3.253 \(\int \frac{1}{x^{10} (a+b x)^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{4 b^7}{a^9 x^2}-\frac{7 b^6}{3 a^8 x^3}+\frac{3 b^5}{2 a^7 x^4}-\frac{b^4}{a^6 x^5}+\frac{2 b^3}{3 a^5 x^6}-\frac{3 b^2}{7 a^4 x^7}-\frac{b^9}{a^{10} (a+b x)}-\frac{9 b^8}{a^{10} x}-\frac{10 b^9 \log (x)}{a^{11}}+\frac{10 b^9 \log (a+b x)}{a^{11}}+\frac{b}{4 a^3 x^8}-\frac{1}{9 a^2 x^9} \]

[Out]

-1/(9*a^2*x^9) + b/(4*a^3*x^8) - (3*b^2)/(7*a^4*x^7) + (2*b^3)/(3*a^5*x^6) - b^4/(a^6*x^5) + (3*b^5)/(2*a^7*x^
4) - (7*b^6)/(3*a^8*x^3) + (4*b^7)/(a^9*x^2) - (9*b^8)/(a^10*x) - b^9/(a^10*(a + b*x)) - (10*b^9*Log[x])/a^11
+ (10*b^9*Log[a + b*x])/a^11

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Rubi [A]  time = 0.0896469, antiderivative size = 146, 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} \[ \frac{4 b^7}{a^9 x^2}-\frac{7 b^6}{3 a^8 x^3}+\frac{3 b^5}{2 a^7 x^4}-\frac{b^4}{a^6 x^5}+\frac{2 b^3}{3 a^5 x^6}-\frac{3 b^2}{7 a^4 x^7}-\frac{b^9}{a^{10} (a+b x)}-\frac{9 b^8}{a^{10} x}-\frac{10 b^9 \log (x)}{a^{11}}+\frac{10 b^9 \log (a+b x)}{a^{11}}+\frac{b}{4 a^3 x^8}-\frac{1}{9 a^2 x^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a + b*x)^2),x]

[Out]

-1/(9*a^2*x^9) + b/(4*a^3*x^8) - (3*b^2)/(7*a^4*x^7) + (2*b^3)/(3*a^5*x^6) - b^4/(a^6*x^5) + (3*b^5)/(2*a^7*x^
4) - (7*b^6)/(3*a^8*x^3) + (4*b^7)/(a^9*x^2) - (9*b^8)/(a^10*x) - b^9/(a^10*(a + b*x)) - (10*b^9*Log[x])/a^11
+ (10*b^9*Log[a + b*x])/a^11

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)^2} \, dx &=\int \left (\frac{1}{a^2 x^{10}}-\frac{2 b}{a^3 x^9}+\frac{3 b^2}{a^4 x^8}-\frac{4 b^3}{a^5 x^7}+\frac{5 b^4}{a^6 x^6}-\frac{6 b^5}{a^7 x^5}+\frac{7 b^6}{a^8 x^4}-\frac{8 b^7}{a^9 x^3}+\frac{9 b^8}{a^{10} x^2}-\frac{10 b^9}{a^{11} x}+\frac{b^{10}}{a^{10} (a+b x)^2}+\frac{10 b^{10}}{a^{11} (a+b x)}\right ) \, dx\\ &=-\frac{1}{9 a^2 x^9}+\frac{b}{4 a^3 x^8}-\frac{3 b^2}{7 a^4 x^7}+\frac{2 b^3}{3 a^5 x^6}-\frac{b^4}{a^6 x^5}+\frac{3 b^5}{2 a^7 x^4}-\frac{7 b^6}{3 a^8 x^3}+\frac{4 b^7}{a^9 x^2}-\frac{9 b^8}{a^{10} x}-\frac{b^9}{a^{10} (a+b x)}-\frac{10 b^9 \log (x)}{a^{11}}+\frac{10 b^9 \log (a+b x)}{a^{11}}\\ \end{align*}

