3.254 \(\int \frac{1}{x^{10} (a+b x)^3} \, dx\)

Optimal. Leaf size=163 \[ \frac{18 b^7}{a^{10} x^2}-\frac{28 b^6}{3 a^9 x^3}+\frac{21 b^5}{4 a^8 x^4}-\frac{3 b^4}{a^7 x^5}+\frac{5 b^3}{3 a^6 x^6}-\frac{6 b^2}{7 a^5 x^7}-\frac{10 b^9}{a^{11} (a+b x)}-\frac{b^9}{2 a^{10} (a+b x)^2}-\frac{45 b^8}{a^{11} x}-\frac{55 b^9 \log (x)}{a^{12}}+\frac{55 b^9 \log (a+b x)}{a^{12}}+\frac{3 b}{8 a^4 x^8}-\frac{1}{9 a^3 x^9} \]

[Out]

-1/(9*a^3*x^9) + (3*b)/(8*a^4*x^8) - (6*b^2)/(7*a^5*x^7) + (5*b^3)/(3*a^6*x^6) - (3*b^4)/(a^7*x^5) + (21*b^5)/
(4*a^8*x^4) - (28*b^6)/(3*a^9*x^3) + (18*b^7)/(a^10*x^2) - (45*b^8)/(a^11*x) - b^9/(2*a^10*(a + b*x)^2) - (10*
b^9)/(a^11*(a + b*x)) - (55*b^9*Log[x])/a^12 + (55*b^9*Log[a + b*x])/a^12

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Rubi [A]  time = 0.108238, antiderivative size = 163, 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} \[ \frac{18 b^7}{a^{10} x^2}-\frac{28 b^6}{3 a^9 x^3}+\frac{21 b^5}{4 a^8 x^4}-\frac{3 b^4}{a^7 x^5}+\frac{5 b^3}{3 a^6 x^6}-\frac{6 b^2}{7 a^5 x^7}-\frac{10 b^9}{a^{11} (a+b x)}-\frac{b^9}{2 a^{10} (a+b x)^2}-\frac{45 b^8}{a^{11} x}-\frac{55 b^9 \log (x)}{a^{12}}+\frac{55 b^9 \log (a+b x)}{a^{12}}+\frac{3 b}{8 a^4 x^8}-\frac{1}{9 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a + b*x)^3),x]

[Out]

-1/(9*a^3*x^9) + (3*b)/(8*a^4*x^8) - (6*b^2)/(7*a^5*x^7) + (5*b^3)/(3*a^6*x^6) - (3*b^4)/(a^7*x^5) + (21*b^5)/
(4*a^8*x^4) - (28*b^6)/(3*a^9*x^3) + (18*b^7)/(a^10*x^2) - (45*b^8)/(a^11*x) - b^9/(2*a^10*(a + b*x)^2) - (10*
b^9)/(a^11*(a + b*x)) - (55*b^9*Log[x])/a^12 + (55*b^9*Log[a + b*x])/a^12

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)^3} \, dx &=\int \left (\frac{1}{a^3 x^{10}}-\frac{3 b}{a^4 x^9}+\frac{6 b^2}{a^5 x^8}-\frac{10 b^3}{a^6 x^7}+\frac{15 b^4}{a^7 x^6}-\frac{21 b^5}{a^8 x^5}+\frac{28 b^6}{a^9 x^4}-\frac{36 b^7}{a^{10} x^3}+\frac{45 b^8}{a^{11} x^2}-\frac{55 b^9}{a^{12} x}+\frac{b^{10}}{a^{10} (a+b x)^3}+\frac{10 b^{10}}{a^{11} (a+b x)^2}+\frac{55 b^{10}}{a^{12} (a+b x)}\right ) \, dx\\ &=-\frac{1}{9 a^3 x^9}+\frac{3 b}{8 a^4 x^8}-\frac{6 b^2}{7 a^5 x^7}+\frac{5 b^3}{3 a^6 x^6}-\frac{3 b^4}{a^7 x^5}+\frac{21 b^5}{4 a^8 x^4}-\frac{28 b^6}{3 a^9 x^3}+\frac{18 b^7}{a^{10} x^2}-\frac{45 b^8}{a^{11} x}-\frac{b^9}{2 a^{10} (a+b x)^2}-\frac{10 b^9}{a^{11} (a+b x)}-\frac{55 b^9 \log (x)}{a^{12}}+\frac{55 b^9 \log (a+b x)}{a^{12}}\\ \end{align*}

