Optimal. Leaf size=113 \[ -\frac {c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac {c^2 (4 c d-3 b e) \log (b+c x)}{b^5}-\frac {c^2 (c d-b e)}{b^4 (b+c x)}-\frac {c (3 c d-2 b e)}{b^4 x}+\frac {2 c d-b e}{2 b^3 x^2}-\frac {d}{3 b^2 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 113, 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 (c d-b e)}{b^4 (b+c x)}-\frac {c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac {c^2 (4 c d-3 b e) \log (b+c x)}{b^5}+\frac {2 c d-b e}{2 b^3 x^2}-\frac {c (3 c d-2 b e)}{b^4 x}-\frac {d}{3 b^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {d}{b^2 x^4}+\frac {-2 c d+b e}{b^3 x^3}-\frac {c (-3 c d+2 b e)}{b^4 x^2}+\frac {c^2 (-4 c d+3 b e)}{b^5 x}-\frac {c^3 (-c d+b e)}{b^4 (b+c x)^2}-\frac {c^3 (-4 c d+3 b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {d}{3 b^2 x^3}+\frac {2 c d-b e}{2 b^3 x^2}-\frac {c (3 c d-2 b e)}{b^4 x}-\frac {c^2 (c d-b e)}{b^4 (b+c x)}-\frac {c^2 (4 c d-3 b e) \log (x)}{b^5}+\frac {c^2 (4 c d-3 b e) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 106, normalized size = 0.94 \begin {gather*} \frac {-\frac {2 b^3 d}{x^3}-\frac {3 b^2 (b e-2 c d)}{x^2}+\frac {6 b c^2 (b e-c d)}{b+c x}+6 c^2 \log (x) (3 b e-4 c d)+6 c^2 (4 c d-3 b e) \log (b+c x)+\frac {6 b c (2 b e-3 c d)}{x}}{6 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 {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 180, normalized size = 1.59 \begin {gather*} -\frac {2 \, b^{4} d + 6 \, {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \, {\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} - {\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x - 6 \, {\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} + {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} + {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (b^{5} c x^{4} + b^{6} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 139, normalized size = 1.23 \begin {gather*} -\frac {{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {2 \, b^{4} d + 6 \, {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \, {\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} - {\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x}{6 \, {\left (c x + b\right )} b^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 134, normalized size = 1.19 \begin {gather*} \frac {c^{2} e}{\left (c x +b \right ) b^{3}}-\frac {c^{3} d}{\left (c x +b \right ) b^{4}}+\frac {3 c^{2} e \ln \relax (x )}{b^{4}}-\frac {3 c^{2} e \ln \left (c x +b \right )}{b^{4}}-\frac {4 c^{3} d \ln \relax (x )}{b^{5}}+\frac {4 c^{3} d \ln \left (c x +b \right )}{b^{5}}+\frac {2 c e}{b^{3} x}-\frac {3 c^{2} d}{b^{4} x}-\frac {e}{2 b^{2} x^{2}}+\frac {c d}{b^{3} x^{2}}-\frac {d}{3 b^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 129, normalized size = 1.14 \begin {gather*} -\frac {2 \, b^{3} d + 6 \, {\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} x^{3} + 3 \, {\left (4 \, b c^{2} d - 3 \, b^{2} c e\right )} x^{2} - {\left (4 \, b^{2} c d - 3 \, b^{3} e\right )} x}{6 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}} + \frac {{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{5}} - \frac {{\left (4 \, c^{3} d - 3 \, b c^{2} e\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 = 1.08, size = 132, normalized size = 1.17 \begin {gather*} \frac {2\,c^2\,\mathrm {atanh}\left (\frac {c^2\,\left (3\,b\,e-4\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (4\,c^3\,d-3\,b\,c^2\,e\right )}\right )\,\left (3\,b\,e-4\,c\,d\right )}{b^5}-\frac {\frac {d}{3\,b}+\frac {x\,\left (3\,b\,e-4\,c\,d\right )}{6\,b^2}-\frac {c\,x^2\,\left (3\,b\,e-4\,c\,d\right )}{2\,b^3}-\frac {c^2\,x^3\,\left (3\,b\,e-4\,c\,d\right )}{b^4}}{c\,x^4+b\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.66, size = 219, normalized size = 1.94 \begin {gather*} \frac {- 2 b^{3} d + x^{3} \left (18 b c^{2} e - 24 c^{3} d\right ) + x^{2} \left (9 b^{2} c e - 12 b c^{2} d\right ) + x \left (- 3 b^{3} e + 4 b^{2} c d\right )}{6 b^{5} x^{3} + 6 b^{4} c x^{4}} + \frac {c^{2} \left (3 b e - 4 c d\right ) \log {\left (x + \frac {3 b^{2} c^{2} e - 4 b c^{3} d - b c^{2} \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} - \frac {c^{2} \left (3 b e - 4 c d\right ) \log {\left (x + \frac {3 b^{2} c^{2} e - 4 b c^{3} d + b c^{2} \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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