Optimal. Leaf size=43 \[ -\frac{\log (x) (c d-b e)}{b^2}+\frac{(c d-b e) \log (b+c x)}{b^2}-\frac{d}{b x} \]
[Out]
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Rubi [A] time = 0.0829217, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\log (x) (c d-b e)}{b^2}+\frac{(c d-b e) \log (b+c x)}{b^2}-\frac{d}{b x} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(x*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 10.9489, size = 34, normalized size = 0.79 \[ - \frac{d}{b x} + \frac{\left (b e - c d\right ) \log{\left (x \right )}}{b^{2}} - \frac{\left (b e - c d\right ) \log{\left (b + c x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/x/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.0290369, size = 42, normalized size = 0.98 \[ \frac{\log (x) (b e-c d)}{b^2}+\frac{(c d-b e) \log (b+c x)}{b^2}-\frac{d}{b x} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(x*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.012, size = 51, normalized size = 1.2 \[ -{\frac{d}{bx}}+{\frac{e\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( x \right ) cd}{{b}^{2}}}-{\frac{\ln \left ( cx+b \right ) e}{b}}+{\frac{\ln \left ( cx+b \right ) cd}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/x/(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.708108, size = 58, normalized size = 1.35 \[ \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{b^{2}} - \frac{{\left (c d - b e\right )} \log \left (x\right )}{b^{2}} - \frac{d}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + b*x)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276406, size = 55, normalized size = 1.28 \[ \frac{{\left (c d - b e\right )} x \log \left (c x + b\right ) -{\left (c d - b e\right )} x \log \left (x\right ) - b d}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + b*x)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.99914, size = 95, normalized size = 2.21 \[ - \frac{d}{b x} + \frac{\left (b e - c d\right ) \log{\left (x + \frac{b^{2} e - b c d - b \left (b e - c d\right )}{2 b c e - 2 c^{2} d} \right )}}{b^{2}} - \frac{\left (b e - c d\right ) \log{\left (x + \frac{b^{2} e - b c d + b \left (b e - c d\right )}{2 b c e - 2 c^{2} d} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/x/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.271629, size = 72, normalized size = 1.67 \[ -\frac{{\left (c d - b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} - \frac{d}{b x} + \frac{{\left (c^{2} d - b c e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + b*x)*x),x, algorithm="giac")
[Out]