\(\int \frac {1}{x^2 (a+b x)^5} \, dx\) [58]

Optimal result

Integrand size = 11, antiderivative size = 77 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=\frac {-\frac {125 b}{12 a^2}-\frac {1}{a x}-\frac {65 b^2 x}{3 a^3}-\frac {35 b^3 x^2}{2 a^4}-\frac {5 b^4 x^3}{a^5}}{(a+b x)^4}+\frac {5 b \log \left (\frac {a+b x}{x}\right )}{a^6} \]

[Out]

(-1/a/x-125/12*b/a^2-65/3*b^2*x/a^3-35/2*b^3*x^2/a^4-5*b^4*x^3/a^5)/(b*x+a)^4+5*b/a^6*ln((b*x+a)/x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=-\frac {5 b \log (x)}{a^6}+\frac {5 b \log (a+b x)}{a^6}-\frac {4 b}{a^5 (a+b x)}-\frac {1}{a^5 x}-\frac {3 b}{2 a^4 (a+b x)^2}-\frac {2 b}{3 a^3 (a+b x)^3}-\frac {b}{4 a^2 (a+b x)^4} \]

[In]

Int[1/(x^2*(a + b*x)^5),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=-\frac {\frac {a \left (12 a^4+125 a^3 b x+260 a^2 b^2 x^2+210 a b^3 x^3+60 b^4 x^4\right )}{x (a+b x)^4}+60 b \log (x)-60 b \log (a+b x)}{12 a^6} \]

[In]

Integrate[1/(x^2*(a + b*x)^5),x]

[Out]

-1/12*((a*(12*a^4 + 125*a^3*b*x + 260*a^2*b^2*x^2 + 210*a*b^3*x^3 + 60*b^4*x^4))/(x*(a + b*x)^4) + 60*b*Log[x]
- 60*b*Log[a + b*x])/a^6

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06

method	result	size
default	\(-\frac {b}{4 a^{2} \left (b x +a \right )^{4}}+\frac {5 b \ln \left (b x +a \right )}{a^{6}}-\frac {4 b}{a^{5} \left (b x +a \right )}-\frac {3 b}{2 a^{4} \left (b x +a \right )^{2}}-\frac {2 b}{3 a^{3} \left (b x +a \right )^{3}}-\frac {1}{a^{5} x}-\frac {5 b \ln \left (x \right )}{a^{6}}\)	\(82\)
risch	\(\frac {-\frac {5 b^{4} x^{4}}{a^{5}}-\frac {35 b^{3} x^{3}}{2 a^{4}}-\frac {65 b^{2} x^{2}}{3 a^{3}}-\frac {125 b x}{12 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{4}}+\frac {5 b \ln \left (-b x -a \right )}{a^{6}}-\frac {5 b \ln \left (x \right )}{a^{6}}\)	\(82\)
norman	\(\frac {-\frac {1}{a}+\frac {20 b^{2} x^{2}}{a^{3}}+\frac {45 b^{3} x^{3}}{a^{4}}+\frac {110 b^{4} x^{4}}{3 a^{5}}+\frac {125 b^{5} x^{5}}{12 a^{6}}}{x \left (b x +a \right )^{4}}-\frac {5 b \ln \left (x \right )}{a^{6}}+\frac {5 b \ln \left (b x +a \right )}{a^{6}}\)	\(83\)
parallelrisch	\(-\frac {60 \ln \left (x \right ) x^{5} b^{5}-60 \ln \left (b x +a \right ) x^{5} b^{5}+240 \ln \left (x \right ) x^{4} a \,b^{4}-240 \ln \left (b x +a \right ) x^{4} a \,b^{4}-125 x^{5} b^{5}+360 \ln \left (x \right ) x^{3} a^{2} b^{3}-360 \ln \left (b x +a \right ) x^{3} a^{2} b^{3}-440 x^{4} a \,b^{4}+240 \ln \left (x \right ) x^{2} a^{3} b^{2}-240 \ln \left (b x +a \right ) x^{2} a^{3} b^{2}-540 b^{3} a^{2} x^{3}+60 \ln \left (x \right ) x \,a^{4} b -60 \ln \left (b x +a \right ) x \,a^{4} b -240 b^{2} a^{3} x^{2}+12 a^{5}}{12 a^{6} x \left (b x +a \right )^{4}}\)	\(193\)

[In]

int(1/x^2/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (72) = 144\).

Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.56 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=-\frac {60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (x\right )}{12 \, {\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )}} \]

[In]

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

[Out]

-1/12*(60*a*b^4*x^4 + 210*a^2*b^3*x^3 + 260*a^3*b^2*x^2 + 125*a^4*b*x + 12*a^5 - 60*(b^5*x^5 + 4*a*b^4*x^4 + 6
*a^2*b^3*x^3 + 4*a^3*b^2*x^2 + a^4*b*x)*log(b*x + a) + 60*(b^5*x^5 + 4*a*b^4*x^4 + 6*a^2*b^3*x^3 + 4*a^3*b^2*x
^2 + a^4*b*x)*log(x))/(a^6*b^4*x^5 + 4*a^7*b^3*x^4 + 6*a^8*b^2*x^3 + 4*a^9*b*x^2 + a^10*x)

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=\frac {- 12 a^{4} - 125 a^{3} b x - 260 a^{2} b^{2} x^{2} - 210 a b^{3} x^{3} - 60 b^{4} x^{4}}{12 a^{9} x + 48 a^{8} b x^{2} + 72 a^{7} b^{2} x^{3} + 48 a^{6} b^{3} x^{4} + 12 a^{5} b^{4} x^{5}} + \frac {5 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} \]

[In]

integrate(1/x**2/(b*x+a)**5,x)

[Out]

(-12*a**4 - 125*a**3*b*x - 260*a**2*b**2*x**2 - 210*a*b**3*x**3 - 60*b**4*x**4)/(12*a**9*x + 48*a**8*b*x**2 +
72*a**7*b**2*x**3 + 48*a**6*b**3*x**4 + 12*a**5*b**4*x**5) + 5*b*(-log(x) + log(a/b + x))/a**6

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=-\frac {60 \, b^{4} x^{4} + 210 \, a b^{3} x^{3} + 260 \, a^{2} b^{2} x^{2} + 125 \, a^{3} b x + 12 \, a^{4}}{12 \, {\left (a^{5} b^{4} x^{5} + 4 \, a^{6} b^{3} x^{4} + 6 \, a^{7} b^{2} x^{3} + 4 \, a^{8} b x^{2} + a^{9} x\right )}} + \frac {5 \, b \log \left (b x + a\right )}{a^{6}} - \frac {5 \, b \log \left (x\right )}{a^{6}} \]

[In]

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

[Out]

-1/12*(60*b^4*x^4 + 210*a*b^3*x^3 + 260*a^2*b^2*x^2 + 125*a^3*b*x + 12*a^4)/(a^5*b^4*x^5 + 4*a^6*b^3*x^4 + 6*a
^7*b^2*x^3 + 4*a^8*b*x^2 + a^9*x) + 5*b*log(b*x + a)/a^6 - 5*b*log(x)/a^6

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=-\frac {5 \, b \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{6}} + \frac {b}{a^{6} {\left (\frac {a}{b x + a} - 1\right )}} - \frac {\frac {48 \, a^{3} b^{9}}{b x + a} + \frac {18 \, a^{4} b^{9}}{{\left (b x + a\right )}^{2}} + \frac {8 \, a^{5} b^{9}}{{\left (b x + a\right )}^{3}} + \frac {3 \, a^{6} b^{9}}{{\left (b x + a\right )}^{4}}}{12 \, a^{8} b^{8}} \]

[In]

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

[Out]

-5*b*log(abs(-a/(b*x + a) + 1))/a^6 + b/(a^6*(a/(b*x + a) - 1)) - 1/12*(48*a^3*b^9/(b*x + a) + 18*a^4*b^9/(b*x
+ a)^2 + 8*a^5*b^9/(b*x + a)^3 + 3*a^6*b^9/(b*x + a)^4)/(a^8*b^8)

Mupad [B] (verification not implemented)

Time = 16.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^2 (a+b x)^5} \, dx=\frac {10\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6}-\frac {\frac {1}{a}+\frac {65\,b^2\,x^2}{3\,a^3}+\frac {35\,b^3\,x^3}{2\,a^4}+\frac {5\,b^4\,x^4}{a^5}+\frac {125\,b\,x}{12\,a^2}}{a^4\,x+4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^3+4\,a\,b^3\,x^4+b^4\,x^5} \]

[In]

int(1/(x^2*(a + b*x)^5),x)

[Out]

(10*b*atanh((2*b*x)/a + 1))/a^6 - (1/a + (65*b^2*x^2)/(3*a^3) + (35*b^3*x^3)/(2*a^4) + (5*b^4*x^4)/a^5 + (125*
b*x)/(12*a^2))/(a^4*x + b^4*x^5 + 4*a^3*b*x^2 + 4*a*b^3*x^4 + 6*a^2*b^2*x^3)