Mathematica [A]  time = 0.139999, size = 145, normalized size = 0.89 \[ -\frac{\frac{a \left (110 a^8 b^2 x^2-165 a^7 b^3 x^3+264 a^6 b^4 x^4-462 a^5 b^5 x^5+924 a^4 b^6 x^6-2310 a^3 b^7 x^7+9240 a^2 b^8 x^8-77 a^9 b x+56 a^{10}+41580 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^9 (a+b x)^2}-27720 b^9 \log (a+b x)+27720 b^9 \log (x)}{504 a^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(a + b*x)^3),x]

[Out]

-((a*(56*a^10 - 77*a^9*b*x + 110*a^8*b^2*x^2 - 165*a^7*b^3*x^3 + 264*a^6*b^4*x^4 - 462*a^5*b^5*x^5 + 924*a^4*b
^6*x^6 - 2310*a^3*b^7*x^7 + 9240*a^2*b^8*x^8 + 41580*a*b^9*x^9 + 27720*b^10*x^10))/(x^9*(a + b*x)^2) + 27720*b
^9*Log[x] - 27720*b^9*Log[a + b*x])/(504*a^12)

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Maple [A]  time = 0.013, size = 150, normalized size = 0.9 \begin{align*} -{\frac{1}{9\,{a}^{3}{x}^{9}}}+{\frac{3\,b}{8\,{a}^{4}{x}^{8}}}-{\frac{6\,{b}^{2}}{7\,{a}^{5}{x}^{7}}}+{\frac{5\,{b}^{3}}{3\,{a}^{6}{x}^{6}}}-3\,{\frac{{b}^{4}}{{a}^{7}{x}^{5}}}+{\frac{21\,{b}^{5}}{4\,{a}^{8}{x}^{4}}}-{\frac{28\,{b}^{6}}{3\,{a}^{9}{x}^{3}}}+18\,{\frac{{b}^{7}}{{a}^{10}{x}^{2}}}-45\,{\frac{{b}^{8}}{{a}^{11}x}}-{\frac{{b}^{9}}{2\,{a}^{10} \left ( bx+a \right ) ^{2}}}-10\,{\frac{{b}^{9}}{{a}^{11} \left ( bx+a \right ) }}-55\,{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{12}}}+55\,{\frac{{b}^{9}\ln \left ( bx+a \right ) }{{a}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(b*x+a)^3,x)

[Out]

-1/9/a^3/x^9+3/8*b/a^4/x^8-6/7*b^2/a^5/x^7+5/3*b^3/a^6/x^6-3*b^4/a^7/x^5+21/4*b^5/a^8/x^4-28/3*b^6/a^9/x^3+18*
b^7/a^10/x^2-45*b^8/a^11/x-1/2*b^9/a^10/(b*x+a)^2-10*b^9/a^11/(b*x+a)-55*b^9*ln(x)/a^12+55*b^9*ln(b*x+a)/a^12

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Maxima [A]  time = 1.07396, size = 220, normalized size = 1.35 \begin{align*} -\frac{27720 \, b^{10} x^{10} + 41580 \, a b^{9} x^{9} + 9240 \, a^{2} b^{8} x^{8} - 2310 \, a^{3} b^{7} x^{7} + 924 \, a^{4} b^{6} x^{6} - 462 \, a^{5} b^{5} x^{5} + 264 \, a^{6} b^{4} x^{4} - 165 \, a^{7} b^{3} x^{3} + 110 \, a^{8} b^{2} x^{2} - 77 \, a^{9} b x + 56 \, a^{10}}{504 \,{\left (a^{11} b^{2} x^{11} + 2 \, a^{12} b x^{10} + a^{13} x^{9}\right )}} + \frac{55 \, b^{9} \log \left (b x + a\right )}{a^{12}} - \frac{55 \, b^{9} \log \left (x\right )}{a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/504*(27720*b^10*x^10 + 41580*a*b^9*x^9 + 9240*a^2*b^8*x^8 - 2310*a^3*b^7*x^7 + 924*a^4*b^6*x^6 - 462*a^5*b^
5*x^5 + 264*a^6*b^4*x^4 - 165*a^7*b^3*x^3 + 110*a^8*b^2*x^2 - 77*a^9*b*x + 56*a^10)/(a^11*b^2*x^11 + 2*a^12*b*
x^10 + a^13*x^9) + 55*b^9*log(b*x + a)/a^12 - 55*b^9*log(x)/a^12

