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

Optimal. Leaf size=85 \[ \frac{c \log (x) (3 c d-2 b e)}{b^4}-\frac{c (3 c d-2 b e) \log (b+c x)}{b^4}+\frac{2 c d-b e}{b^3 x}+\frac{c (c d-b e)}{b^3 (b+c x)}-\frac{d}{2 b^2 x^2} \]

[Out]

-d/(2*b^2*x^2) + (2*c*d - b*e)/(b^3*x) + (c*(c*d - b*e))/(b^3*(b + c*x)) + (c*(3
*c*d - 2*b*e)*Log[x])/b^4 - (c*(3*c*d - 2*b*e)*Log[b + c*x])/b^4

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Rubi [A]  time = 0.162202, antiderivative size = 85, 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 \[ \frac{c \log (x) (3 c d-2 b e)}{b^4}-\frac{c (3 c d-2 b e) \log (b+c x)}{b^4}+\frac{2 c d-b e}{b^3 x}+\frac{c (c d-b e)}{b^3 (b+c x)}-\frac{d}{2 b^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-d/(2*b^2*x^2) + (2*c*d - b*e)/(b^3*x) + (c*(c*d - b*e))/(b^3*(b + c*x)) + (c*(3
*c*d - 2*b*e)*Log[x])/b^4 - (c*(3*c*d - 2*b*e)*Log[b + c*x])/b^4

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Rubi in Sympy [A]  time = 20.2121, size = 80, normalized size = 0.94 \[ - \frac{d}{2 b^{2} x^{2}} - \frac{c \left (b e - c d\right )}{b^{3} \left (b + c x\right )} - \frac{b e - 2 c d}{b^{3} x} - \frac{c \left (2 b e - 3 c d\right ) \log{\left (x \right )}}{b^{4}} + \frac{c \left (2 b e - 3 c d\right ) \log{\left (b + c x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.136775, size = 85, normalized size = 1. \[ \frac{-\frac{b \left (b^2 (d+2 e x)+b c x (4 e x-3 d)-6 c^2 d x^2\right )}{x^2 (b+c x)}+2 c \log (x) (3 c d-2 b e)+2 c (2 b e-3 c d) \log (b+c x)}{2 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(-((b*(-6*c^2*d*x^2 + b^2*(d + 2*e*x) + b*c*x*(-3*d + 4*e*x)))/(x^2*(b + c*x)))
+ 2*c*(3*c*d - 2*b*e)*Log[x] + 2*c*(-3*c*d + 2*b*e)*Log[b + c*x])/(2*b^4)

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Maple [A]  time = 0.016, size = 107, normalized size = 1.3 \[ -{\frac{d}{2\,{b}^{2}{x}^{2}}}-{\frac{e}{{b}^{2}x}}+2\,{\frac{cd}{{b}^{3}x}}-2\,{\frac{c\ln \left ( x \right ) e}{{b}^{3}}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{4}}}+2\,{\frac{c\ln \left ( cx+b \right ) e}{{b}^{3}}}-3\,{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{4}}}-{\frac{ce}{{b}^{2} \left ( cx+b \right ) }}+{\frac{{c}^{2}d}{{b}^{3} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/x/(c*x^2+b*x)^2,x)

[Out]

-1/2*d/b^2/x^2-1/b^2/x*e+2/b^3/x*c*d-2*c/b^3*ln(x)*e+3*c^2/b^4*ln(x)*d+2*c/b^3*l
n(c*x+b)*e-3*c^2/b^4*ln(c*x+b)*d-c/b^2/(c*x+b)*e+c^2/b^3/(c*x+b)*d

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Maxima [A]  time = 0.689559, size = 135, normalized size = 1.59 \[ -\frac{b^{2} d - 2 \,{\left (3 \, c^{2} d - 2 \, b c e\right )} x^{2} -{\left (3 \, b c d - 2 \, b^{2} e\right )} x}{2 \,{\left (b^{3} c x^{3} + b^{4} x^{2}\right )}} - \frac{{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (c x + b\right )}{b^{4}} + \frac{{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (x\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^2*d - 2*(3*c^2*d - 2*b*c*e)*x^2 - (3*b*c*d - 2*b^2*e)*x)/(b^3*c*x^3 + b^
4*x^2) - (3*c^2*d - 2*b*c*e)*log(c*x + b)/b^4 + (3*c^2*d - 2*b*c*e)*log(x)/b^4

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Fricas [A]  time = 0.286897, size = 204, normalized size = 2.4 \[ -\frac{b^{3} d - 2 \,{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} -{\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x + 2 \,{\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} +{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} +{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{4} c x^{3} + b^{5} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^3*d - 2*(3*b*c^2*d - 2*b^2*c*e)*x^2 - (3*b^2*c*d - 2*b^3*e)*x + 2*((3*c^
3*d - 2*b*c^2*e)*x^3 + (3*b*c^2*d - 2*b^2*c*e)*x^2)*log(c*x + b) - 2*((3*c^3*d -
2*b*c^2*e)*x^3 + (3*b*c^2*d - 2*b^2*c*e)*x^2)*log(x))/(b^4*c*x^3 + b^5*x^2)

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Sympy [A]  time = 2.89348, size = 184, normalized size = 2.16 \[ - \frac{b^{2} d + x^{2} \left (4 b c e - 6 c^{2} d\right ) + x \left (2 b^{2} e - 3 b c d\right )}{2 b^{4} x^{2} + 2 b^{3} c x^{3}} - \frac{c \left (2 b e - 3 c d\right ) \log{\left (x + \frac{2 b^{2} c e - 3 b c^{2} d - b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} + \frac{c \left (2 b e - 3 c d\right ) \log{\left (x + \frac{2 b^{2} c e - 3 b c^{2} d + b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268901, size = 150, normalized size = 1.76 \[ \frac{{\left (3 \, c^{2} d - 2 \, b c e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} - \frac{{\left (3 \, c^{3} d - 2 \, b c^{2} e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac{b^{3} d - 2 \,{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} -{\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x}{2 \,{\left (c x + b\right )} b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*c^2*d - 2*b*c*e)*ln(abs(x))/b^4 - (3*c^3*d - 2*b*c^2*e)*ln(abs(c*x + b))/(b^4
*c) - 1/2*(b^3*d - 2*(3*b*c^2*d - 2*b^2*c*e)*x^2 - (3*b^2*c*d - 2*b^3*e)*x)/((c*
x + b)*b^4*x^2)