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

Optimal. Leaf size=65 \[ -\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{c d-b e}{b^2 (b+c x)}-\frac{d}{b^2 x} \]

[Out]

-(d/(b^2*x)) - (c*d - b*e)/(b^2*(b + c*x)) - ((2*c*d - b*e)*Log[x])/b^3 + ((2*c*
d - b*e)*Log[b + c*x])/b^3

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Rubi [A]  time = 0.116887, antiderivative size = 65, 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 \[ -\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{c d-b e}{b^2 (b+c x)}-\frac{d}{b^2 x} \]

Antiderivative was successfully verified.

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

[Out]

-(d/(b^2*x)) - (c*d - b*e)/(b^2*(b + c*x)) - ((2*c*d - b*e)*Log[x])/b^3 + ((2*c*
d - b*e)*Log[b + c*x])/b^3

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Rubi in Sympy [A]  time = 15.8927, size = 54, normalized size = 0.83 \[ - \frac{d}{b^{2} x} + \frac{b e - c d}{b^{2} \left (b + c x\right )} + \frac{\left (b e - 2 c d\right ) \log{\left (x \right )}}{b^{3}} - \frac{\left (b e - 2 c d\right ) \log{\left (b + c x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(b**2*x) + (b*e - c*d)/(b**2*(b + c*x)) + (b*e - 2*c*d)*log(x)/b**3 - (b*e -
2*c*d)*log(b + c*x)/b**3

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Mathematica [A]  time = 0.0746549, size = 56, normalized size = 0.86 \[ \frac{\frac{b (b e-c d)}{b+c x}+\log (x) (b e-2 c d)+(2 c d-b e) \log (b+c x)-\frac{b d}{x}}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(-((b*d)/x) + (b*(-(c*d) + b*e))/(b + c*x) + (-2*c*d + b*e)*Log[x] + (2*c*d - b*
e)*Log[b + c*x])/b^3

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Maple [A]  time = 0.001, size = 78, normalized size = 1.2 \[ -{\frac{d}{{b}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) cd}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) e}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) cd}{{b}^{3}}}+{\frac{e}{b \left ( cx+b \right ) }}-{\frac{cd}{{b}^{2} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+b*x)^2,x)

[Out]

-d/b^2/x+1/b^2*ln(x)*e-2/b^3*ln(x)*c*d-1/b^2*ln(c*x+b)*e+2/b^3*ln(c*x+b)*c*d+1/b
/(c*x+b)*e-c*d/b^2/(c*x+b)

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Maxima [A]  time = 0.698705, size = 93, normalized size = 1.43 \[ -\frac{b d +{\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac{{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac{{\left (2 \, c d - b e\right )} \log \left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*d + (2*c*d - b*e)*x)/(b^2*c*x^2 + b^3*x) + (2*c*d - b*e)*log(c*x + b)/b^3 -
(2*c*d - b*e)*log(x)/b^3

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Fricas [A]  time = 0.280949, size = 150, normalized size = 2.31 \[ -\frac{b^{2} d +{\left (2 \, b c d - b^{2} e\right )} x -{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^2*d + (2*b*c*d - b^2*e)*x - ((2*c^2*d - b*c*e)*x^2 + (2*b*c*d - b^2*e)*x)*lo
g(c*x + b) + ((2*c^2*d - b*c*e)*x^2 + (2*b*c*d - b^2*e)*x)*log(x))/(b^3*c*x^2 +
b^4*x)

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Sympy [A]  time = 2.48055, size = 128, normalized size = 1.97 \[ \frac{- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269101, size = 104, normalized size = 1.6 \[ -\frac{{\left (2 \, c d - b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c d x - b x e + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac{{\left (2 \, c^{2} d - b c e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*c*d - b*e)*ln(abs(x))/b^3 - (2*c*d*x - b*x*e + b*d)/((c*x^2 + b*x)*b^2) + (2
*c^2*d - b*c*e)*ln(abs(c*x + b))/(b^3*c)