Optimal. Leaf size=30 \[ \frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c} \]
[Out]
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Rubi [A] time = 0.0681589, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 9.12636, size = 22, normalized size = 0.73 \[ \frac{d \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right ) \log{\left (b + c x \right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.0153534, size = 29, normalized size = 0.97 \[ \frac{(b e-c d) \log (b+c x)}{b c}+\frac{d \log (x)}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.009, size = 32, normalized size = 1.1 \[{\frac{d\ln \left ( x \right ) }{b}}+{\frac{\ln \left ( cx+b \right ) e}{c}}-{\frac{\ln \left ( cx+b \right ) d}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.699417, size = 41, normalized size = 1.37 \[ \frac{d \log \left (x\right )}{b} - \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221339, size = 39, normalized size = 1.3 \[ \frac{c d \log \left (x\right ) -{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.31604, size = 41, normalized size = 1.37 \[ \frac{d \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right ) \log{\left (x + \frac{- b d + \frac{b \left (b e - c d\right )}{c}}{b e - 2 c d} \right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.204343, size = 45, normalized size = 1.5 \[ \frac{d{\rm ln}\left ({\left | x \right |}\right )}{b} - \frac{{\left (c d - b e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x),x, algorithm="giac")
[Out]