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Fricas [A]  time = 1.92144, size = 490, normalized size = 3.01 \begin{align*} -\frac{27720 \, a b^{10} x^{10} + 41580 \, a^{2} b^{9} x^{9} + 9240 \, a^{3} b^{8} x^{8} - 2310 \, a^{4} b^{7} x^{7} + 924 \, a^{5} b^{6} x^{6} - 462 \, a^{6} b^{5} x^{5} + 264 \, a^{7} b^{4} x^{4} - 165 \, a^{8} b^{3} x^{3} + 110 \, a^{9} b^{2} x^{2} - 77 \, a^{10} b x + 56 \, a^{11} - 27720 \,{\left (b^{11} x^{11} + 2 \, a b^{10} x^{10} + a^{2} b^{9} x^{9}\right )} \log \left (b x + a\right ) + 27720 \,{\left (b^{11} x^{11} + 2 \, a b^{10} x^{10} + a^{2} b^{9} x^{9}\right )} \log \left (x\right )}{504 \,{\left (a^{12} b^{2} x^{11} + 2 \, a^{13} b x^{10} + a^{14} x^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/504*(27720*a*b^10*x^10 + 41580*a^2*b^9*x^9 + 9240*a^3*b^8*x^8 - 2310*a^4*b^7*x^7 + 924*a^5*b^6*x^6 - 462*a^
6*b^5*x^5 + 264*a^7*b^4*x^4 - 165*a^8*b^3*x^3 + 110*a^9*b^2*x^2 - 77*a^10*b*x + 56*a^11 - 27720*(b^11*x^11 + 2
*a*b^10*x^10 + a^2*b^9*x^9)*log(b*x + a) + 27720*(b^11*x^11 + 2*a*b^10*x^10 + a^2*b^9*x^9)*log(x))/(a^12*b^2*x
^11 + 2*a^13*b*x^10 + a^14*x^9)

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Sympy [A]  time = 1.71289, size = 163, normalized size = 1. \begin{align*} - \frac{56 a^{10} - 77 a^{9} b x + 110 a^{8} b^{2} x^{2} - 165 a^{7} b^{3} x^{3} + 264 a^{6} b^{4} x^{4} - 462 a^{5} b^{5} x^{5} + 924 a^{4} b^{6} x^{6} - 2310 a^{3} b^{7} x^{7} + 9240 a^{2} b^{8} x^{8} + 41580 a b^{9} x^{9} + 27720 b^{10} x^{10}}{504 a^{13} x^{9} + 1008 a^{12} b x^{10} + 504 a^{11} b^{2} x^{11}} + \frac{55 b^{9} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(b*x+a)**3,x)

[Out]

-(56*a**10 - 77*a**9*b*x + 110*a**8*b**2*x**2 - 165*a**7*b**3*x**3 + 264*a**6*b**4*x**4 - 462*a**5*b**5*x**5 +
924*a**4*b**6*x**6 - 2310*a**3*b**7*x**7 + 9240*a**2*b**8*x**8 + 41580*a*b**9*x**9 + 27720*b**10*x**10)/(504*
a**13*x**9 + 1008*a**12*b*x**10 + 504*a**11*b**2*x**11) + 55*b**9*(-log(x) + log(a/b + x))/a**12

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Giac [A]  time = 1.20668, size = 205, normalized size = 1.26 \begin{align*} \frac{55 \, b^{9} \log \left ({\left | b x + a \right |}\right )}{a^{12}} - \frac{55 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{12}} - \frac{27720 \, a b^{10} x^{10} + 41580 \, a^{2} b^{9} x^{9} + 9240 \, a^{3} b^{8} x^{8} - 2310 \, a^{4} b^{7} x^{7} + 924 \, a^{5} b^{6} x^{6} - 462 \, a^{6} b^{5} x^{5} + 264 \, a^{7} b^{4} x^{4} - 165 \, a^{8} b^{3} x^{3} + 110 \, a^{9} b^{2} x^{2} - 77 \, a^{10} b x + 56 \, a^{11}}{504 \,{\left (b x + a\right )}^{2} a^{12} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^3,x, algorithm="giac")

[Out]

55*b^9*log(abs(b*x + a))/a^12 - 55*b^9*log(abs(x))/a^12 - 1/504*(27720*a*b^10*x^10 + 41580*a^2*b^9*x^9 + 9240*
a^3*b^8*x^8 - 2310*a^4*b^7*x^7 + 924*a^5*b^6*x^6 - 462*a^6*b^5*x^5 + 264*a^7*b^4*x^4 - 165*a^8*b^3*x^3 + 110*a
^9*b^2*x^2 - 77*a^10*b*x + 56*a^11)/((b*x + a)^2*a^12*x^9)