Optimal. Leaf size=47 \[ \frac {\log (a e+c d x)}{c d^2-a e^2}-\frac {\log (d+e x)}{c d^2-a e^2} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {616, 31} \[ \frac {\log (a e+c d x)}{c d^2-a e^2}-\frac {\log (d+e x)}{c d^2-a e^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rubi steps
\begin {align*} \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=-\frac {(c d e) \int \frac {1}{c d^2+c d e x} \, dx}{c d^2-a e^2}+\frac {(c d e) \int \frac {1}{a e^2+c d e x} \, dx}{c d^2-a e^2}\\ &=\frac {\log (a e+c d x)}{c d^2-a e^2}-\frac {\log (d+e x)}{c d^2-a e^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.70 \[ \frac {\log (a e+c d x)-\log (d+e x)}{c d^2-a e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 33, normalized size = 0.70 \[ \frac {\log \left (c d x + a e\right ) - \log \left (e x + d\right )}{c d^{2} - a e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 75, normalized size = 1.60 \[ \frac {2 \, \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 1.02 \[ \frac {\ln \left (e x +d \right )}{a \,e^{2}-c \,d^{2}}-\frac {\ln \left (c d x +a e \right )}{a \,e^{2}-c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 47, normalized size = 1.00 \[ \frac {\log \left (c d x + a e\right )}{c d^{2} - a e^{2}} - \frac {\log \left (e x + d\right )}{c d^{2} - a e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 51, normalized size = 1.09 \[ \frac {\mathrm {atan}\left (\frac {2{}\mathrm {i}\,c\,d^2+2{}\mathrm {i}\,c\,e\,x\,d}{a\,e^2-c\,d^2}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,e^2-c\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.37, size = 172, normalized size = 3.66 \[ \frac {\log {\left (x + \frac {- \frac {a^{2} e^{4}}{a e^{2} - c d^{2}} + \frac {2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} - \frac {c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} - \frac {\log {\left (x + \frac {\frac {a^{2} e^{4}}{a e^{2} - c d^{2}} - \frac {2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} + \frac {c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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