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

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

[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 = 0.0428869, 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, Rules used = {44} \[ -\frac{3 b^2}{2 a^4 x^2}+\frac{b^4}{a^5 (a+b x)}+\frac{4 b^3}{a^5 x}+\frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}+\frac{2 b}{3 a^3 x^3}-\frac{1}{4 a^2 x^4} \]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 (a+b x)^2} \, dx &=\int \left (\frac{1}{a^2 x^5}-\frac{2 b}{a^3 x^4}+\frac{3 b^2}{a^4 x^3}-\frac{4 b^3}{a^5 x^2}+\frac{5 b^4}{a^6 x}-\frac{b^5}{a^5 (a+b x)^2}-\frac{5 b^5}{a^6 (a+b x)}\right ) \, dx\\ &=-\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{4 b^3}{a^5 x}+\frac{b^4}{a^5 (a+b x)}+\frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.0792331, size = 79, normalized size = 0.94 \[ \frac{\frac{a \left (-10 a^2 b^2 x^2+5 a^3 b x-3 a^4+30 a b^3 x^3+60 b^4 x^4\right )}{x^4 (a+b x)}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{12 a^6} \]

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 = 0.01, size = 79, normalized size = 0.9 \begin{align*} -{\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}}} \end{align*}

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 = 1.05882, size = 116, normalized size = 1.38 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^2,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 = 1.4977, size = 231, normalized size = 2.75 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^2,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 = 0.704667, size = 80, normalized size = 0.95 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.19309, size = 140, normalized size = 1.67 \begin{align*} \frac{5 \, b^{4} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*b^4*log(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)