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

Optimal. Leaf size=113 \[ -\frac{c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac{c^2 (4 c d-3 b e) \log (b+c x)}{b^5}-\frac{c^2 (c d-b e)}{b^4 (b+c x)}-\frac{c (3 c d-2 b e)}{b^4 x}+\frac{2 c d-b e}{2 b^3 x^2}-\frac{d}{3 b^2 x^3} \]

[Out]

-d/(3*b^2*x^3) + (2*c*d - b*e)/(2*b^3*x^2) - (c*(3*c*d - 2*b*e))/(b^4*x) - (c^2*
(c*d - b*e))/(b^4*(b + c*x)) - (c^2*(4*c*d - 3*b*e)*Log[x])/b^5 + (c^2*(4*c*d -
3*b*e)*Log[b + c*x])/b^5

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Rubi [A]  time = 0.221056, antiderivative size = 113, 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 \[ -\frac{c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac{c^2 (4 c d-3 b e) \log (b+c x)}{b^5}-\frac{c^2 (c d-b e)}{b^4 (b+c x)}-\frac{c (3 c d-2 b e)}{b^4 x}+\frac{2 c d-b e}{2 b^3 x^2}-\frac{d}{3 b^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-d/(3*b^2*x^3) + (2*c*d - b*e)/(2*b^3*x^2) - (c*(3*c*d - 2*b*e))/(b^4*x) - (c^2*
(c*d - b*e))/(b^4*(b + c*x)) - (c^2*(4*c*d - 3*b*e)*Log[x])/b^5 + (c^2*(4*c*d -
3*b*e)*Log[b + c*x])/b^5

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Rubi in Sympy [A]  time = 28.173, size = 105, normalized size = 0.93 \[ - \frac{d}{3 b^{2} x^{3}} - \frac{b e - 2 c d}{2 b^{3} x^{2}} + \frac{c^{2} \left (b e - c d\right )}{b^{4} \left (b + c x\right )} + \frac{c \left (2 b e - 3 c d\right )}{b^{4} x} + \frac{c^{2} \left (3 b e - 4 c d\right ) \log{\left (x \right )}}{b^{5}} - \frac{c^{2} \left (3 b e - 4 c d\right ) \log{\left (b + c x \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.227802, size = 106, normalized size = 0.94 \[ \frac{-\frac{2 b^3 d}{x^3}-\frac{3 b^2 (b e-2 c d)}{x^2}+\frac{6 b c^2 (b e-c d)}{b+c x}+6 c^2 \log (x) (3 b e-4 c d)+6 c^2 (4 c d-3 b e) \log (b+c x)+\frac{6 b c (2 b e-3 c d)}{x}}{6 b^5} \]

Antiderivative was successfully verified.

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

[Out]

((-2*b^3*d)/x^3 - (3*b^2*(-2*c*d + b*e))/x^2 + (6*b*c*(-3*c*d + 2*b*e))/x + (6*b
*c^2*(-(c*d) + b*e))/(b + c*x) + 6*c^2*(-4*c*d + 3*b*e)*Log[x] + 6*c^2*(4*c*d -
3*b*e)*Log[b + c*x])/(6*b^5)

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Maple [A]  time = 0.018, size = 134, normalized size = 1.2 \[ -{\frac{d}{3\,{b}^{2}{x}^{3}}}-{\frac{e}{2\,{b}^{2}{x}^{2}}}+{\frac{cd}{{b}^{3}{x}^{2}}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) e}{{b}^{4}}}-4\,{\frac{{c}^{3}\ln \left ( x \right ) d}{{b}^{5}}}+2\,{\frac{ce}{{b}^{3}x}}-3\,{\frac{{c}^{2}d}{{b}^{4}x}}-3\,{\frac{{c}^{2}\ln \left ( cx+b \right ) e}{{b}^{4}}}+4\,{\frac{{c}^{3}\ln \left ( cx+b \right ) d}{{b}^{5}}}+{\frac{e{c}^{2}}{{b}^{3} \left ( cx+b \right ) }}-{\frac{d{c}^{3}}{{b}^{4} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.705094, size = 174, normalized size = 1.54 \[ -\frac{2 \, b^{3} d + 6 \,{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} x^{3} + 3 \,{\left (4 \, b c^{2} d - 3 \, b^{2} c e\right )} x^{2} -{\left (4 \, b^{2} c d - 3 \, b^{3} e\right )} x}{6 \,{\left (b^{4} c x^{4} + b^{5} x^{3}\right )}} + \frac{{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{5}} - \frac{{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (x\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*b^3*d + 6*(4*c^3*d - 3*b*c^2*e)*x^3 + 3*(4*b*c^2*d - 3*b^2*c*e)*x^2 - (4
*b^2*c*d - 3*b^3*e)*x)/(b^4*c*x^4 + b^5*x^3) + (4*c^3*d - 3*b*c^2*e)*log(c*x + b
)/b^5 - (4*c^3*d - 3*b*c^2*e)*log(x)/b^5

