Optimal. Leaf size=62 \[ \frac {c \log (x) (c d-b e)}{b^3}-\frac {c (c d-b e) \log (b+c x)}{b^3}+\frac {c d-b e}{b^2 x}-\frac {d}{2 b x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 62, 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 d-b e}{b^2 x}+\frac {c \log (x) (c d-b e)}{b^3}-\frac {c (c d-b e) \log (b+c x)}{b^3}-\frac {d}{2 b x^2} \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 )} \, dx &=\int \left (\frac {d}{b x^3}+\frac {-c d+b e}{b^2 x^2}-\frac {c (-c d+b e)}{b^3 x}+\frac {c^2 (-c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac {d}{2 b x^2}+\frac {c d-b e}{b^2 x}+\frac {c (c d-b e) \log (x)}{b^3}-\frac {c (c d-b e) \log (b+c x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.94 \begin {gather*} \frac {-\frac {b (b d+2 b e x-2 c d x)}{x^2}+2 c \log (x) (c d-b e)+2 c (b e-c d) \log (b+c x)}{2 b^3} \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 )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 68, normalized size = 1.10 \begin {gather*} -\frac {2 \, {\left (c^{2} d - b c e\right )} x^{2} \log \left (c x + b\right ) - 2 \, {\left (c^{2} d - b c e\right )} x^{2} \log \relax (x) + b^{2} d - 2 \, {\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 78, normalized size = 1.26 \begin {gather*} \frac {{\left (c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {{\left (c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac {b^{2} d - 2 \, {\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 75, normalized size = 1.21 \begin {gather*} -\frac {c e \ln \relax (x )}{b^{2}}+\frac {c e \ln \left (c x +b \right )}{b^{2}}+\frac {c^{2} d \ln \relax (x )}{b^{3}}-\frac {c^{2} d \ln \left (c x +b \right )}{b^{3}}-\frac {e}{b x}+\frac {c d}{b^{2} x}-\frac {d}{2 b \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 63, normalized size = 1.02 \begin {gather*} -\frac {{\left (c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{3}} + \frac {{\left (c^{2} d - b c e\right )} \log \relax (x)}{b^{3}} - \frac {b d - 2 \, {\left (c d - b e\right )} x}{2 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 73, normalized size = 1.18 \begin {gather*} -\frac {\frac {d}{2\,b}+\frac {x\,\left (b\,e-c\,d\right )}{b^2}}{x^2}-\frac {2\,c\,\mathrm {atanh}\left (\frac {c\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (c^2\,d-b\,c\,e\right )}\right )\,\left (b\,e-c\,d\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.45, size = 131, normalized size = 2.11 \begin {gather*} \frac {- b d + x \left (- 2 b e + 2 c d\right )}{2 b^{2} x^{2}} - \frac {c \left (b e - c d\right ) \log {\left (x + \frac {b^{2} c e - b c^{2} d - b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} + \frac {c \left (b e - c d\right ) \log {\left (x + \frac {b^{2} c e - b c^{2} d + b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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