Optimal. Leaf size=65 \[ -\frac {\log (x) (2 c d-b e)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3}-\frac {c d-b e}{b^2 (b+c x)}-\frac {d}{b^2 x} \]
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Rubi [A] time = 0.05, antiderivative size = 65, 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} \[ -\frac {c d-b e}{b^2 (b+c x)}-\frac {\log (x) (2 c d-b e)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3}-\frac {d}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {d}{b^2 x^2}+\frac {-2 c d+b e}{b^3 x}-\frac {c (-c d+b e)}{b^2 (b+c x)^2}-\frac {c (-2 c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac {d}{b^2 x}-\frac {c d-b e}{b^2 (b+c x)}-\frac {(2 c d-b e) \log (x)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.86 \[ \frac {\frac {b (b e-c d)}{b+c x}+\log (x) (b e-2 c d)+(2 c d-b e) \log (b+c x)-\frac {b d}{x}}{b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 111, normalized size = 1.71 \[ -\frac {b^{2} d + {\left (2 \, b c d - b^{2} e\right )} x - {\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} + {\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} + {\left (2 \, b c d - b^{2} e\right )} x\right )} \log \relax (x)}{b^{3} c x^{2} + b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 1.18 \[ -\frac {{\left (2 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {2 \, c d x - b x e + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac {{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 1.20 \[ \frac {e}{\left (c x +b \right ) b}-\frac {c d}{\left (c x +b \right ) b^{2}}+\frac {e \ln \relax (x )}{b^{2}}-\frac {e \ln \left (c x +b \right )}{b^{2}}-\frac {2 c d \ln \relax (x )}{b^{3}}+\frac {2 c d \ln \left (c x +b \right )}{b^{3}}-\frac {d}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 69, normalized size = 1.06 \[ -\frac {b d + {\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac {{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac {{\left (2 \, c d - b e\right )} \log \relax (x)}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 57, normalized size = 0.88 \[ -\frac {\frac {d}{b}-\frac {x\,\left (b\,e-2\,c\,d\right )}{b^2}}{c\,x^2+b\,x}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b\,e-2\,c\,d\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.51, size = 128, normalized size = 1.97 \[ \frac {- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac {\left (b e - 2 c d\right ) \log {\left (x + \frac {b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac {\left (b e - 2 c d\right ) \log {\left (x + \frac {b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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