3.238 \(\int \frac{1}{x^4 (a+b x)^{10}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{165 b^3}{a^{12} (a+b x)}-\frac{60 b^3}{a^{11} (a+b x)^2}-\frac{28 b^3}{a^{10} (a+b x)^3}-\frac{14 b^3}{a^9 (a+b x)^4}-\frac{7 b^3}{a^8 (a+b x)^5}-\frac{10 b^3}{3 a^7 (a+b x)^6}-\frac{10 b^3}{7 a^6 (a+b x)^7}-\frac{b^3}{2 a^5 (a+b x)^8}-\frac{b^3}{9 a^4 (a+b x)^9}-\frac{55 b^2}{a^{12} x}-\frac{220 b^3 \log (x)}{a^{13}}+\frac{220 b^3 \log (a+b x)}{a^{13}}+\frac{5 b}{a^{11} x^2}-\frac{1}{3 a^{10} x^3} \]

[Out]

-1/(3*a^10*x^3) + (5*b)/(a^11*x^2) - (55*b^2)/(a^12*x) - b^3/(9*a^4*(a + b*x)^9) - b^3/(2*a^5*(a + b*x)^8) - (
10*b^3)/(7*a^6*(a + b*x)^7) - (10*b^3)/(3*a^7*(a + b*x)^6) - (7*b^3)/(a^8*(a + b*x)^5) - (14*b^3)/(a^9*(a + b*
x)^4) - (28*b^3)/(a^10*(a + b*x)^3) - (60*b^3)/(a^11*(a + b*x)^2) - (165*b^3)/(a^12*(a + b*x)) - (220*b^3*Log[
x])/a^13 + (220*b^3*Log[a + b*x])/a^13

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Rubi [A]  time = 0.164663, antiderivative size = 198, 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} \[ -\frac{165 b^3}{a^{12} (a+b x)}-\frac{60 b^3}{a^{11} (a+b x)^2}-\frac{28 b^3}{a^{10} (a+b x)^3}-\frac{14 b^3}{a^9 (a+b x)^4}-\frac{7 b^3}{a^8 (a+b x)^5}-\frac{10 b^3}{3 a^7 (a+b x)^6}-\frac{10 b^3}{7 a^6 (a+b x)^7}-\frac{b^3}{2 a^5 (a+b x)^8}-\frac{b^3}{9 a^4 (a+b x)^9}-\frac{55 b^2}{a^{12} x}-\frac{220 b^3 \log (x)}{a^{13}}+\frac{220 b^3 \log (a+b x)}{a^{13}}+\frac{5 b}{a^{11} x^2}-\frac{1}{3 a^{10} x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^10),x]

[Out]

-1/(3*a^10*x^3) + (5*b)/(a^11*x^2) - (55*b^2)/(a^12*x) - b^3/(9*a^4*(a + b*x)^9) - b^3/(2*a^5*(a + b*x)^8) - (
10*b^3)/(7*a^6*(a + b*x)^7) - (10*b^3)/(3*a^7*(a + b*x)^6) - (7*b^3)/(a^8*(a + b*x)^5) - (14*b^3)/(a^9*(a + b*
x)^4) - (28*b^3)/(a^10*(a + b*x)^3) - (60*b^3)/(a^11*(a + b*x)^2) - (165*b^3)/(a^12*(a + b*x)) - (220*b^3*Log[
x])/a^13 + (220*b^3*Log[a + b*x])/a^13

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^4 (a+b x)^{10}} \, dx &=\int \left (\frac{1}{a^{10} x^4}-\frac{10 b}{a^{11} x^3}+\frac{55 b^2}{a^{12} x^2}-\frac{220 b^3}{a^{13} x}+\frac{b^4}{a^4 (a+b x)^{10}}+\frac{4 b^4}{a^5 (a+b x)^9}+\frac{10 b^4}{a^6 (a+b x)^8}+\frac{20 b^4}{a^7 (a+b x)^7}+\frac{35 b^4}{a^8 (a+b x)^6}+\frac{56 b^4}{a^9 (a+b x)^5}+\frac{84 b^4}{a^{10} (a+b x)^4}+\frac{120 b^4}{a^{11} (a+b x)^3}+\frac{165 b^4}{a^{12} (a+b x)^2}+\frac{220 b^4}{a^{13} (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a^{10} x^3}+\frac{5 b}{a^{11} x^2}-\frac{55 b^2}{a^{12} x}-\frac{b^3}{9 a^4 (a+b x)^9}-\frac{b^3}{2 a^5 (a+b x)^8}-\frac{10 b^3}{7 a^6 (a+b x)^7}-\frac{10 b^3}{3 a^7 (a+b x)^6}-\frac{7 b^3}{a^8 (a+b x)^5}-\frac{14 b^3}{a^9 (a+b x)^4}-\frac{28 b^3}{a^{10} (a+b x)^3}-\frac{60 b^3}{a^{11} (a+b x)^2}-\frac{165 b^3}{a^{12} (a+b x)}-\frac{220 b^3 \log (x)}{a^{13}}+\frac{220 b^3 \log (a+b x)}{a^{13}}\\ \end{align*}