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Fricas [A]  time = 0.283254, size = 243, normalized size = 2.15 \[ -\frac{2 \, b^{4} d + 6 \,{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \,{\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} -{\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x - 6 \,{\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} +{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} +{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (b^{5} c x^{4} + b^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*b^4*d + 6*(4*b*c^3*d - 3*b^2*c^2*e)*x^3 + 3*(4*b^2*c^2*d - 3*b^3*c*e)*x^
2 - (4*b^3*c*d - 3*b^4*e)*x - 6*((4*c^4*d - 3*b*c^3*e)*x^4 + (4*b*c^3*d - 3*b^2*
c^2*e)*x^3)*log(c*x + b) + 6*((4*c^4*d - 3*b*c^3*e)*x^4 + (4*b*c^3*d - 3*b^2*c^2
*e)*x^3)*log(x))/(b^5*c*x^4 + b^6*x^3)

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Sympy [A]  time = 3.3156, size = 219, normalized size = 1.94 \[ \frac{- 2 b^{3} d + x^{3} \left (18 b c^{2} e - 24 c^{3} d\right ) + x^{2} \left (9 b^{2} c e - 12 b c^{2} d\right ) + x \left (- 3 b^{3} e + 4 b^{2} c d\right )}{6 b^{5} x^{3} + 6 b^{4} c x^{4}} + \frac{c^{2} \left (3 b e - 4 c d\right ) \log{\left (x + \frac{3 b^{2} c^{2} e - 4 b c^{3} d - b c^{2} \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} - \frac{c^{2} \left (3 b e - 4 c d\right ) \log{\left (x + \frac{3 b^{2} c^{2} e - 4 b c^{3} d + b c^{2} \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*b**3*d + x**3*(18*b*c**2*e - 24*c**3*d) + x**2*(9*b**2*c*e - 12*b*c**2*d) +
x*(-3*b**3*e + 4*b**2*c*d))/(6*b**5*x**3 + 6*b**4*c*x**4) + c**2*(3*b*e - 4*c*d)
*log(x + (3*b**2*c**2*e - 4*b*c**3*d - b*c**2*(3*b*e - 4*c*d))/(6*b*c**3*e - 8*c
**4*d))/b**5 - c**2*(3*b*e - 4*c*d)*log(x + (3*b**2*c**2*e - 4*b*c**3*d + b*c**2
*(3*b*e - 4*c*d))/(6*b*c**3*e - 8*c**4*d))/b**5

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GIAC/XCAS [A]  time = 0.273714, size = 188, normalized size = 1.66 \[ -\frac{{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{{\left (4 \, c^{4} d - 3 \, b c^{3} e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac{2 \, b^{4} d + 6 \,{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \,{\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} -{\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x}{6 \,{\left (c x + b\right )} b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*c^3*d - 3*b*c^2*e)*ln(abs(x))/b^5 + (4*c^4*d - 3*b*c^3*e)*ln(abs(c*x + b))/(
b^5*c) - 1/6*(2*b^4*d + 6*(4*b*c^3*d - 3*b^2*c^2*e)*x^3 + 3*(4*b^2*c^2*d - 3*b^3
*c*e)*x^2 - (4*b^3*c*d - 3*b^4*e)*x)/((c*x + b)*b^5*x^3)