3.48 \(\int \frac{d+e x}{x \left (b x+c x^2\right )} \, dx\)

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]

-(d/(b*x)) - ((c*d - b*e)*Log[x])/b^2 + ((c*d - b*e)*Log[b + c*x])/b^2

_______________________________________________________________________________________

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]

-(d/(b*x)) - ((c*d - b*e)*Log[x])/b^2 + ((c*d - b*e)*Log[b + c*x])/b^2

_______________________________________________________________________________________

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]

-d/(b*x) + (b*e - c*d)*log(x)/b**2 - (b*e - c*d)*log(b + c*x)/b**2

_______________________________________________________________________________________

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]

-(d/(b*x)) + ((-(c*d) + b*e)*Log[x])/b^2 + ((c*d - b*e)*Log[b + c*x])/b^2

_______________________________________________________________________________________

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]

-d/b/x+1/b*ln(x)*e-1/b^2*ln(x)*c*d-1/b*ln(c*x+b)*e+1/b^2*ln(c*x+b)*c*d

_______________________________________________________________________________________

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]

(c*d - b*e)*log(c*x + b)/b^2 - (c*d - b*e)*log(x)/b^2 - d/(b*x)

_______________________________________________________________________________________

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]

((c*d - b*e)*x*log(c*x + b) - (c*d - b*e)*x*log(x) - b*d)/(b^2*x)

_______________________________________________________________________________________

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]

-d/(b*x) + (b*e - c*d)*log(x + (b**2*e - b*c*d - b*(b*e - c*d))/(2*b*c*e - 2*c**
2*d))/b**2 - (b*e - c*d)*log(x + (b**2*e - b*c*d + b*(b*e - c*d))/(2*b*c*e - 2*c
**2*d))/b**2

_______________________________________________________________________________________

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]

-(c*d - b*e)*ln(abs(x))/b^2 - d/(b*x) + (c^2*d - b*c*e)*ln(abs(c*x + b))/(b^2*c)