\(\int \frac {1}{x^{10} (a+b x)} \, dx\) [252]

Optimal result

Integrand size = 11, antiderivative size = 134 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=-\frac {1}{9 a x^9}+\frac {b}{8 a^2 x^8}-\frac {b^2}{7 a^3 x^7}+\frac {b^3}{6 a^4 x^6}-\frac {b^4}{5 a^5 x^5}+\frac {b^5}{4 a^6 x^4}-\frac {b^6}{3 a^7 x^3}+\frac {b^7}{2 a^8 x^2}-\frac {b^8}{a^9 x}-\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 134, 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} \[ \int \frac {1}{x^{10} (a+b x)} \, dx=-\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}}-\frac {b^8}{a^9 x}+\frac {b^7}{2 a^8 x^2}-\frac {b^6}{3 a^7 x^3}+\frac {b^5}{4 a^6 x^4}-\frac {b^4}{5 a^5 x^5}+\frac {b^3}{6 a^4 x^6}-\frac {b^2}{7 a^3 x^7}+\frac {b}{8 a^2 x^8}-\frac {1}{9 a x^9} \]

[In]

Int[1/(x^10*(a + b*x)),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=-\frac {1}{9 a x^9}+\frac {b}{8 a^2 x^8}-\frac {b^2}{7 a^3 x^7}+\frac {b^3}{6 a^4 x^6}-\frac {b^4}{5 a^5 x^5}+\frac {b^5}{4 a^6 x^4}-\frac {b^6}{3 a^7 x^3}+\frac {b^7}{2 a^8 x^2}-\frac {b^8}{a^9 x}-\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}} \]

[In]

Integrate[1/(x^10*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {1}{9 a \,x^{9}}+\frac {b}{8 a^{2} x^{8}}-\frac {b^{2}}{7 a^{3} x^{7}}+\frac {b^{3}}{6 a^{4} x^{6}}-\frac {b^{4}}{5 a^{5} x^{5}}+\frac {b^{5}}{4 a^{6} x^{4}}-\frac {b^{6}}{3 a^{7} x^{3}}+\frac {b^{7}}{2 a^{8} x^{2}}-\frac {b^{8}}{a^{9} x}-\frac {b^{9} \ln \left (x \right )}{a^{10}}+\frac {b^{9} \ln \left (b x +a \right )}{a^{10}}\)	\(119\)
norman	\(\frac {-\frac {1}{9 a}+\frac {b x}{8 a^{2}}-\frac {b^{2} x^{2}}{7 a^{3}}+\frac {b^{3} x^{3}}{6 a^{4}}-\frac {b^{4} x^{4}}{5 a^{5}}+\frac {b^{5} x^{5}}{4 a^{6}}-\frac {b^{6} x^{6}}{3 a^{7}}+\frac {b^{7} x^{7}}{2 a^{8}}-\frac {b^{8} x^{8}}{a^{9}}}{x^{9}}+\frac {b^{9} \ln \left (b x +a \right )}{a^{10}}-\frac {b^{9} \ln \left (x \right )}{a^{10}}\)	\(119\)
parallelrisch	\(-\frac {2520 \ln \left (x \right ) x^{9} b^{9}-2520 \ln \left (b x +a \right ) x^{9} b^{9}+2520 a \,x^{8} b^{8}-1260 a^{2} x^{7} b^{7}+840 x^{6} a^{3} b^{6}-630 a^{4} x^{5} b^{5}+504 a^{5} b^{4} x^{4}-420 a^{6} b^{3} x^{3}+360 a^{7} b^{2} x^{2}-315 a^{8} b x +280 a^{9}}{2520 a^{10} x^{9}}\)	\(121\)
risch	\(\frac {-\frac {1}{9 a}+\frac {b x}{8 a^{2}}-\frac {b^{2} x^{2}}{7 a^{3}}+\frac {b^{3} x^{3}}{6 a^{4}}-\frac {b^{4} x^{4}}{5 a^{5}}+\frac {b^{5} x^{5}}{4 a^{6}}-\frac {b^{6} x^{6}}{3 a^{7}}+\frac {b^{7} x^{7}}{2 a^{8}}-\frac {b^{8} x^{8}}{a^{9}}}{x^{9}}-\frac {b^{9} \ln \left (x \right )}{a^{10}}+\frac {b^{9} \ln \left (-b x -a \right )}{a^{10}}\)	\(122\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=\frac {2520 \, b^{9} x^{9} \log \left (b x + a\right ) - 2520 \, b^{9} x^{9} \log \left (x\right ) - 2520 \, a b^{8} x^{8} + 1260 \, a^{2} b^{7} x^{7} - 840 \, a^{3} b^{6} x^{6} + 630 \, a^{4} b^{5} x^{5} - 504 \, a^{5} b^{4} x^{4} + 420 \, a^{6} b^{3} x^{3} - 360 \, a^{7} b^{2} x^{2} + 315 \, a^{8} b x - 280 \, a^{9}}{2520 \, a^{10} x^{9}} \]

