\(\int \frac {1}{x (a+b x)^{10}} \, dx\) [235]

Optimal result

Integrand size = 11, antiderivative size = 141 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=\frac {1}{9 a (a+b x)^9}+\frac {1}{8 a^2 (a+b x)^8}+\frac {1}{7 a^3 (a+b x)^7}+\frac {1}{6 a^4 (a+b x)^6}+\frac {1}{5 a^5 (a+b x)^5}+\frac {1}{4 a^6 (a+b x)^4}+\frac {1}{3 a^7 (a+b x)^3}+\frac {1}{2 a^8 (a+b x)^2}+\frac {1}{a^9 (a+b x)}+\frac {\log (x)}{a^{10}}-\frac {\log (a+b x)}{a^{10}} \]

[Out]

1/9/a/(b*x+a)^9+1/8/a^2/(b*x+a)^8+1/7/a^3/(b*x+a)^7+1/6/a^4/(b*x+a)^6+1/5/a^5/(b*x+a)^5+1/4/a^6/(b*x+a)^4+1/3/
a^7/(b*x+a)^3+1/2/a^8/(b*x+a)^2+1/a^9/(b*x+a)+ln(x)/a^10-ln(b*x+a)/a^10

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x (a+b x)^{10}} \, dx=-\frac {\log (a+b x)}{a^{10}}+\frac {\log (x)}{a^{10}}+\frac {1}{a^9 (a+b x)}+\frac {1}{2 a^8 (a+b x)^2}+\frac {1}{3 a^7 (a+b x)^3}+\frac {1}{4 a^6 (a+b x)^4}+\frac {1}{5 a^5 (a+b x)^5}+\frac {1}{6 a^4 (a+b x)^6}+\frac {1}{7 a^3 (a+b x)^7}+\frac {1}{8 a^2 (a+b x)^8}+\frac {1}{9 a (a+b x)^9} \]

[In]

Int[1/(x*(a + b*x)^10),x]

[Out]

1/(9*a*(a + b*x)^9) + 1/(8*a^2*(a + b*x)^8) + 1/(7*a^3*(a + b*x)^7) + 1/(6*a^4*(a + b*x)^6) + 1/(5*a^5*(a + b*
x)^5) + 1/(4*a^6*(a + b*x)^4) + 1/(3*a^7*(a + b*x)^3) + 1/(2*a^8*(a + b*x)^2) + 1/(a^9*(a + b*x)) + Log[x]/a^1
0 - Log[a + b*x]/a^10

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^{10} x}-\frac {b}{a (a+b x)^{10}}-\frac {b}{a^2 (a+b x)^9}-\frac {b}{a^3 (a+b x)^8}-\frac {b}{a^4 (a+b x)^7}-\frac {b}{a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {b}{a^7 (a+b x)^4}-\frac {b}{a^8 (a+b x)^3}-\frac {b}{a^9 (a+b x)^2}-\frac {b}{a^{10} (a+b x)}\right ) \, dx \\ & = \frac {1}{9 a (a+b x)^9}+\frac {1}{8 a^2 (a+b x)^8}+\frac {1}{7 a^3 (a+b x)^7}+\frac {1}{6 a^4 (a+b x)^6}+\frac {1}{5 a^5 (a+b x)^5}+\frac {1}{4 a^6 (a+b x)^4}+\frac {1}{3 a^7 (a+b x)^3}+\frac {1}{2 a^8 (a+b x)^2}+\frac {1}{a^9 (a+b x)}+\frac {\log (x)}{a^{10}}-\frac {\log (a+b x)}{a^{10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=\frac {280 a^8+315 a^7 (a+b x)+360 a^6 (a+b x)^2+420 a^5 (a+b x)^3+504 a^4 (a+b x)^4+630 a^3 (a+b x)^5+840 a^2 (a+b x)^6+1260 a (a+b x)^7+2520 (a+b x)^8}{2520 a^9 (a+b x)^9}+\frac {\log (x)}{a^{10}}-\frac {\log (a+b x)}{a^{10}} \]

[In]

Integrate[1/(x*(a + b*x)^10),x]

[Out]

(280*a^8 + 315*a^7*(a + b*x) + 360*a^6*(a + b*x)^2 + 420*a^5*(a + b*x)^3 + 504*a^4*(a + b*x)^4 + 630*a^3*(a +
b*x)^5 + 840*a^2*(a + b*x)^6 + 1260*a*(a + b*x)^7 + 2520*(a + b*x)^8)/(2520*a^9*(a + b*x)^9) + Log[x]/a^10 - L
og[a + b*x]/a^10