Mathematica [A]  time = 0.170529, size = 156, normalized size = 0.79 \[ -\frac{\frac{a \left (2772 a^9 b^2 x^2+78419 a^8 b^3 x^3+456291 a^7 b^4 x^4+1326204 a^6 b^5 x^5+2318316 a^5 b^6 x^6+2604294 a^4 b^7 x^7+1905750 a^3 b^8 x^8+882420 a^2 b^9 x^9-252 a^{10} b x+42 a^{11}+235620 a b^{10} x^{10}+27720 b^{11} x^{11}\right )}{x^3 (a+b x)^9}-27720 b^3 \log (a+b x)+27720 b^3 \log (x)}{126 a^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^10),x]

[Out]

-((a*(42*a^11 - 252*a^10*b*x + 2772*a^9*b^2*x^2 + 78419*a^8*b^3*x^3 + 456291*a^7*b^4*x^4 + 1326204*a^6*b^5*x^5
+ 2318316*a^5*b^6*x^6 + 2604294*a^4*b^7*x^7 + 1905750*a^3*b^8*x^8 + 882420*a^2*b^9*x^9 + 235620*a*b^10*x^10 +
27720*b^11*x^11))/(x^3*(a + b*x)^9) + 27720*b^3*Log[x] - 27720*b^3*Log[a + b*x])/(126*a^13)

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Maple [A]  time = 0.016, size = 189, normalized size = 1. \begin{align*} -{\frac{1}{3\,{a}^{10}{x}^{3}}}+5\,{\frac{b}{{a}^{11}{x}^{2}}}-55\,{\frac{{b}^{2}}{{a}^{12}x}}-{\frac{{b}^{3}}{9\,{a}^{4} \left ( bx+a \right ) ^{9}}}-{\frac{{b}^{3}}{2\,{a}^{5} \left ( bx+a \right ) ^{8}}}-{\frac{10\,{b}^{3}}{7\,{a}^{6} \left ( bx+a \right ) ^{7}}}-{\frac{10\,{b}^{3}}{3\,{a}^{7} \left ( bx+a \right ) ^{6}}}-7\,{\frac{{b}^{3}}{{a}^{8} \left ( bx+a \right ) ^{5}}}-14\,{\frac{{b}^{3}}{{a}^{9} \left ( bx+a \right ) ^{4}}}-28\,{\frac{{b}^{3}}{{a}^{10} \left ( bx+a \right ) ^{3}}}-60\,{\frac{{b}^{3}}{{a}^{11} \left ( bx+a \right ) ^{2}}}-165\,{\frac{{b}^{3}}{{a}^{12} \left ( bx+a \right ) }}-220\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{13}}}+220\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^10,x)

[Out]

-1/3/a^10/x^3+5*b/a^11/x^2-55*b^2/a^12/x-1/9*b^3/a^4/(b*x+a)^9-1/2*b^3/a^5/(b*x+a)^8-10/7*b^3/a^6/(b*x+a)^7-10
/3*b^3/a^7/(b*x+a)^6-7*b^3/a^8/(b*x+a)^5-14*b^3/a^9/(b*x+a)^4-28*b^3/a^10/(b*x+a)^3-60*b^3/a^11/(b*x+a)^2-165*
b^3/a^12/(b*x+a)-220*b^3*ln(x)/a^13+220*b^3*ln(b*x+a)/a^13

