^2 (b B-A c)}{b^3 (b+c x)^3}+\frac {c^2 (2 b B-3 A c)}{b^4 (b+c x)^2}+\frac {3 c^2 (b B-2 A c)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {A}{2 b^3 x^2}-\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c)}{2 b^3 (b+c x)^2}-\frac {c (2 b B-3 A c)}{b^4 (b+c x)}-\frac {3 c (b B-2 A c) \log (x)}{b^5}+\frac {3 c (b B-2 A c) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 106, normalized size = 0.97 \begin {gather*} \frac {-\frac {b \left (A \left (b^3-4 b^2 c x-18 b c^2 x^2-12 c^3 x^3\right )+b B x \left (2 b^2+9 b c x+6 c^2 x^2\right )\right )}{x^2 (b+c x)^2}+6 c \log (x) (2 A c-b B)+6 c (b B-2 A c) \log (b+c x)}{2 b^5} \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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 225, normalized size = 2.06 \begin {gather*} -\frac {A b^{4} + 6 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + 9 \, {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 2 \, {\left (B b^{4} - 2 \, A b^{3} c\right )} x - 6 \, {\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{4} + 2 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{4} + 2 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 124, normalized size = 1.14 \begin {gather*} -\frac {3 \, {\left (B b c - 2 \, A c^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {3 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{2} x^{3} - 12 \, A c^{3} x^{3} + 9 \, B b^{2} c x^{2} - 18 \, A b c^{2} x^{2} + 2 \, B b^{3} x - 4 \, A b^{2} c x + A b^{3}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 138, normalized size = 1.27 \begin {gather*} \frac {A \,c^{2}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {B c}{2 \left (c x +b \right )^{2} b^{2}}+\frac {3 A \,c^{2}}{\left (c x +b \right ) b^{4}}+\frac {6 A \,c^{2} \ln \relax (x )}{b^{5}}-\frac {6 A \,c^{2} \ln \left (c x +b \right )}{b^{5}}-\frac {2 B c}{\left (c x +b \right ) b^{3}}-\frac {3 B c \ln \relax (x )}{b^{4}}+\frac {3 B c \ln \left (c x +b \right )}{b^{4}}+\frac {3 A c}{b^{4} x}-\frac {B}{b^{3} x}-\frac {A}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 131, normalized size = 1.20 \begin {gather*} -\frac {A b^{3} + 6 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 9 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 2 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} + \frac {3 \, {\left (B b c - 2 \, A c^{2}\right )} \log \left (c x + b\right )}{b^{5}} - \frac {3 \, {\left (B b c - 2 \, A c^{2}\right )} \log \relax (x)}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 136, normalized size = 1.25 \begin {gather*} \frac {\frac {x\,\left (2\,A\,c-B\,b\right )}{b^2}-\frac {A}{2\,b}+\frac {3\,c^2\,x^3\,\left (2\,A\,c-B\,b\right )}{b^4}+\frac {9\,c\,x^2\,\left (2\,A\,c-B\,b\right )}{2\,b^3}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,\left (2\,A\,c-B\,b\right )\,\left (b+2\,c\,x\right )}{b\,\left (6\,A\,c^2-3\,B\,b\,c\right )}\right )\,\left (2\,A\,c-B\,b\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.72, size = 219, normalized size = 2.01 \begin {gather*} \frac {- A b^{3} + x^{3} \left (12 A c^{3} - 6 B b c^{2}\right ) + x^{2} \left (18 A b c^{2} - 9 B b^{2} c\right ) + x \left (4 A b^{2} c - 2 B b^{3}\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac {3 c \left (- 2 A c + B b\right ) \log {\left (x + \frac {- 6 A b c^{2} + 3 B b^{2} c - 3 b c \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{b^{5}} + \frac {3 c \left (- 2 A c + B b\right ) \log {\left (x + \frac {- 6 A b c^{2} + 3 B b^{2} c + 3 b c \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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