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {\frac {b^{8} x^{8}}{a^{9}}+\frac {17 b^{7} x^{7}}{2 a^{8}}+\frac {191 b^{6} x^{6}}{6 a^{7}}+\frac {275 b^{5} x^{5}}{4 a^{6}}+\frac {1879 b^{4} x^{4}}{20 a^{5}}+\frac {2509 b^{3} x^{3}}{30 a^{4}}+\frac {3349 b^{2} x^{2}}{70 a^{3}}+\frac {4609 b x}{280 a^{2}}+\frac {7129}{2520 a}}{\left (b x +a \right )^{9}}-\frac {\ln \left (b x +a \right )}{a^{10}}+\frac {\ln \left (-x \right )}{a^{10}}\)	\(118\)
norman	\(\frac {-\frac {9 b x}{a^{2}}-\frac {54 b^{2} x^{2}}{a^{3}}-\frac {154 b^{3} x^{3}}{a^{4}}-\frac {525 b^{4} x^{4}}{2 a^{5}}-\frac {2877 b^{5} x^{5}}{10 a^{6}}-\frac {1029 b^{6} x^{6}}{5 a^{7}}-\frac {3267 b^{7} x^{7}}{35 a^{8}}-\frac {6849 b^{8} x^{8}}{280 a^{9}}-\frac {7129 b^{9} x^{9}}{2520 a^{10}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (x \right )}{a^{10}}-\frac {\ln \left (b x +a \right )}{a^{10}}\)	\(123\)
default	\(\frac {1}{9 a \left (b x +a \right )^{9}}+\frac {1}{8 a^{2} \left (b x +a \right )^{8}}+\frac {1}{7 a^{3} \left (b x +a \right )^{7}}+\frac {1}{6 a^{4} \left (b x +a \right )^{6}}+\frac {1}{5 a^{5} \left (b x +a \right )^{5}}+\frac {1}{4 a^{6} \left (b x +a \right )^{4}}+\frac {1}{3 a^{7} \left (b x +a \right )^{3}}+\frac {1}{2 a^{8} \left (b x +a \right )^{2}}+\frac {1}{a^{9} \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{10}}-\frac {\ln \left (b x +a \right )}{a^{10}}\)	\(126\)
parallelrisch	\(\frac {2520 \ln \left (x \right ) a^{9}+90720 \ln \left (x \right ) x^{2} a^{7} b^{2}+22680 \ln \left (x \right ) x \,a^{8} b -2520 \ln \left (b x +a \right ) a^{9}+22680 \ln \left (x \right ) x^{8} a \,b^{8}-22680 \ln \left (b x +a \right ) x^{8} a \,b^{8}+90720 \ln \left (x \right ) x^{7} a^{2} b^{7}-90720 \ln \left (b x +a \right ) x^{7} a^{2} b^{7}+211680 \ln \left (x \right ) x^{6} a^{3} b^{6}+317520 \ln \left (x \right ) x^{5} a^{4} b^{5}+317520 \ln \left (x \right ) x^{4} a^{5} b^{4}+211680 \ln \left (x \right ) x^{3} a^{6} b^{3}-22680 \ln \left (b x +a \right ) x \,a^{8} b -211680 \ln \left (b x +a \right ) x^{6} a^{3} b^{6}-317520 \ln \left (b x +a \right ) x^{5} a^{4} b^{5}-317520 \ln \left (b x +a \right ) x^{4} a^{5} b^{4}-211680 \ln \left (b x +a \right ) x^{3} a^{6} b^{3}-90720 \ln \left (b x +a \right ) x^{2} a^{7} b^{2}-7129 b^{9} x^{9}-725004 a^{4} x^{5} b^{5}+2520 \ln \left (x \right ) x^{9} b^{9}-2520 \ln \left (b x +a \right ) x^{9} b^{9}-518616 x^{6} a^{3} b^{6}-61641 a \,x^{8} b^{8}-661500 a^{5} b^{4} x^{4}-388080 a^{6} b^{3} x^{3}-136080 a^{7} b^{2} x^{2}-22680 a^{8} b x -235224 a^{2} x^{7} b^{7}}{2520 a^{10} \left (b x +a \right )^{9}}\)	\(374\)

[In]

int(1/x/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

(b^8/a^9*x^8+17/2*b^7/a^8*x^7+191/6*b^6/a^7*x^6+275/4*b^5/a^6*x^5+1879/20*b^4/a^5*x^4+2509/30*b^3/a^4*x^3+3349
/70*b^2/a^3*x^2+4609/280*b/a^2*x+7129/2520/a)/(b*x+a)^9-ln(b*x+a)/a^10+1/a^10*ln(-x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (125) = 250\).

