Optimal. Leaf size=86 \[ -\frac {c^2 \log (x) (c d-b e)}{b^4}+\frac {c^2 (c d-b e) \log (b+c x)}{b^4}-\frac {c (c d-b e)}{b^3 x}+\frac {c d-b e}{2 b^2 x^2}-\frac {d}{3 b x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 86, 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 {c^2 \log (x) (c d-b e)}{b^4}+\frac {c^2 (c d-b e) \log (b+c x)}{b^4}+\frac {c d-b e}{2 b^2 x^2}-\frac {c (c d-b e)}{b^3 x}-\frac {d}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {d+e x}{x^3 \left (b x+c x^2\right )} \, dx &=\int \left (\frac {d}{b x^4}+\frac {-c d+b e}{b^2 x^3}-\frac {c (-c d+b e)}{b^3 x^2}+\frac {c^2 (-c d+b e)}{b^4 x}-\frac {c^3 (-c d+b e)}{b^4 (b+c x)}\right ) \, dx\\ &=-\frac {d}{3 b x^3}+\frac {c d-b e}{2 b^2 x^2}-\frac {c (c d-b e)}{b^3 x}-\frac {c^2 (c d-b e) \log (x)}{b^4}+\frac {c^2 (c d-b e) \log (b+c x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 0.94 \begin {gather*} \frac {\frac {b \left (-\left (b^2 (2 d+3 e x)\right )+3 b c x (d+2 e x)-6 c^2 d x^2\right )}{x^3}+6 c^2 \log (x) (b e-c d)+6 c^2 (c d-b e) \log (b+c x)}{6 b^4} \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 {d+e x}{x^3 \left (b x+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 94, normalized size = 1.09 \begin {gather*} \frac {6 \, {\left (c^{3} d - b c^{2} e\right )} x^{3} \log \left (c x + b\right ) - 6 \, {\left (c^{3} d - b c^{2} e\right )} x^{3} \log \relax (x) - 2 \, b^{3} d - 6 \, {\left (b c^{2} d - b^{2} c e\right )} x^{2} + 3 \, {\left (b^{2} c d - b^{3} e\right )} x}{6 \, b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 103, normalized size = 1.20 \begin {gather*} -\frac {{\left (c^{3} d - b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} + \frac {{\left (c^{4} d - b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac {2 \, b^{3} d + 6 \, {\left (b c^{2} d - b^{2} c e\right )} x^{2} - 3 \, {\left (b^{2} c d - b^{3} e\right )} x}{6 \, b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 101, normalized size = 1.17 \begin {gather*} \frac {c^{2} e \ln \relax (x )}{b^{3}}-\frac {c^{2} e \ln \left (c x +b \right )}{b^{3}}-\frac {c^{3} d \ln \relax (x )}{b^{4}}+\frac {c^{3} d \ln \left (c x +b \right )}{b^{4}}+\frac {c e}{b^{2} x}-\frac {c^{2} d}{b^{3} x}-\frac {e}{2 b \,x^{2}}+\frac {c d}{2 b^{2} x^{2}}-\frac {d}{3 b \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 89, normalized size = 1.03 \begin {gather*} \frac {{\left (c^{3} d - b c^{2} e\right )} \log \left (c x + b\right )}{b^{4}} - \frac {{\left (c^{3} d - b c^{2} e\right )} \log \relax (x)}{b^{4}} - \frac {2 \, b^{2} d + 6 \, {\left (c^{2} d - b c e\right )} x^{2} - 3 \, {\left (b c d - b^{2} e\right )} x}{6 \, b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 97, normalized size = 1.13 \begin {gather*} \frac {2\,c^2\,\mathrm {atanh}\left (\frac {c^2\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (c^3\,d-b\,c^2\,e\right )}\right )\,\left (b\,e-c\,d\right )}{b^4}-\frac {\frac {d}{3\,b}+\frac {x\,\left (b\,e-c\,d\right )}{2\,b^2}-\frac {c\,x^2\,\left (b\,e-c\,d\right )}{b^3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.50, size = 165, normalized size = 1.92 \begin {gather*} \frac {- 2 b^{2} d + x^{2} \left (6 b c e - 6 c^{2} d\right ) + x \left (- 3 b^{2} e + 3 b c d\right )}{6 b^{3} x^{3}} + \frac {c^{2} \left (b e - c d\right ) \log {\left (x + \frac {b^{2} c^{2} e - b c^{3} d - b c^{2} \left (b e - c d\right )}{2 b c^{3} e - 2 c^{4} d} \right )}}{b^{4}} - \frac {c^{2} \left (b e - c d\right ) \log {\left (x + \frac {b^{2} c^{2} e - b c^{3} d + b c^{2} \left (b e - c d\right )}{2 b c^{3} e - 2 c^{4} d} \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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