[In]

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

[Out]

1/2520*(2520*b^9*x^9*log(b*x + a) - 2520*b^9*x^9*log(x) - 2520*a*b^8*x^8 + 1260*a^2*b^7*x^7 - 840*a^3*b^6*x^6
+ 630*a^4*b^5*x^5 - 504*a^5*b^4*x^4 + 420*a^6*b^3*x^3 - 360*a^7*b^2*x^2 + 315*a^8*b*x - 280*a^9)/(a^10*x^9)

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=\frac {- 280 a^{8} + 315 a^{7} b x - 360 a^{6} b^{2} x^{2} + 420 a^{5} b^{3} x^{3} - 504 a^{4} b^{4} x^{4} + 630 a^{3} b^{5} x^{5} - 840 a^{2} b^{6} x^{6} + 1260 a b^{7} x^{7} - 2520 b^{8} x^{8}}{2520 a^{9} x^{9}} + \frac {b^{9} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{10}} \]

[In]

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

[Out]

(-280*a**8 + 315*a**7*b*x - 360*a**6*b**2*x**2 + 420*a**5*b**3*x**3 - 504*a**4*b**4*x**4 + 630*a**3*b**5*x**5
- 840*a**2*b**6*x**6 + 1260*a*b**7*x**7 - 2520*b**8*x**8)/(2520*a**9*x**9) + b**9*(-log(x) + log(a/b + x))/a**
10

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=\frac {b^{9} \log \left (b x + a\right )}{a^{10}} - \frac {b^{9} \log \left (x\right )}{a^{10}} - \frac {2520 \, b^{8} x^{8} - 1260 \, a b^{7} x^{7} + 840 \, a^{2} b^{6} x^{6} - 630 \, a^{3} b^{5} x^{5} + 504 \, a^{4} b^{4} x^{4} - 420 \, a^{5} b^{3} x^{3} + 360 \, a^{6} b^{2} x^{2} - 315 \, a^{7} b x + 280 \, a^{8}}{2520 \, a^{9} x^{9}} \]

[In]

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

[Out]

b^9*log(b*x + a)/a^10 - b^9*log(x)/a^10 - 1/2520*(2520*b^8*x^8 - 1260*a*b^7*x^7 + 840*a^2*b^6*x^6 - 630*a^3*b^
5*x^5 + 504*a^4*b^4*x^4 - 420*a^5*b^3*x^3 + 360*a^6*b^2*x^2 - 315*a^7*b*x + 280*a^8)/(a^9*x^9)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=\frac {b^{9} \log \left ({\left | b x + a \right |}\right )}{a^{10}} - \frac {b^{9} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac {2520 \, a b^{8} x^{8} - 1260 \, a^{2} b^{7} x^{7} + 840 \, a^{3} b^{6} x^{6} - 630 \, a^{4} b^{5} x^{5} + 504 \, a^{5} b^{4} x^{4} - 420 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 315 \, a^{8} b x + 280 \, a^{9}}{2520 \, a^{10} x^{9}} \]

[In]

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

[Out]

b^9*log(abs(b*x + a))/a^10 - b^9*log(abs(x))/a^10 - 1/2520*(2520*a*b^8*x^8 - 1260*a^2*b^7*x^7 + 840*a^3*b^6*x^
6 - 630*a^4*b^5*x^5 + 504*a^5*b^4*x^4 - 420*a^6*b^3*x^3 + 360*a^7*b^2*x^2 - 315*a^8*b*x + 280*a^9)/(a^10*x^9)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^{10} (a+b x)} \, dx=-\frac {280\,a^9+2520\,a\,b^8\,x^8-5040\,b^9\,x^9\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )+360\,a^7\,b^2\,x^2-420\,a^6\,b^3\,x^3+504\,a^5\,b^4\,x^4-630\,a^4\,b^5\,x^5+840\,a^3\,b^6\,x^6-1260\,a^2\,b^7\,x^7-315\,a^8\,b\,x}{2520\,a^{10}\,x^9} \]

[In]

int(1/(x^10*(a + b*x)),x)

[Out]

-(280*a^9 + 2520*a*b^8*x^8 - 5040*b^9*x^9*atanh((2*b*x)/a + 1) + 360*a^7*b^2*x^2 - 420*a^6*b^3*x^3 + 504*a^5*b
^4*x^4 - 630*a^4*b^5*x^5 + 840*a^3*b^6*x^6 - 1260*a^2*b^7*x^7 - 315*a^8*b*x)/(2520*a^10*x^9)