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

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

[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*Log[x])/a^10 + (b^9*Log[a + b*x])/a^10

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Rubi [A]  time = 0.0608035, 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, Rules used = {44} \[ \frac{b^7}{2 a^8 x^2}-\frac{b^6}{3 a^7 x^3}+\frac{b^5}{4 a^6 x^4}-\frac{b^4}{5 a^5 x^5}+\frac{b^3}{6 a^4 x^6}-\frac{b^2}{7 a^3 x^7}-\frac{b^8}{a^9 x}-\frac{b^9 \log (x)}{a^{10}}+\frac{b^9 \log (a+b x)}{a^{10}}+\frac{b}{8 a^2 x^8}-\frac{1}{9 a x^9} \]

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*Log[x])/a^10 + (b^9*Log[a + b*x])/a^10

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^{10} (a+b x)} \, dx &=\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]  time = 0.0060532, size = 134, normalized size = 1. \[ \frac{b^7}{2 a^8 x^2}-\frac{b^6}{3 a^7 x^3}+\frac{b^5}{4 a^6 x^4}-\frac{b^4}{5 a^5 x^5}+\frac{b^3}{6 a^4 x^6}-\frac{b^2}{7 a^3 x^7}-\frac{b^8}{a^9 x}-\frac{b^9 \log (x)}{a^{10}}+\frac{b^9 \log (a+b x)}{a^{10}}+\frac{b}{8 a^2 x^8}-\frac{1}{9 a x^9} \]

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*Log[x])/a^10 + (b^9*Log[a + b*x])/a^10

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Maple [A]  time = 0.01, size = 119, normalized size = 0.9 \begin{align*} -{\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}}} \end{align*}

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 = 1.0785, size = 158, normalized size = 1.18 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.78598, size = 296, normalized size = 2.21 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 1.00563, size = 116, normalized size = 0.87 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.19382, size = 165, normalized size = 1.23 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)