Optimal. Leaf size=85 \[ \frac {c \log (x) (3 c d-2 b e)}{b^4}-\frac {c (3 c d-2 b e) \log (b+c x)}{b^4}+\frac {2 c d-b e}{b^3 x}+\frac {c (c d-b e)}{b^3 (b+c x)}-\frac {d}{2 b^2 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 85, 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 {2 c d-b e}{b^3 x}+\frac {c (c d-b e)}{b^3 (b+c x)}+\frac {c \log (x) (3 c d-2 b e)}{b^4}-\frac {c (3 c d-2 b e) \log (b+c x)}{b^4}-\frac {d}{2 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {d}{b^2 x^3}+\frac {-2 c d+b e}{b^3 x^2}-\frac {c (-3 c d+2 b e)}{b^4 x}+\frac {c^2 (-c d+b e)}{b^3 (b+c x)^2}+\frac {c^2 (-3 c d+2 b e)}{b^4 (b+c x)}\right ) \, dx\\ &=-\frac {d}{2 b^2 x^2}+\frac {2 c d-b e}{b^3 x}+\frac {c (c d-b e)}{b^3 (b+c x)}+\frac {c (3 c d-2 b e) \log (x)}{b^4}-\frac {c (3 c d-2 b e) \log (b+c x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 85, normalized size = 1.00 \begin {gather*} \frac {-\frac {b \left (b^2 (d+2 e x)+b c x (4 e x-3 d)-6 c^2 d x^2\right )}{x^2 (b+c x)}+2 c \log (x) (3 c d-2 b e)+2 c (2 b e-3 c d) \log (b+c x)}{2 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 \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 = 0.40, size = 151, normalized size = 1.78 \begin {gather*} -\frac {b^{3} d - 2 \, {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} - {\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x + 2 \, {\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} + {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} + {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{4} c x^{3} + b^{5} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 111, normalized size = 1.31 \begin {gather*} \frac {{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {{\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac {b^{3} d - 2 \, {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} - {\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x}{2 \, {\left (c x + b\right )} b^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 107, normalized size = 1.26 \begin {gather*} -\frac {c e}{\left (c x +b \right ) b^{2}}+\frac {c^{2} d}{\left (c x +b \right ) b^{3}}-\frac {2 c e \ln \relax (x )}{b^{3}}+\frac {2 c e \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d \ln \relax (x )}{b^{4}}-\frac {3 c^{2} d \ln \left (c x +b \right )}{b^{4}}-\frac {e}{b^{2} x}+\frac {2 c d}{b^{3} x}-\frac {d}{2 b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 100, normalized size = 1.18 \begin {gather*} -\frac {b^{2} d - 2 \, {\left (3 \, c^{2} d - 2 \, b c e\right )} x^{2} - {\left (3 \, b c d - 2 \, b^{2} e\right )} x}{2 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}} - \frac {{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (c x + b\right )}{b^{4}} + \frac {{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \relax (x)}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 105, normalized size = 1.24 \begin {gather*} -\frac {\frac {d}{2\,b}+\frac {x\,\left (2\,b\,e-3\,c\,d\right )}{2\,b^2}+\frac {c\,x^2\,\left (2\,b\,e-3\,c\,d\right )}{b^3}}{c\,x^3+b\,x^2}-\frac {2\,c\,\mathrm {atanh}\left (\frac {c\,\left (2\,b\,e-3\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (3\,c^2\,d-2\,b\,c\,e\right )}\right )\,\left (2\,b\,e-3\,c\,d\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.59, size = 184, normalized size = 2.16 \begin {gather*} \frac {- b^{2} d + x^{2} \left (- 4 b c e + 6 c^{2} d\right ) + x \left (- 2 b^{2} e + 3 b c d\right )}{2 b^{4} x^{2} + 2 b^{3} c x^{3}} - \frac {c \left (2 b e - 3 c d\right ) \log {\left (x + \frac {2 b^{2} c e - 3 b c^{2} d - b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} + \frac {c \left (2 b e - 3 c d\right ) \log {\left (x + \frac {2 b^{2} c e - 3 b c^{2} d + b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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