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

Optimal. Leaf size=141 \[ \frac{1}{a^9 (a+b x)}+\frac{1}{2 a^8 (a+b x)^2}+\frac{1}{3 a^7 (a+b x)^3}+\frac{1}{4 a^6 (a+b x)^4}+\frac{1}{5 a^5 (a+b x)^5}+\frac{1}{6 a^4 (a+b x)^6}+\frac{1}{7 a^3 (a+b x)^7}+\frac{1}{8 a^2 (a+b x)^8}-\frac{\log (a+b x)}{a^{10}}+\frac{\log (x)}{a^{10}}+\frac{1}{9 a (a+b x)^9} \]

[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

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Rubi [A]  time = 0.0748266, antiderivative size = 141, 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} \[ \frac{1}{a^9 (a+b x)}+\frac{1}{2 a^8 (a+b x)^2}+\frac{1}{3 a^7 (a+b x)^3}+\frac{1}{4 a^6 (a+b x)^4}+\frac{1}{5 a^5 (a+b x)^5}+\frac{1}{6 a^4 (a+b x)^6}+\frac{1}{7 a^3 (a+b x)^7}+\frac{1}{8 a^2 (a+b x)^8}-\frac{\log (a+b x)}{a^{10}}+\frac{\log (x)}{a^{10}}+\frac{1}{9 a (a+b x)^9} \]

Antiderivative was successfully verified.

[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 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 (a+b x)^{10}} \, dx &=\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]  time = 0.148634, size = 127, normalized size = 0.9 \[ \frac{315 a^7 (a+b x)+360 a^6 (a+b x)^2+420 a^5 (a+b x)^3+504 a^4 (a+b x)^4+630 a^3 (a+b x)^5+840 a^2 (a+b x)^6+280 a^8+1260 a (a+b x)^7+2520 (a+b x)^8}{2520 a^9 (a+b x)^9}-\frac{\log (a+b x)}{a^{10}}+\frac{\log (x)}{a^{10}} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.011, size = 126, normalized size = 0.9 \begin{align*}{\frac{1}{9\,a \left ( bx+a \right ) ^{9}}}+{\frac{1}{8\,{a}^{2} \left ( bx+a \right ) ^{8}}}+{\frac{1}{7\,{a}^{3} \left ( bx+a \right ) ^{7}}}+{\frac{1}{6\,{a}^{4} \left ( bx+a \right ) ^{6}}}+{\frac{1}{5\,{a}^{5} \left ( bx+a \right ) ^{5}}}+{\frac{1}{4\,{a}^{6} \left ( bx+a \right ) ^{4}}}+{\frac{1}{3\,{a}^{7} \left ( bx+a \right ) ^{3}}}+{\frac{1}{2\,{a}^{8} \left ( bx+a \right ) ^{2}}}+{\frac{1}{{a}^{9} \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{10}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.13768, size = 277, normalized size = 1.96 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.57749, size = 896, normalized size = 6.35 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 2.11026, size = 212, normalized size = 1.5 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.19564, size = 162, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)