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

Optimal. Leaf size=158 \[ -\frac{9 b}{a^{10} (a+b x)}-\frac{4 b}{a^9 (a+b x)^2}-\frac{7 b}{3 a^8 (a+b x)^3}-\frac{3 b}{2 a^7 (a+b x)^4}-\frac{b}{a^6 (a+b x)^5}-\frac{2 b}{3 a^5 (a+b x)^6}-\frac{3 b}{7 a^4 (a+b x)^7}-\frac{b}{4 a^3 (a+b x)^8}-\frac{b}{9 a^2 (a+b x)^9}-\frac{10 b \log (x)}{a^{11}}+\frac{10 b \log (a+b x)}{a^{11}}-\frac{1}{a^{10} x} \]

[Out]

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

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Rubi [A]  time = 0.124898, antiderivative size = 158, 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} \[ -\frac{9 b}{a^{10} (a+b x)}-\frac{4 b}{a^9 (a+b x)^2}-\frac{7 b}{3 a^8 (a+b x)^3}-\frac{3 b}{2 a^7 (a+b x)^4}-\frac{b}{a^6 (a+b x)^5}-\frac{2 b}{3 a^5 (a+b x)^6}-\frac{3 b}{7 a^4 (a+b x)^7}-\frac{b}{4 a^3 (a+b x)^8}-\frac{b}{9 a^2 (a+b x)^9}-\frac{10 b \log (x)}{a^{11}}+\frac{10 b \log (a+b x)}{a^{11}}-\frac{1}{a^{10} x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^10),x]

[Out]

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

Mathematica [A]  time = 0.162486, size = 130, normalized size = 0.82 \[ -\frac{\frac{a \left (41481 a^7 b^2 x^2+120564 a^6 b^3 x^3+210756 a^5 b^4 x^4+236754 a^4 b^5 x^5+173250 a^3 b^6 x^6+80220 a^2 b^7 x^7+7129 a^8 b x+252 a^9+21420 a b^8 x^8+2520 b^9 x^9\right )}{x (a+b x)^9}-2520 b \log (a+b x)+2520 b \log (x)}{252 a^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^10),x]

[Out]

-((a*(252*a^9 + 7129*a^8*b*x + 41481*a^7*b^2*x^2 + 120564*a^6*b^3*x^3 + 210756*a^5*b^4*x^4 + 236754*a^4*b^5*x^
5 + 173250*a^3*b^6*x^6 + 80220*a^2*b^7*x^7 + 21420*a*b^8*x^8 + 2520*b^9*x^9))/(x*(a + b*x)^9) + 2520*b*Log[x]
- 2520*b*Log[a + b*x])/(252*a^11)

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Maple [A]  time = 0.014, size = 147, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{10}x}}-{\frac{b}{9\,{a}^{2} \left ( bx+a \right ) ^{9}}}-{\frac{b}{4\,{a}^{3} \left ( bx+a \right ) ^{8}}}-{\frac{3\,b}{7\,{a}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{2\,b}{3\,{a}^{5} \left ( bx+a \right ) ^{6}}}-{\frac{b}{{a}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{3\,b}{2\,{a}^{7} \left ( bx+a \right ) ^{4}}}-{\frac{7\,b}{3\,{a}^{8} \left ( bx+a \right ) ^{3}}}-4\,{\frac{b}{{a}^{9} \left ( bx+a \right ) ^{2}}}-9\,{\frac{b}{{a}^{10} \left ( bx+a \right ) }}-10\,{\frac{b\ln \left ( x \right ) }{{a}^{11}}}+10\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10074, size = 301, normalized size = 1.91 \begin{align*} -\frac{2520 \, b^{9} x^{9} + 21420 \, a b^{8} x^{8} + 80220 \, a^{2} b^{7} x^{7} + 173250 \, a^{3} b^{6} x^{6} + 236754 \, a^{4} b^{5} x^{5} + 210756 \, a^{5} b^{4} x^{4} + 120564 \, a^{6} b^{3} x^{3} + 41481 \, a^{7} b^{2} x^{2} + 7129 \, a^{8} b x + 252 \, a^{9}}{252 \,{\left (a^{10} b^{9} x^{10} + 9 \, a^{11} b^{8} x^{9} + 36 \, a^{12} b^{7} x^{8} + 84 \, a^{13} b^{6} x^{7} + 126 \, a^{14} b^{5} x^{6} + 126 \, a^{15} b^{4} x^{5} + 84 \, a^{16} b^{3} x^{4} + 36 \, a^{17} b^{2} x^{3} + 9 \, a^{18} b x^{2} + a^{19} x\right )}} + \frac{10 \, b \log \left (b x + a\right )}{a^{11}} - \frac{10 \, b \log \left (x\right )}{a^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/252*(2520*b^9*x^9 + 21420*a*b^8*x^8 + 80220*a^2*b^7*x^7 + 173250*a^3*b^6*x^6 + 236754*a^4*b^5*x^5 + 210756*
a^5*b^4*x^4 + 120564*a^6*b^3*x^3 + 41481*a^7*b^2*x^2 + 7129*a^8*b*x + 252*a^9)/(a^10*b^9*x^10 + 9*a^11*b^8*x^9
+ 36*a^12*b^7*x^8 + 84*a^13*b^6*x^7 + 126*a^14*b^5*x^6 + 126*a^15*b^4*x^5 + 84*a^16*b^3*x^4 + 36*a^17*b^2*x^3
+ 9*a^18*b*x^2 + a^19*x) + 10*b*log(b*x + a)/a^11 - 10*b*log(x)/a^11

