3.192 \(\int \frac{1}{x^5 (a+b x)^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{3 b^2}{a^5 x^2}+\frac{5 b^4}{a^6 (a+b x)}+\frac{b^4}{2 a^5 (a+b x)^2}+\frac{10 b^3}{a^6 x}+\frac{15 b^4 \log (x)}{a^7}-\frac{15 b^4 \log (a+b x)}{a^7}+\frac{b}{a^4 x^3}-\frac{1}{4 a^3 x^4} \]

[Out]

-1/(4*a^3*x^4) + b/(a^4*x^3) - (3*b^2)/(a^5*x^2) + (10*b^3)/(a^6*x) + b^4/(2*a^5*(a + b*x)^2) + (5*b^4)/(a^6*(
a + b*x)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a + b*x])/a^7

________________________________________________________________________________________

Rubi [A]  time = 0.052223, antiderivative size = 97, 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} \[ -\frac{3 b^2}{a^5 x^2}+\frac{5 b^4}{a^6 (a+b x)}+\frac{b^4}{2 a^5 (a+b x)^2}+\frac{10 b^3}{a^6 x}+\frac{15 b^4 \log (x)}{a^7}-\frac{15 b^4 \log (a+b x)}{a^7}+\frac{b}{a^4 x^3}-\frac{1}{4 a^3 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)^3),x]

[Out]

-1/(4*a^3*x^4) + b/(a^4*x^3) - (3*b^2)/(a^5*x^2) + (10*b^3)/(a^6*x) + b^4/(2*a^5*(a + b*x)^2) + (5*b^4)/(a^6*(
a + b*x)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a + b*x])/a^7

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)^3} \, dx &=\int \left (\frac{1}{a^3 x^5}-\frac{3 b}{a^4 x^4}+\frac{6 b^2}{a^5 x^3}-\frac{10 b^3}{a^6 x^2}+\frac{15 b^4}{a^7 x}-\frac{b^5}{a^5 (a+b x)^3}-\frac{5 b^5}{a^6 (a+b x)^2}-\frac{15 b^5}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac{1}{4 a^3 x^4}+\frac{b}{a^4 x^3}-\frac{3 b^2}{a^5 x^2}+\frac{10 b^3}{a^6 x}+\frac{b^4}{2 a^5 (a+b x)^2}+\frac{5 b^4}{a^6 (a+b x)}+\frac{15 b^4 \log (x)}{a^7}-\frac{15 b^4 \log (a+b x)}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0793184, size = 90, normalized size = 0.93 \[ \frac{\frac{a \left (-5 a^3 b^2 x^2+20 a^2 b^3 x^3+2 a^4 b x-a^5+90 a b^4 x^4+60 b^5 x^5\right )}{x^4 (a+b x)^2}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{4 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)^3),x]

[Out]

((a*(-a^5 + 2*a^4*b*x - 5*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 90*a*b^4*x^4 + 60*b^5*x^5))/(x^4*(a + b*x)^2) + 60*b^
4*Log[x] - 60*b^4*Log[a + b*x])/(4*a^7)

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 94, normalized size = 1. \begin{align*} -{\frac{1}{4\,{a}^{3}{x}^{4}}}+{\frac{b}{{a}^{4}{x}^{3}}}-3\,{\frac{{b}^{2}}{{a}^{5}{x}^{2}}}+10\,{\frac{{b}^{3}}{{a}^{6}x}}+{\frac{{b}^{4}}{2\,{a}^{5} \left ( bx+a \right ) ^{2}}}+5\,{\frac{{b}^{4}}{{a}^{6} \left ( bx+a \right ) }}+15\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{7}}}-15\,{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x+a)^3,x)

[Out]

-1/4/a^3/x^4+b/a^4/x^3-3*b^2/a^5/x^2+10*b^3/a^6/x+1/2*b^4/a^5/(b*x+a)^2+5*b^4/a^6/(b*x+a)+15*b^4*ln(x)/a^7-15*
b^4*ln(b*x+a)/a^7

________________________________________________________________________________________

Maxima [A]  time = 1.08144, size = 146, normalized size = 1.51 \begin{align*} \frac{60 \, b^{5} x^{5} + 90 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 5 \, a^{3} b^{2} x^{2} + 2 \, a^{4} b x - a^{5}}{4 \,{\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} - \frac{15 \, b^{4} \log \left (b x + a\right )}{a^{7}} + \frac{15 \, b^{4} \log \left (x\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^3,x, algorithm="maxima")

[Out]

1/4*(60*b^5*x^5 + 90*a*b^4*x^4 + 20*a^2*b^3*x^3 - 5*a^3*b^2*x^2 + 2*a^4*b*x - a^5)/(a^6*b^2*x^6 + 2*a^7*b*x^5
+ a^8*x^4) - 15*b^4*log(b*x + a)/a^7 + 15*b^4*log(x)/a^7

________________________________________________________________________________________

Fricas [A]  time = 1.62782, size = 313, normalized size = 3.23 \begin{align*} \frac{60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6} - 60 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(60*a*b^5*x^5 + 90*a^2*b^4*x^4 + 20*a^3*b^3*x^3 - 5*a^4*b^2*x^2 + 2*a^5*b*x - a^6 - 60*(b^6*x^6 + 2*a*b^5*
x^5 + a^2*b^4*x^4)*log(b*x + a) + 60*(b^6*x^6 + 2*a*b^5*x^5 + a^2*b^4*x^4)*log(x))/(a^7*b^2*x^6 + 2*a^8*b*x^5
+ a^9*x^4)

________________________________________________________________________________________

Sympy [A]  time = 0.814769, size = 102, normalized size = 1.05 \begin{align*} \frac{- a^{5} + 2 a^{4} b x - 5 a^{3} b^{2} x^{2} + 20 a^{2} b^{3} x^{3} + 90 a b^{4} x^{4} + 60 b^{5} x^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{5} + 4 a^{6} b^{2} x^{6}} + \frac{15 b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x+a)**3,x)

[Out]

(-a**5 + 2*a**4*b*x - 5*a**3*b**2*x**2 + 20*a**2*b**3*x**3 + 90*a*b**4*x**4 + 60*b**5*x**5)/(4*a**8*x**4 + 8*a
**7*b*x**5 + 4*a**6*b**2*x**6) + 15*b**4*(log(x) - log(a/b + x))/a**7

________________________________________________________________________________________

Giac [A]  time = 1.17565, size = 131, normalized size = 1.35 \begin{align*} -\frac{15 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{7}} + \frac{15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6}}{4 \,{\left (b x + a\right )}^{2} a^{7} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^3,x, algorithm="giac")

[Out]

-15*b^4*log(abs(b*x + a))/a^7 + 15*b^4*log(abs(x))/a^7 + 1/4*(60*a*b^5*x^5 + 90*a^2*b^4*x^4 + 20*a^3*b^3*x^3 -
5*a^4*b^2*x^2 + 2*a^5*b*x - a^6)/((b*x + a)^2*a^7*x^4)