3.189 \(\int \frac{1}{x^2 (a+b x)^3} \, dx\)

Optimal. Leaf size=57 \[ -\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}-\frac{2 b}{a^3 (a+b x)}-\frac{1}{a^3 x}-\frac{b}{2 a^2 (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0664493, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}-\frac{2 b}{a^3 (a+b x)}-\frac{1}{a^3 x}-\frac{b}{2 a^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.7211, size = 54, normalized size = 0.95 \[ - \frac{b}{2 a^{2} \left (a + b x\right )^{2}} - \frac{2 b}{a^{3} \left (a + b x\right )} - \frac{1}{a^{3} x} - \frac{3 b \log{\left (x \right )}}{a^{4}} + \frac{3 b \log{\left (a + b x \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(2*a**2*(a + b*x)**2) - 2*b/(a**3*(a + b*x)) - 1/(a**3*x) - 3*b*log(x)/a**4 +
3*b*log(a + b*x)/a**4

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Mathematica [A]  time = 0.0803801, size = 53, normalized size = 0.93 \[ -\frac{\frac{a \left (2 a^2+9 a b x+6 b^2 x^2\right )}{x (a+b x)^2}-6 b \log (a+b x)+6 b \log (x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)^3),x]

[Out]

-((a*(2*a^2 + 9*a*b*x + 6*b^2*x^2))/(x*(a + b*x)^2) + 6*b*Log[x] - 6*b*Log[a + b
*x])/(2*a^4)

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Maple [A]  time = 0.016, size = 56, normalized size = 1. \[ -{\frac{1}{{a}^{3}x}}-{\frac{b}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}-2\,{\frac{b}{{a}^{3} \left ( bx+a \right ) }}-3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}}}+3\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^3,x)

[Out]

-1/a^3/x-1/2*b/a^2/(b*x+a)^2-2*b/a^3/(b*x+a)-3*b*ln(x)/a^4+3*b*ln(b*x+a)/a^4

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Maxima [A]  time = 1.34609, size = 93, normalized size = 1.63 \[ -\frac{6 \, b^{2} x^{2} + 9 \, a b x + 2 \, a^{2}}{2 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} + \frac{3 \, b \log \left (b x + a\right )}{a^{4}} - \frac{3 \, b \log \left (x\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^2),x, algorithm="maxima")

[Out]

-1/2*(6*b^2*x^2 + 9*a*b*x + 2*a^2)/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x) + 3*b*log
(b*x + a)/a^4 - 3*b*log(x)/a^4

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Fricas [A]  time = 0.218736, size = 147, normalized size = 2.58 \[ -\frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^2),x, algorithm="fricas")

[Out]

-1/2*(6*a*b^2*x^2 + 9*a^2*b*x + 2*a^3 - 6*(b^3*x^3 + 2*a*b^2*x^2 + a^2*b*x)*log(
b*x + a) + 6*(b^3*x^3 + 2*a*b^2*x^2 + a^2*b*x)*log(x))/(a^4*b^2*x^3 + 2*a^5*b*x^
2 + a^6*x)

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Sympy [A]  time = 1.90753, size = 65, normalized size = 1.14 \[ - \frac{2 a^{2} + 9 a b x + 6 b^{2} x^{2}}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac{3 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a)**3,x)

[Out]

-(2*a**2 + 9*a*b*x + 6*b**2*x**2)/(2*a**5*x + 4*a**4*b*x**2 + 2*a**3*b**2*x**3)
+ 3*b*(-log(x) + log(a/b + x))/a**4

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GIAC/XCAS [A]  time = 0.208565, size = 81, normalized size = 1.42 \[ \frac{3 \, b{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{3 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{2 \,{\left (b x + a\right )}^{2} a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^2),x, algorithm="giac")

[Out]

3*b*ln(abs(b*x + a))/a^4 - 3*b*ln(abs(x))/a^4 - 1/2*(6*a*b^2*x^2 + 9*a^2*b*x + 2
*a^3)/((b*x + a)^2*a^4*x)