3.179 \(\int \frac{1}{x^5 (a+b x)^2} \, dx\)

Optimal. Leaf size=84 \[ \frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}+\frac{b^4}{a^5 (a+b x)}+\frac{4 b^3}{a^5 x}-\frac{3 b^2}{2 a^4 x^2}+\frac{2 b}{3 a^3 x^3}-\frac{1}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0946695, antiderivative size = 84, 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 \[ \frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}+\frac{b^4}{a^5 (a+b x)}+\frac{4 b^3}{a^5 x}-\frac{3 b^2}{2 a^4 x^2}+\frac{2 b}{3 a^3 x^3}-\frac{1}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.5668, size = 83, normalized size = 0.99 \[ - \frac{1}{4 a^{2} x^{4}} + \frac{2 b}{3 a^{3} x^{3}} - \frac{3 b^{2}}{2 a^{4} x^{2}} + \frac{b^{4}}{a^{5} \left (a + b x\right )} + \frac{4 b^{3}}{a^{5} x} + \frac{5 b^{4} \log{\left (x \right )}}{a^{6}} - \frac{5 b^{4} \log{\left (a + b x \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0794422, size = 79, normalized size = 0.94 \[ \frac{\frac{a \left (-3 a^4+5 a^3 b x-10 a^2 b^2 x^2+30 a b^3 x^3+60 b^4 x^4\right )}{x^4 (a+b x)}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{12 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x)^2),x]

[Out]

((a*(-3*a^4 + 5*a^3*b*x - 10*a^2*b^2*x^2 + 30*a*b^3*x^3 + 60*b^4*x^4))/(x^4*(a +
b*x)) + 60*b^4*Log[x] - 60*b^4*Log[a + b*x])/(12*a^6)

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Maple [A]  time = 0.017, size = 79, normalized size = 0.9 \[ -{\frac{1}{4\,{a}^{2}{x}^{4}}}+{\frac{2\,b}{3\,{a}^{3}{x}^{3}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}{x}^{2}}}+4\,{\frac{{b}^{3}}{{a}^{5}x}}+{\frac{{b}^{4}}{{a}^{5} \left ( bx+a \right ) }}+5\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{6}}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.32318, size = 116, normalized size = 1.38 \[ \frac{60 \, b^{4} x^{4} + 30 \, a b^{3} x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x - 3 \, a^{4}}{12 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac{5 \, b^{4} \log \left (b x + a\right )}{a^{6}} + \frac{5 \, b^{4} \log \left (x\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^5),x, algorithm="maxima")

[Out]

1/12*(60*b^4*x^4 + 30*a*b^3*x^3 - 10*a^2*b^2*x^2 + 5*a^3*b*x - 3*a^4)/(a^5*b*x^5
+ a^6*x^4) - 5*b^4*log(b*x + a)/a^6 + 5*b^4*log(x)/a^6

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Fricas [A]  time = 0.208307, size = 146, normalized size = 1.74 \[ \frac{60 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x - 3 \, a^{5} - 60 \,{\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^5),x, algorithm="fricas")

[Out]

1/12*(60*a*b^4*x^4 + 30*a^2*b^3*x^3 - 10*a^3*b^2*x^2 + 5*a^4*b*x - 3*a^5 - 60*(b
^5*x^5 + a*b^4*x^4)*log(b*x + a) + 60*(b^5*x^5 + a*b^4*x^4)*log(x))/(a^6*b*x^5 +
a^7*x^4)

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Sympy [A]  time = 2.02078, size = 80, normalized size = 0.95 \[ \frac{- 3 a^{4} + 5 a^{3} b x - 10 a^{2} b^{2} x^{2} + 30 a b^{3} x^{3} + 60 b^{4} x^{4}}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} + \frac{5 b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x+a)**2,x)

[Out]

(-3*a**4 + 5*a**3*b*x - 10*a**2*b**2*x**2 + 30*a*b**3*x**3 + 60*b**4*x**4)/(12*a
**6*x**4 + 12*a**5*b*x**5) + 5*b**4*(log(x) - log(a/b + x))/a**6

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GIAC/XCAS [A]  time = 0.214089, size = 140, normalized size = 1.67 \[ \frac{5 \, b^{4}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{6}} + \frac{b^{4}}{{\left (b x + a\right )} a^{5}} - \frac{\frac{260 \, a b^{4}}{b x + a} - \frac{300 \, a^{2} b^{4}}{{\left (b x + a\right )}^{2}} + \frac{120 \, a^{3} b^{4}}{{\left (b x + a\right )}^{3}} - 77 \, b^{4}}{12 \, a^{6}{\left (\frac{a}{b x + a} - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^5),x, algorithm="giac")

[Out]

5*b^4*ln(abs(-a/(b*x + a) + 1))/a^6 + b^4/((b*x + a)*a^5) - 1/12*(260*a*b^4/(b*x
+ a) - 300*a^2*b^4/(b*x + a)^2 + 120*a^3*b^4/(b*x + a)^3 - 77*b^4)/(a^6*(a/(b*x
+ a) - 1)^4)