Optimal. Leaf size=62 \[ \frac{c \log (x) (c d-b e)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}+\frac{c d-b e}{b^2 x}-\frac{d}{2 b x^2} \]
[Out]
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Rubi [A] time = 0.105663, antiderivative size = 62, 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 \[ \frac{c \log (x) (c d-b e)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}+\frac{c d-b e}{b^2 x}-\frac{d}{2 b x^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(x^2*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 14.7635, size = 53, normalized size = 0.85 \[ - \frac{d}{2 b x^{2}} - \frac{b e - c d}{b^{2} x} - \frac{c \left (b e - c d\right ) \log{\left (x \right )}}{b^{3}} + \frac{c \left (b e - c d\right ) \log{\left (b + c x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/x**2/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.0602944, size = 58, normalized size = 0.94 \[ \frac{-\frac{b (b d+2 b e x-2 c d x)}{x^2}+2 c \log (x) (c d-b e)+2 c (b e-c d) \log (b+c x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(x^2*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.013, size = 75, normalized size = 1.2 \[ -{\frac{d}{2\,b{x}^{2}}}-{\frac{e}{bx}}+{\frac{cd}{{b}^{2}x}}-{\frac{c\ln \left ( x \right ) e}{{b}^{2}}}+{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{3}}}+{\frac{c\ln \left ( cx+b \right ) e}{{b}^{2}}}-{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/x^2/(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.712533, size = 85, normalized size = 1.37 \[ -\frac{{\left (c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (c^{2} d - b c e\right )} \log \left (x\right )}{b^{3}} - \frac{b d - 2 \,{\left (c d - b e\right )} x}{2 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + b*x)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286297, size = 92, normalized size = 1.48 \[ -\frac{2 \,{\left (c^{2} d - b c e\right )} x^{2} \log \left (c x + b\right ) - 2 \,{\left (c^{2} d - b c e\right )} x^{2} \log \left (x\right ) + b^{2} d - 2 \,{\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + b*x)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.35881, size = 131, normalized size = 2.11 \[ - \frac{b d + x \left (2 b e - 2 c d\right )}{2 b^{2} x^{2}} - \frac{c \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c e - b c^{2} d - b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} + \frac{c \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c e - b c^{2} d + b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/x**2/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.269919, size = 105, normalized size = 1.69 \[ \frac{{\left (c^{2} d - b c e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (c^{3} d - b c^{2} e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac{b^{2} d - 2 \,{\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + b*x)*x^2),x, algorithm="giac")
[Out]