Time = 0.23 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=\frac {2520 \, a b^{8} x^{8} + 21420 \, a^{2} b^{7} x^{7} + 80220 \, a^{3} b^{6} x^{6} + 173250 \, a^{4} b^{5} x^{5} + 236754 \, a^{5} b^{4} x^{4} + 210756 \, a^{6} b^{3} x^{3} + 120564 \, a^{7} b^{2} x^{2} + 41481 \, a^{8} b x + 7129 \, a^{9} - 2520 \, {\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \left (x\right )}{2520 \, {\left (a^{10} b^{9} x^{9} + 9 \, a^{11} b^{8} x^{8} + 36 \, a^{12} b^{7} x^{7} + 84 \, a^{13} b^{6} x^{6} + 126 \, a^{14} b^{5} x^{5} + 126 \, a^{15} b^{4} x^{4} + 84 \, a^{16} b^{3} x^{3} + 36 \, a^{17} b^{2} x^{2} + 9 \, a^{18} b x + a^{19}\right )}} \]

[In]

integrate(1/x/(b*x+a)^10,x, algorithm="fricas")

[Out]

1/2520*(2520*a*b^8*x^8 + 21420*a^2*b^7*x^7 + 80220*a^3*b^6*x^6 + 173250*a^4*b^5*x^5 + 236754*a^5*b^4*x^4 + 210
756*a^6*b^3*x^3 + 120564*a^7*b^2*x^2 + 41481*a^8*b*x + 7129*a^9 - 2520*(b^9*x^9 + 9*a*b^8*x^8 + 36*a^2*b^7*x^7
+ 84*a^3*b^6*x^6 + 126*a^4*b^5*x^5 + 126*a^5*b^4*x^4 + 84*a^6*b^3*x^3 + 36*a^7*b^2*x^2 + 9*a^8*b*x + a^9)*log
(b*x + a) + 2520*(b^9*x^9 + 9*a*b^8*x^8 + 36*a^2*b^7*x^7 + 84*a^3*b^6*x^6 + 126*a^4*b^5*x^5 + 126*a^5*b^4*x^4
+ 84*a^6*b^3*x^3 + 36*a^7*b^2*x^2 + 9*a^8*b*x + a^9)*log(x))/(a^10*b^9*x^9 + 9*a^11*b^8*x^8 + 36*a^12*b^7*x^7
+ 84*a^13*b^6*x^6 + 126*a^14*b^5*x^5 + 126*a^15*b^4*x^4 + 84*a^16*b^3*x^3 + 36*a^17*b^2*x^2 + 9*a^18*b*x + a^1
9)

Sympy [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=\frac {7129 a^{8} + 41481 a^{7} b x + 120564 a^{6} b^{2} x^{2} + 210756 a^{5} b^{3} x^{3} + 236754 a^{4} b^{4} x^{4} + 173250 a^{3} b^{5} x^{5} + 80220 a^{2} b^{6} x^{6} + 21420 a b^{7} x^{7} + 2520 b^{8} x^{8}}{2520 a^{18} + 22680 a^{17} b x + 90720 a^{16} b^{2} x^{2} + 211680 a^{15} b^{3} x^{3} + 317520 a^{14} b^{4} x^{4} + 317520 a^{13} b^{5} x^{5} + 211680 a^{12} b^{6} x^{6} + 90720 a^{11} b^{7} x^{7} + 22680 a^{10} b^{8} x^{8} + 2520 a^{9} b^{9} x^{9}} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a^{10}} \]

[In]

integrate(1/x/(b*x+a)**10,x)

[Out]

(7129*a**8 + 41481*a**7*b*x + 120564*a**6*b**2*x**2 + 210756*a**5*b**3*x**3 + 236754*a**4*b**4*x**4 + 173250*a
**3*b**5*x**5 + 80220*a**2*b**6*x**6 + 21420*a*b**7*x**7 + 2520*b**8*x**8)/(2520*a**18 + 22680*a**17*b*x + 907
20*a**16*b**2*x**2 + 211680*a**15*b**3*x**3 + 317520*a**14*b**4*x**4 + 317520*a**13*b**5*x**5 + 211680*a**12*b
**6*x**6 + 90720*a**11*b**7*x**7 + 22680*a**10*b**8*x**8 + 2520*a**9*b**9*x**9) + (log(x) - log(a/b + x))/a**1
0

