3.282 \(\int \frac{1}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=76 \[ \frac{6 c^2 \log (x)}{b^5}-\frac{6 c^2 \log (b+c x)}{b^5}+\frac{3 c^2}{b^4 (b+c x)}+\frac{3 c}{b^4 x}+\frac{c^2}{2 b^3 (b+c x)^2}-\frac{1}{2 b^3 x^2} \]

[Out]

-1/(2*b^3*x^2) + (3*c)/(b^4*x) + c^2/(2*b^3*(b + c*x)^2) + (3*c^2)/(b^4*(b + c*x
)) + (6*c^2*Log[x])/b^5 - (6*c^2*Log[b + c*x])/b^5

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Rubi [A]  time = 0.108613, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{6 c^2 \log (x)}{b^5}-\frac{6 c^2 \log (b+c x)}{b^5}+\frac{3 c^2}{b^4 (b+c x)}+\frac{3 c}{b^4 x}+\frac{c^2}{2 b^3 (b+c x)^2}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-3),x]

[Out]

-1/(2*b^3*x^2) + (3*c)/(b^4*x) + c^2/(2*b^3*(b + c*x)^2) + (3*c^2)/(b^4*(b + c*x
)) + (6*c^2*Log[x])/b^5 - (6*c^2*Log[b + c*x])/b^5

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Rubi in Sympy [A]  time = 8.31271, size = 70, normalized size = 0.92 \[ - \frac{b + 2 c x}{2 b^{2} \left (b x + c x^{2}\right )^{2}} + \frac{3 c \left (b + 2 c x\right )}{b^{4} \left (b x + c x^{2}\right )} + \frac{6 c^{2} \log{\left (x \right )}}{b^{5}} - \frac{6 c^{2} \log{\left (b + c x \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0805922, size = 68, normalized size = 0.89 \[ \frac{\frac{b \left (-b^3+4 b^2 c x+18 b c^2 x^2+12 c^3 x^3\right )}{x^2 (b+c x)^2}-12 c^2 \log (b+c x)+12 c^2 \log (x)}{2 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-3),x]

[Out]

((b*(-b^3 + 4*b^2*c*x + 18*b*c^2*x^2 + 12*c^3*x^3))/(x^2*(b + c*x)^2) + 12*c^2*L
og[x] - 12*c^2*Log[b + c*x])/(2*b^5)

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Maple [A]  time = 0.015, size = 73, normalized size = 1. \[ -{\frac{1}{2\,{b}^{3}{x}^{2}}}+3\,{\frac{c}{{b}^{4}x}}+{\frac{{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+3\,{\frac{{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}+6\,{\frac{{c}^{2}\ln \left ( x \right ) }{{b}^{5}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^3,x)

[Out]

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

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Maxima [A]  time = 0.687055, size = 116, normalized size = 1.53 \[ \frac{12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} + 4 \, b^{2} c x - b^{3}}{2 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac{6 \, c^{2} \log \left (c x + b\right )}{b^{5}} + \frac{6 \, c^{2} \log \left (x\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-3),x, algorithm="maxima")

[Out]

1/2*(12*c^3*x^3 + 18*b*c^2*x^2 + 4*b^2*c*x - b^3)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b
^6*x^2) - 6*c^2*log(c*x + b)/b^5 + 6*c^2*log(x)/b^5

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Fricas [A]  time = 0.231483, size = 176, normalized size = 2.32 \[ \frac{12 \, b c^{3} x^{3} + 18 \, b^{2} c^{2} x^{2} + 4 \, b^{3} c x - b^{4} - 12 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + b^{2} c^{2} x^{2}\right )} \log \left (c x + b\right ) + 12 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + b^{2} c^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-3),x, algorithm="fricas")

[Out]

1/2*(12*b*c^3*x^3 + 18*b^2*c^2*x^2 + 4*b^3*c*x - b^4 - 12*(c^4*x^4 + 2*b*c^3*x^3
+ b^2*c^2*x^2)*log(c*x + b) + 12*(c^4*x^4 + 2*b*c^3*x^3 + b^2*c^2*x^2)*log(x))/
(b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)

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Sympy [A]  time = 2.03974, size = 78, normalized size = 1.03 \[ \frac{- b^{3} + 4 b^{2} c x + 18 b c^{2} x^{2} + 12 c^{3} x^{3}}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} + \frac{6 c^{2} \left (\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x)**3,x)

[Out]

(-b**3 + 4*b**2*c*x + 18*b*c**2*x**2 + 12*c**3*x**3)/(2*b**6*x**2 + 4*b**5*c*x**
3 + 2*b**4*c**2*x**4) + 6*c**2*(log(x) - log(b/c + x))/b**5

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GIAC/XCAS [A]  time = 0.208433, size = 99, normalized size = 1.3 \[ -\frac{6 \, c^{2}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5}} + \frac{6 \, c^{2}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} + 4 \, b^{2} c x - b^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-3),x, algorithm="giac")

[Out]

-6*c^2*ln(abs(c*x + b))/b^5 + 6*c^2*ln(abs(x))/b^5 + 1/2*(12*c^3*x^3 + 18*b*c^2*
x^2 + 4*b^2*c*x - b^3)/((c*x^2 + b*x)^2*b^4)