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

Optimal. Leaf size=134 \[ -\frac{b^9 \log (x)}{a^{10}}+\frac{b^9 \log (a+b x)}{a^{10}}-\frac{b^8}{a^9 x}+\frac{b^7}{2 a^8 x^2}-\frac{b^6}{3 a^7 x^3}+\frac{b^5}{4 a^6 x^4}-\frac{b^4}{5 a^5 x^5}+\frac{b^3}{6 a^4 x^6}-\frac{b^2}{7 a^3 x^7}+\frac{b}{8 a^2 x^8}-\frac{1}{9 a x^9} \]

[Out]

-1/(9*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*Lo
g[x])/a^10 + (b^9*Log[a + b*x])/a^10

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Rubi [A]  time = 0.130101, antiderivative size = 134, 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 \[ -\frac{b^9 \log (x)}{a^{10}}+\frac{b^9 \log (a+b x)}{a^{10}}-\frac{b^8}{a^9 x}+\frac{b^7}{2 a^8 x^2}-\frac{b^6}{3 a^7 x^3}+\frac{b^5}{4 a^6 x^4}-\frac{b^4}{5 a^5 x^5}+\frac{b^3}{6 a^4 x^6}-\frac{b^2}{7 a^3 x^7}+\frac{b}{8 a^2 x^8}-\frac{1}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*(a + b*x)),x]

[Out]

-1/(9*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*Lo
g[x])/a^10 + (b^9*Log[a + b*x])/a^10

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Rubi in Sympy [A]  time = 20.8213, size = 121, normalized size = 0.9 \[ - \frac{1}{9 a x^{9}} + \frac{b}{8 a^{2} x^{8}} - \frac{b^{2}}{7 a^{3} x^{7}} + \frac{b^{3}}{6 a^{4} x^{6}} - \frac{b^{4}}{5 a^{5} x^{5}} + \frac{b^{5}}{4 a^{6} x^{4}} - \frac{b^{6}}{3 a^{7} x^{3}} + \frac{b^{7}}{2 a^{8} x^{2}} - \frac{b^{8}}{a^{9} x} - \frac{b^{9} \log{\left (x \right )}}{a^{10}} + \frac{b^{9} \log{\left (a + b x \right )}}{a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(b*x+a),x)

[Out]

-1/(9*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

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Mathematica [A]  time = 0.00893041, size = 134, normalized size = 1. \[ -\frac{b^9 \log (x)}{a^{10}}+\frac{b^9 \log (a+b x)}{a^{10}}-\frac{b^8}{a^9 x}+\frac{b^7}{2 a^8 x^2}-\frac{b^6}{3 a^7 x^3}+\frac{b^5}{4 a^6 x^4}-\frac{b^4}{5 a^5 x^5}+\frac{b^3}{6 a^4 x^6}-\frac{b^2}{7 a^3 x^7}+\frac{b}{8 a^2 x^8}-\frac{1}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*(a + b*x)),x]

[Out]

-1/(9*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*Lo
g[x])/a^10 + (b^9*Log[a + b*x])/a^10

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Maple [A]  time = 0.016, size = 119, normalized size = 0.9 \[ -{\frac{1}{9\,a{x}^{9}}}+{\frac{b}{8\,{a}^{2}{x}^{8}}}-{\frac{{b}^{2}}{7\,{a}^{3}{x}^{7}}}+{\frac{{b}^{3}}{6\,{a}^{4}{x}^{6}}}-{\frac{{b}^{4}}{5\,{a}^{5}{x}^{5}}}+{\frac{{b}^{5}}{4\,{a}^{6}{x}^{4}}}-{\frac{{b}^{6}}{3\,{a}^{7}{x}^{3}}}+{\frac{{b}^{7}}{2\,{a}^{8}{x}^{2}}}-{\frac{{b}^{8}}{{a}^{9}x}}-{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{10}}}+{\frac{{b}^{9}\ln \left ( bx+a \right ) }{{a}^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(b*x+a),x)

[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

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Maxima [A]  time = 1.35016, size = 158, normalized size = 1.18 \[ \frac{b^{9} \log \left (b x + a\right )}{a^{10}} - \frac{b^{9} \log \left (x\right )}{a^{10}} - \frac{2520 \, b^{8} x^{8} - 1260 \, a b^{7} x^{7} + 840 \, a^{2} b^{6} x^{6} - 630 \, a^{3} b^{5} x^{5} + 504 \, a^{4} b^{4} x^{4} - 420 \, a^{5} b^{3} x^{3} + 360 \, a^{6} b^{2} x^{2} - 315 \, a^{7} b x + 280 \, a^{8}}{2520 \, a^{9} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*x^10),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)

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Fricas [A]  time = 0.21854, size = 162, normalized size = 1.21 \[ \frac{2520 \, b^{9} x^{9} \log \left (b x + a\right ) - 2520 \, b^{9} x^{9} \log \left (x\right ) - 2520 \, a b^{8} x^{8} + 1260 \, a^{2} b^{7} x^{7} - 840 \, a^{3} b^{6} x^{6} + 630 \, a^{4} b^{5} x^{5} - 504 \, a^{5} b^{4} x^{4} + 420 \, a^{6} b^{3} x^{3} - 360 \, a^{7} b^{2} x^{2} + 315 \, a^{8} b x - 280 \, a^{9}}{2520 \, a^{10} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*x^10),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)

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Sympy [A]  time = 2.8164, size = 116, normalized size = 0.87 \[ - \frac{280 a^{8} - 315 a^{7} b x + 360 a^{6} b^{2} x^{2} - 420 a^{5} b^{3} x^{3} + 504 a^{4} b^{4} x^{4} - 630 a^{3} b^{5} x^{5} + 840 a^{2} b^{6} x^{6} - 1260 a b^{7} x^{7} + 2520 b^{8} x^{8}}{2520 a^{9} x^{9}} + \frac{b^{9} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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GIAC/XCAS [A]  time = 0.204695, size = 165, normalized size = 1.23 \[ \frac{b^{9}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{10}} - \frac{b^{9}{\rm ln}\left ({\left | x \right |}\right )}{a^{10}} - \frac{2520 \, a b^{8} x^{8} - 1260 \, a^{2} b^{7} x^{7} + 840 \, a^{3} b^{6} x^{6} - 630 \, a^{4} b^{5} x^{5} + 504 \, a^{5} b^{4} x^{4} - 420 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 315 \, a^{8} b x + 280 \, a^{9}}{2520 \, a^{10} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^9*ln(abs(b*x + a))/a^10 - b^9*ln(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)