3.264 \(\int \frac{1}{(d+e x) \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=53 \[ -\frac{c \log (b+c x)}{b (c d-b e)}+\frac{e \log (d+e x)}{d (c d-b e)}+\frac{\log (x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.113781, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{c \log (b+c x)}{b (c d-b e)}+\frac{e \log (d+e x)}{d (c d-b e)}+\frac{\log (x)}{b d} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(b*x + c*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.8043, size = 39, normalized size = 0.74 \[ - \frac{e \log{\left (d + e x \right )}}{d \left (b e - c d\right )} + \frac{c \log{\left (b + c x \right )}}{b \left (b e - c d\right )} + \frac{\log{\left (x \right )}}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.0388552, size = 48, normalized size = 0.91 \[ \frac{-c d \log (b+c x)+b e \log (d+e x)-b e \log (x)+c d \log (x)}{b c d^2-b^2 d e} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 54, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{bd}}+{\frac{c\ln \left ( cx+b \right ) }{ \left ( be-cd \right ) b}}-{\frac{e\ln \left ( ex+d \right ) }{d \left ( be-cd \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 0.6889, size = 72, normalized size = 1.36 \[ -\frac{c \log \left (c x + b\right )}{b c d - b^{2} e} + \frac{e \log \left (e x + d\right )}{c d^{2} - b d e} + \frac{\log \left (x\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.316618, size = 68, normalized size = 1.28 \[ -\frac{c d \log \left (c x + b\right ) - b e \log \left (e x + d\right ) -{\left (c d - b e\right )} \log \left (x\right )}{b c d^{2} - b^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 59.6379, size = 583, normalized size = 11. \[ - \frac{e \log{\left (x + \frac{- \frac{2 b^{6} e^{6}}{\left (b e - c d\right )^{2}} + \frac{6 b^{5} c d e^{5}}{\left (b e - c d\right )^{2}} - \frac{8 b^{4} c^{2} d^{2} e^{4}}{\left (b e - c d\right )^{2}} + \frac{3 b^{4} c d e^{4}}{b e - c d} + 2 b^{4} e^{4} + \frac{6 b^{3} c^{3} d^{3} e^{3}}{\left (b e - c d\right )^{2}} - \frac{6 b^{3} c^{2} d^{2} e^{3}}{b e - c d} - 3 b^{3} c d e^{3} - \frac{2 b^{2} c^{4} d^{4} e^{2}}{\left (b e - c d\right )^{2}} + \frac{3 b^{2} c^{3} d^{3} e^{2}}{b e - c d} + 2 b^{2} c^{2} d^{2} e^{2} - 3 b c^{3} d^{3} e + 2 c^{4} d^{4}}{2 b^{3} c e^{4} - 3 b^{2} c^{2} d e^{3} - 3 b c^{3} d^{2} e^{2} + 2 c^{4} d^{3} e} \right )}}{d \left (b e - c d\right )} + \frac{c \log{\left (x + \frac{- \frac{2 b^{4} c^{2} d^{2} e^{4}}{\left (b e - c d\right )^{2}} + 2 b^{4} e^{4} + \frac{6 b^{3} c^{3} d^{3} e^{3}}{\left (b e - c d\right )^{2}} - \frac{3 b^{3} c^{2} d^{2} e^{3}}{b e - c d} - 3 b^{3} c d e^{3} - \frac{8 b^{2} c^{4} d^{4} e^{2}}{\left (b e - c d\right )^{2}} + \frac{6 b^{2} c^{3} d^{3} e^{2}}{b e - c d} + 2 b^{2} c^{2} d^{2} e^{2} + \frac{6 b c^{5} d^{5} e}{\left (b e - c d\right )^{2}} - \frac{3 b c^{4} d^{4} e}{b e - c d} - 3 b c^{3} d^{3} e - \frac{2 c^{6} d^{6}}{\left (b e - c d\right )^{2}} + 2 c^{4} d^{4}}{2 b^{3} c e^{4} - 3 b^{2} c^{2} d e^{3} - 3 b c^{3} d^{2} e^{2} + 2 c^{4} d^{3} e} \right )}}{b \left (b e - c d\right )} + \frac{\log{\left (x \right )}}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x),x)

[Out]

-e*log(x + (-2*b**6*e**6/(b*e - c*d)**2 + 6*b**5*c*d*e**5/(b*e - c*d)**2 - 8*b**
4*c**2*d**2*e**4/(b*e - c*d)**2 + 3*b**4*c*d*e**4/(b*e - c*d) + 2*b**4*e**4 + 6*
b**3*c**3*d**3*e**3/(b*e - c*d)**2 - 6*b**3*c**2*d**2*e**3/(b*e - c*d) - 3*b**3*
c*d*e**3 - 2*b**2*c**4*d**4*e**2/(b*e - c*d)**2 + 3*b**2*c**3*d**3*e**2/(b*e - c
*d) + 2*b**2*c**2*d**2*e**2 - 3*b*c**3*d**3*e + 2*c**4*d**4)/(2*b**3*c*e**4 - 3*
b**2*c**2*d*e**3 - 3*b*c**3*d**2*e**2 + 2*c**4*d**3*e))/(d*(b*e - c*d)) + c*log(
x + (-2*b**4*c**2*d**2*e**4/(b*e - c*d)**2 + 2*b**4*e**4 + 6*b**3*c**3*d**3*e**3
/(b*e - c*d)**2 - 3*b**3*c**2*d**2*e**3/(b*e - c*d) - 3*b**3*c*d*e**3 - 8*b**2*c
**4*d**4*e**2/(b*e - c*d)**2 + 6*b**2*c**3*d**3*e**2/(b*e - c*d) + 2*b**2*c**2*d
**2*e**2 + 6*b*c**5*d**5*e/(b*e - c*d)**2 - 3*b*c**4*d**4*e/(b*e - c*d) - 3*b*c*
*3*d**3*e - 2*c**6*d**6/(b*e - c*d)**2 + 2*c**4*d**4)/(2*b**3*c*e**4 - 3*b**2*c*
*2*d*e**3 - 3*b*c**3*d**2*e**2 + 2*c**4*d**3*e))/(b*(b*e - c*d)) + log(x)/(b*d)

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GIAC/XCAS [A]  time = 0.220973, size = 169, normalized size = 3.19 \[ -\frac{{\left (c d + b e\right )}{\rm ln}\left (\frac{{\left | 2 \, c x e + c d + b e -{\left | c d - b e \right |} \right |}}{{\left | 2 \, c x e + c d + b e +{\left | c d - b e \right |} \right |}}\right )}{2 \, b d{\left | c d - b e \right |}} - \frac{{\rm ln}\left ({\left | c x^{2} e + c d x + b x e + b d \right |}\right )}{2 \, b d} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)),x, algorithm="giac")

[Out]

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