Mathematica [A]  time = 0.150389, size = 134, normalized size = 0.92 \[ -\frac{\frac{a \left (45 a^7 b^2 x^2-60 a^6 b^3 x^3+84 a^5 b^4 x^4-126 a^4 b^5 x^5+210 a^3 b^6 x^6-420 a^2 b^7 x^7-35 a^8 b x+28 a^9+1260 a b^8 x^8+2520 b^9 x^9\right )}{x^9 (a+b x)}-2520 b^9 \log (a+b x)+2520 b^9 \log (x)}{252 a^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(a + b*x)^2),x]

[Out]

-((a*(28*a^9 - 35*a^8*b*x + 45*a^7*b^2*x^2 - 60*a^6*b^3*x^3 + 84*a^5*b^4*x^4 - 126*a^4*b^5*x^5 + 210*a^3*b^6*x
^6 - 420*a^2*b^7*x^7 + 1260*a*b^8*x^8 + 2520*b^9*x^9))/(x^9*(a + b*x)) + 2520*b^9*Log[x] - 2520*b^9*Log[a + b*
x])/(252*a^11)

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Maple [A]  time = 0.013, size = 135, normalized size = 0.9 \begin{align*} -{\frac{1}{9\,{a}^{2}{x}^{9}}}+{\frac{b}{4\,{a}^{3}{x}^{8}}}-{\frac{3\,{b}^{2}}{7\,{a}^{4}{x}^{7}}}+{\frac{2\,{b}^{3}}{3\,{a}^{5}{x}^{6}}}-{\frac{{b}^{4}}{{a}^{6}{x}^{5}}}+{\frac{3\,{b}^{5}}{2\,{a}^{7}{x}^{4}}}-{\frac{7\,{b}^{6}}{3\,{a}^{8}{x}^{3}}}+4\,{\frac{{b}^{7}}{{a}^{9}{x}^{2}}}-9\,{\frac{{b}^{8}}{{a}^{10}x}}-{\frac{{b}^{9}}{{a}^{10} \left ( bx+a \right ) }}-10\,{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{11}}}+10\,{\frac{{b}^{9}\ln \left ( bx+a \right ) }{{a}^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(b*x+a)^2,x)

[Out]

-1/9/a^2/x^9+1/4*b/a^3/x^8-3/7*b^2/a^4/x^7+2/3*b^3/a^5/x^6-b^4/a^6/x^5+3/2*b^5/a^7/x^4-7/3*b^6/a^8/x^3+4*b^7/a
^9/x^2-9*b^8/a^10/x-b^9/a^10/(b*x+a)-10*b^9*ln(x)/a^11+10*b^9*ln(b*x+a)/a^11

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Maxima [A]  time = 1.1136, size = 190, normalized size = 1.3 \begin{align*} -\frac{2520 \, b^{9} x^{9} + 1260 \, a b^{8} x^{8} - 420 \, a^{2} b^{7} x^{7} + 210 \, a^{3} b^{6} x^{6} - 126 \, a^{4} b^{5} x^{5} + 84 \, a^{5} b^{4} x^{4} - 60 \, a^{6} b^{3} x^{3} + 45 \, a^{7} b^{2} x^{2} - 35 \, a^{8} b x + 28 \, a^{9}}{252 \,{\left (a^{10} b x^{10} + a^{11} x^{9}\right )}} + \frac{10 \, b^{9} \log \left (b x + a\right )}{a^{11}} - \frac{10 \, b^{9} \log \left (x\right )}{a^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/252*(2520*b^9*x^9 + 1260*a*b^8*x^8 - 420*a^2*b^7*x^7 + 210*a^3*b^6*x^6 - 126*a^4*b^5*x^5 + 84*a^5*b^4*x^4 -
60*a^6*b^3*x^3 + 45*a^7*b^2*x^2 - 35*a^8*b*x + 28*a^9)/(a^10*b*x^10 + a^11*x^9) + 10*b^9*log(b*x + a)/a^11 -
10*b^9*log(x)/a^11