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=\frac {2520 \, b^{8} x^{8} + 21420 \, a b^{7} x^{7} + 80220 \, a^{2} b^{6} x^{6} + 173250 \, a^{3} b^{5} x^{5} + 236754 \, a^{4} b^{4} x^{4} + 210756 \, a^{5} b^{3} x^{3} + 120564 \, a^{6} b^{2} x^{2} + 41481 \, a^{7} b x + 7129 \, a^{8}}{2520 \, {\left (a^{9} b^{9} x^{9} + 9 \, a^{10} b^{8} x^{8} + 36 \, a^{11} b^{7} x^{7} + 84 \, a^{12} b^{6} x^{6} + 126 \, a^{13} b^{5} x^{5} + 126 \, a^{14} b^{4} x^{4} + 84 \, a^{15} b^{3} x^{3} + 36 \, a^{16} b^{2} x^{2} + 9 \, a^{17} b x + a^{18}\right )}} - \frac {\log \left (b x + a\right )}{a^{10}} + \frac {\log \left (x\right )}{a^{10}} \]

[In]

integrate(1/x/(b*x+a)^10,x, algorithm="maxima")

[Out]

1/2520*(2520*b^8*x^8 + 21420*a*b^7*x^7 + 80220*a^2*b^6*x^6 + 173250*a^3*b^5*x^5 + 236754*a^4*b^4*x^4 + 210756*
a^5*b^3*x^3 + 120564*a^6*b^2*x^2 + 41481*a^7*b*x + 7129*a^8)/(a^9*b^9*x^9 + 9*a^10*b^8*x^8 + 36*a^11*b^7*x^7 +
84*a^12*b^6*x^6 + 126*a^13*b^5*x^5 + 126*a^14*b^4*x^4 + 84*a^15*b^3*x^3 + 36*a^16*b^2*x^2 + 9*a^17*b*x + a^18
) - log(b*x + a)/a^10 + log(x)/a^10

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{10}} + \frac {\log \left ({\left | x \right |}\right )}{a^{10}} + \frac {2520 \, a b^{8} x^{8} + 21420 \, a^{2} b^{7} x^{7} + 80220 \, a^{3} b^{6} x^{6} + 173250 \, a^{4} b^{5} x^{5} + 236754 \, a^{5} b^{4} x^{4} + 210756 \, a^{6} b^{3} x^{3} + 120564 \, a^{7} b^{2} x^{2} + 41481 \, a^{8} b x + 7129 \, a^{9}}{2520 \, {\left (b x + a\right )}^{9} a^{10}} \]

[In]

integrate(1/x/(b*x+a)^10,x, algorithm="giac")

[Out]

-log(abs(b*x + a))/a^10 + log(abs(x))/a^10 + 1/2520*(2520*a*b^8*x^8 + 21420*a^2*b^7*x^7 + 80220*a^3*b^6*x^6 +
173250*a^4*b^5*x^5 + 236754*a^5*b^4*x^4 + 210756*a^6*b^3*x^3 + 120564*a^7*b^2*x^2 + 41481*a^8*b*x + 7129*a^9)/
((b*x + a)^9*a^10)

Mupad [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x (a+b x)^{10}} \, dx=\frac {1}{9\,a\,{\left (a+b\,x\right )}^9}-\frac {\ln \left (\frac {a+b\,x}{x}\right )-\frac {14\,b^2\,x^2}{{\left (a+b\,x\right )}^2}+\frac {56\,b^3\,x^3}{3\,{\left (a+b\,x\right )}^3}-\frac {35\,b^4\,x^4}{2\,{\left (a+b\,x\right )}^4}+\frac {56\,b^5\,x^5}{5\,{\left (a+b\,x\right )}^5}-\frac {14\,b^6\,x^6}{3\,{\left (a+b\,x\right )}^6}+\frac {8\,b^7\,x^7}{7\,{\left (a+b\,x\right )}^7}-\frac {b^8\,x^8}{8\,{\left (a+b\,x\right )}^8}+\frac {8\,b\,x}{a+b\,x}}{a^{10}} \]

[In]

int(1/(x*(a + b*x)^10),x)

[Out]

1/(9*a*(a + b*x)^9) - (log((a + b*x)/x) - (14*b^2*x^2)/(a + b*x)^2 + (56*b^3*x^3)/(3*(a + b*x)^3) - (35*b^4*x^
4)/(2*(a + b*x)^4) + (56*b^5*x^5)/(5*(a + b*x)^5) - (14*b^6*x^6)/(3*(a + b*x)^6) + (8*b^7*x^7)/(7*(a + b*x)^7)
- (b^8*x^8)/(8*(a + b*x)^8) + (8*b*x)/(a + b*x))/a^10