Optimal. Leaf size=73 \[ -\frac {2 d \log (x) (c d-b e)}{b^3}+\frac {2 d (c d-b e) \log (b+c x)}{b^3}-\frac {(c d-b e)^2}{b^2 c (b+c x)}-\frac {d^2}{b^2 x} \]
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Rubi [A] time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {(c d-b e)^2}{b^2 c (b+c x)}-\frac {2 d \log (x) (c d-b e)}{b^3}+\frac {2 d (c d-b e) \log (b+c x)}{b^3}-\frac {d^2}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {d^2}{b^2 x^2}+\frac {2 d (-c d+b e)}{b^3 x}+\frac {(-c d+b e)^2}{b^2 (b+c x)^2}-\frac {2 c d (-c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac {d^2}{b^2 x}-\frac {(c d-b e)^2}{b^2 c (b+c x)}-\frac {2 d (c d-b e) \log (x)}{b^3}+\frac {2 d (c d-b e) \log (b+c x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 67, normalized size = 0.92 \[ \frac {-\frac {b (c d-b e)^2}{c (b+c x)}+2 d \log (x) (b e-c d)+2 d (c d-b e) \log (b+c x)-\frac {b d^2}{x}}{b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 149, normalized size = 2.04 \[ -\frac {b^{2} c d^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x - 2 \, {\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \relax (x)}{b^{3} c^{2} x^{2} + b^{4} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 101, normalized size = 1.38 \[ -\frac {2 \, {\left (c d^{2} - b d e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {2 \, {\left (c^{2} d^{2} - b c d e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac {2 \, c^{2} d^{2} x - 2 \, b c d x e + b c d^{2} + b^{2} x e^{2}}{{\left (c x^{2} + b x\right )} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 106, normalized size = 1.45 \[ \frac {2 d e}{\left (c x +b \right ) b}-\frac {c \,d^{2}}{\left (c x +b \right ) b^{2}}+\frac {2 d e \ln \relax (x )}{b^{2}}-\frac {2 d e \ln \left (c x +b \right )}{b^{2}}-\frac {2 c \,d^{2} \ln \relax (x )}{b^{3}}+\frac {2 c \,d^{2} \ln \left (c x +b \right )}{b^{3}}-\frac {e^{2}}{\left (c x +b \right ) c}-\frac {d^{2}}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 93, normalized size = 1.27 \[ -\frac {b c d^{2} + {\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} + \frac {2 \, {\left (c d^{2} - b d e\right )} \log \left (c x + b\right )}{b^{3}} - \frac {2 \, {\left (c d^{2} - b d e\right )} \log \relax (x)}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 101, normalized size = 1.38 \[ \frac {4\,d\,\mathrm {atanh}\left (\frac {2\,d\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (2\,c\,d^2-2\,b\,d\,e\right )}\right )\,\left (b\,e-c\,d\right )}{b^3}-\frac {\frac {d^2}{b}+\frac {x\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,c}}{c\,x^2+b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.75, size = 173, normalized size = 2.37 \[ \frac {- b c d^{2} + x \left (- b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac {2 d \left (b e - c d\right ) \log {\left (x + \frac {2 b^{2} d e - 2 b c d^{2} - 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} - \frac {2 d \left (b e - c d\right ) \log {\left (x + \frac {2 b^{2} d e - 2 b c d^{2} + 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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