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Fricas [A]  time = 1.79665, size = 377, normalized size = 2.58 \begin{align*} -\frac{2520 \, a b^{9} x^{9} + 1260 \, a^{2} b^{8} x^{8} - 420 \, a^{3} b^{7} x^{7} + 210 \, a^{4} b^{6} x^{6} - 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} - 60 \, a^{7} b^{3} x^{3} + 45 \, a^{8} b^{2} x^{2} - 35 \, a^{9} b x + 28 \, a^{10} - 2520 \,{\left (b^{10} x^{10} + a b^{9} x^{9}\right )} \log \left (b x + a\right ) + 2520 \,{\left (b^{10} x^{10} + a b^{9} x^{9}\right )} \log \left (x\right )}{252 \,{\left (a^{11} b x^{10} + a^{12} x^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/252*(2520*a*b^9*x^9 + 1260*a^2*b^8*x^8 - 420*a^3*b^7*x^7 + 210*a^4*b^6*x^6 - 126*a^5*b^5*x^5 + 84*a^6*b^4*x
^4 - 60*a^7*b^3*x^3 + 45*a^8*b^2*x^2 - 35*a^9*b*x + 28*a^10 - 2520*(b^10*x^10 + a*b^9*x^9)*log(b*x + a) + 2520
*(b^10*x^10 + a*b^9*x^9)*log(x))/(a^11*b*x^10 + a^12*x^9)

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Sympy [A]  time = 1.25739, size = 139, normalized size = 0.95 \begin{align*} - \frac{28 a^{9} - 35 a^{8} b x + 45 a^{7} b^{2} x^{2} - 60 a^{6} b^{3} x^{3} + 84 a^{5} b^{4} x^{4} - 126 a^{4} b^{5} x^{5} + 210 a^{3} b^{6} x^{6} - 420 a^{2} b^{7} x^{7} + 1260 a b^{8} x^{8} + 2520 b^{9} x^{9}}{252 a^{11} x^{9} + 252 a^{10} b x^{10}} + \frac{10 b^{9} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(b*x+a)**2,x)

[Out]

-(28*a**9 - 35*a**8*b*x + 45*a**7*b**2*x**2 - 60*a**6*b**3*x**3 + 84*a**5*b**4*x**4 - 126*a**4*b**5*x**5 + 210
*a**3*b**6*x**6 - 420*a**2*b**7*x**7 + 1260*a*b**8*x**8 + 2520*b**9*x**9)/(252*a**11*x**9 + 252*a**10*b*x**10)
+ 10*b**9*(-log(x) + log(a/b + x))/a**11

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Giac [A]  time = 1.23207, size = 243, normalized size = 1.66 \begin{align*} -\frac{10 \, b^{9} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{11}} - \frac{b^{9}}{{\left (b x + a\right )} a^{10}} - \frac{\frac{41481 \, a b^{9}}{b x + a} - \frac{155844 \, a^{2} b^{9}}{{\left (b x + a\right )}^{2}} + \frac{337176 \, a^{3} b^{9}}{{\left (b x + a\right )}^{3}} - \frac{460404 \, a^{4} b^{9}}{{\left (b x + a\right )}^{4}} + \frac{407484 \, a^{5} b^{9}}{{\left (b x + a\right )}^{5}} - \frac{229320 \, a^{6} b^{9}}{{\left (b x + a\right )}^{6}} + \frac{75600 \, a^{7} b^{9}}{{\left (b x + a\right )}^{7}} - \frac{11340 \, a^{8} b^{9}}{{\left (b x + a\right )}^{8}} - 4861 \, b^{9}}{252 \, a^{11}{\left (\frac{a}{b x + a} - 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^2,x, algorithm="giac")

[Out]

-10*b^9*log(abs(-a/(b*x + a) + 1))/a^11 - b^9/((b*x + a)*a^10) - 1/252*(41481*a*b^9/(b*x + a) - 155844*a^2*b^9
/(b*x + a)^2 + 337176*a^3*b^9/(b*x + a)^3 - 460404*a^4*b^9/(b*x + a)^4 + 407484*a^5*b^9/(b*x + a)^5 - 229320*a
^6*b^9/(b*x + a)^6 + 75600*a^7*b^9/(b*x + a)^7 - 11340*a^8*b^9/(b*x + a)^8 - 4861*b^9)/(a^11*(a/(b*x + a) - 1)
^9)