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Fricas [B]  time = 1.64411, size = 956, normalized size = 6.05 \begin{align*} -\frac{2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10} - 2520 \,{\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (b x + a\right ) + 2520 \,{\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (x\right )}{252 \,{\left (a^{11} b^{9} x^{10} + 9 \, a^{12} b^{8} x^{9} + 36 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{7} + 126 \, a^{15} b^{5} x^{6} + 126 \, a^{16} b^{4} x^{5} + 84 \, a^{17} b^{3} x^{4} + 36 \, a^{18} b^{2} x^{3} + 9 \, a^{19} b x^{2} + a^{20} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/252*(2520*a*b^9*x^9 + 21420*a^2*b^8*x^8 + 80220*a^3*b^7*x^7 + 173250*a^4*b^6*x^6 + 236754*a^5*b^5*x^5 + 210
756*a^6*b^4*x^4 + 120564*a^7*b^3*x^3 + 41481*a^8*b^2*x^2 + 7129*a^9*b*x + 252*a^10 - 2520*(b^10*x^10 + 9*a*b^9
*x^9 + 36*a^2*b^8*x^8 + 84*a^3*b^7*x^7 + 126*a^4*b^6*x^6 + 126*a^5*b^5*x^5 + 84*a^6*b^4*x^4 + 36*a^7*b^3*x^3 +
9*a^8*b^2*x^2 + a^9*b*x)*log(b*x + a) + 2520*(b^10*x^10 + 9*a*b^9*x^9 + 36*a^2*b^8*x^8 + 84*a^3*b^7*x^7 + 126
*a^4*b^6*x^6 + 126*a^5*b^5*x^5 + 84*a^6*b^4*x^4 + 36*a^7*b^3*x^3 + 9*a^8*b^2*x^2 + a^9*b*x)*log(x))/(a^11*b^9*
x^10 + 9*a^12*b^8*x^9 + 36*a^13*b^7*x^8 + 84*a^14*b^6*x^7 + 126*a^15*b^5*x^6 + 126*a^16*b^4*x^5 + 84*a^17*b^3*
x^4 + 36*a^18*b^2*x^3 + 9*a^19*b*x^2 + a^20*x)

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Sympy [A]  time = 2.64632, size = 231, normalized size = 1.46 \begin{align*} - \frac{252 a^{9} + 7129 a^{8} b x + 41481 a^{7} b^{2} x^{2} + 120564 a^{6} b^{3} x^{3} + 210756 a^{5} b^{4} x^{4} + 236754 a^{4} b^{5} x^{5} + 173250 a^{3} b^{6} x^{6} + 80220 a^{2} b^{7} x^{7} + 21420 a b^{8} x^{8} + 2520 b^{9} x^{9}}{252 a^{19} x + 2268 a^{18} b x^{2} + 9072 a^{17} b^{2} x^{3} + 21168 a^{16} b^{3} x^{4} + 31752 a^{15} b^{4} x^{5} + 31752 a^{14} b^{5} x^{6} + 21168 a^{13} b^{6} x^{7} + 9072 a^{12} b^{7} x^{8} + 2268 a^{11} b^{8} x^{9} + 252 a^{10} b^{9} x^{10}} + \frac{10 b \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**2/(b*x+a)**10,x)

[Out]

-(252*a**9 + 7129*a**8*b*x + 41481*a**7*b**2*x**2 + 120564*a**6*b**3*x**3 + 210756*a**5*b**4*x**4 + 236754*a**
4*b**5*x**5 + 173250*a**3*b**6*x**6 + 80220*a**2*b**7*x**7 + 21420*a*b**8*x**8 + 2520*b**9*x**9)/(252*a**19*x
+ 2268*a**18*b*x**2 + 9072*a**17*b**2*x**3 + 21168*a**16*b**3*x**4 + 31752*a**15*b**4*x**5 + 31752*a**14*b**5*
x**6 + 21168*a**13*b**6*x**7 + 9072*a**12*b**7*x**8 + 2268*a**11*b**8*x**9 + 252*a**10*b**9*x**10) + 10*b*(-lo
g(x) + log(a/b + x))/a**11

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Giac [A]  time = 1.18441, size = 185, normalized size = 1.17 \begin{align*} \frac{10 \, b \log \left ({\left | b x + a \right |}\right )}{a^{11}} - \frac{10 \, b \log \left ({\left | x \right |}\right )}{a^{11}} - \frac{2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10}}{252 \,{\left (b x + a\right )}^{9} a^{11} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*b*log(abs(b*x + a))/a^11 - 10*b*log(abs(x))/a^11 - 1/252*(2520*a*b^9*x^9 + 21420*a^2*b^8*x^8 + 80220*a^3*b^
7*x^7 + 173250*a^4*b^6*x^6 + 236754*a^5*b^5*x^5 + 210756*a^6*b^4*x^4 + 120564*a^7*b^3*x^3 + 41481*a^8*b^2*x^2
+ 7129*a^9*b*x + 252*a^10)/((b*x + a)^9*a^11*x)