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Maxima [A]  time = 1.19602, size = 339, normalized size = 1.71 \begin{align*} -\frac{27720 \, b^{11} x^{11} + 235620 \, a b^{10} x^{10} + 882420 \, a^{2} b^{9} x^{9} + 1905750 \, a^{3} b^{8} x^{8} + 2604294 \, a^{4} b^{7} x^{7} + 2318316 \, a^{5} b^{6} x^{6} + 1326204 \, a^{6} b^{5} x^{5} + 456291 \, a^{7} b^{4} x^{4} + 78419 \, a^{8} b^{3} x^{3} + 2772 \, a^{9} b^{2} x^{2} - 252 \, a^{10} b x + 42 \, a^{11}}{126 \,{\left (a^{12} b^{9} x^{12} + 9 \, a^{13} b^{8} x^{11} + 36 \, a^{14} b^{7} x^{10} + 84 \, a^{15} b^{6} x^{9} + 126 \, a^{16} b^{5} x^{8} + 126 \, a^{17} b^{4} x^{7} + 84 \, a^{18} b^{3} x^{6} + 36 \, a^{19} b^{2} x^{5} + 9 \, a^{20} b x^{4} + a^{21} x^{3}\right )}} + \frac{220 \, b^{3} \log \left (b x + a\right )}{a^{13}} - \frac{220 \, b^{3} \log \left (x\right )}{a^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/126*(27720*b^11*x^11 + 235620*a*b^10*x^10 + 882420*a^2*b^9*x^9 + 1905750*a^3*b^8*x^8 + 2604294*a^4*b^7*x^7
+ 2318316*a^5*b^6*x^6 + 1326204*a^6*b^5*x^5 + 456291*a^7*b^4*x^4 + 78419*a^8*b^3*x^3 + 2772*a^9*b^2*x^2 - 252*
a^10*b*x + 42*a^11)/(a^12*b^9*x^12 + 9*a^13*b^8*x^11 + 36*a^14*b^7*x^10 + 84*a^15*b^6*x^9 + 126*a^16*b^5*x^8 +
126*a^17*b^4*x^7 + 84*a^18*b^3*x^6 + 36*a^19*b^2*x^5 + 9*a^20*b*x^4 + a^21*x^3) + 220*b^3*log(b*x + a)/a^13 -
220*b^3*log(x)/a^13

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Fricas [B]  time = 1.67768, size = 1054, normalized size = 5.32 \begin{align*} -\frac{27720 \, a b^{11} x^{11} + 235620 \, a^{2} b^{10} x^{10} + 882420 \, a^{3} b^{9} x^{9} + 1905750 \, a^{4} b^{8} x^{8} + 2604294 \, a^{5} b^{7} x^{7} + 2318316 \, a^{6} b^{6} x^{6} + 1326204 \, a^{7} b^{5} x^{5} + 456291 \, a^{8} b^{4} x^{4} + 78419 \, a^{9} b^{3} x^{3} + 2772 \, a^{10} b^{2} x^{2} - 252 \, a^{11} b x + 42 \, a^{12} - 27720 \,{\left (b^{12} x^{12} + 9 \, a b^{11} x^{11} + 36 \, a^{2} b^{10} x^{10} + 84 \, a^{3} b^{9} x^{9} + 126 \, a^{4} b^{8} x^{8} + 126 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{5} + 9 \, a^{8} b^{4} x^{4} + a^{9} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 27720 \,{\left (b^{12} x^{12} + 9 \, a b^{11} x^{11} + 36 \, a^{2} b^{10} x^{10} + 84 \, a^{3} b^{9} x^{9} + 126 \, a^{4} b^{8} x^{8} + 126 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{5} + 9 \, a^{8} b^{4} x^{4} + a^{9} b^{3} x^{3}\right )} \log \left (x\right )}{126 \,{\left (a^{13} b^{9} x^{12} + 9 \, a^{14} b^{8} x^{11} + 36 \, a^{15} b^{7} x^{10} + 84 \, a^{16} b^{6} x^{9} + 126 \, a^{17} b^{5} x^{8} + 126 \, a^{18} b^{4} x^{7} + 84 \, a^{19} b^{3} x^{6} + 36 \, a^{20} b^{2} x^{5} + 9 \, a^{21} b x^{4} + a^{22} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/126*(27720*a*b^11*x^11 + 235620*a^2*b^10*x^10 + 882420*a^3*b^9*x^9 + 1905750*a^4*b^8*x^8 + 2604294*a^5*b^7*
x^7 + 2318316*a^6*b^6*x^6 + 1326204*a^7*b^5*x^5 + 456291*a^8*b^4*x^4 + 78419*a^9*b^3*x^3 + 2772*a^10*b^2*x^2 -
252*a^11*b*x + 42*a^12 - 27720*(b^12*x^12 + 9*a*b^11*x^11 + 36*a^2*b^10*x^10 + 84*a^3*b^9*x^9 + 126*a^4*b^8*x
^8 + 126*a^5*b^7*x^7 + 84*a^6*b^6*x^6 + 36*a^7*b^5*x^5 + 9*a^8*b^4*x^4 + a^9*b^3*x^3)*log(b*x + a) + 27720*(b^
12*x^12 + 9*a*b^11*x^11 + 36*a^2*b^10*x^10 + 84*a^3*b^9*x^9 + 126*a^4*b^8*x^8 + 126*a^5*b^7*x^7 + 84*a^6*b^6*x
^6 + 36*a^7*b^5*x^5 + 9*a^8*b^4*x^4 + a^9*b^3*x^3)*log(x))/(a^13*b^9*x^12 + 9*a^14*b^8*x^11 + 36*a^15*b^7*x^10
+ 84*a^16*b^6*x^9 + 126*a^17*b^5*x^8 + 126*a^18*b^4*x^7 + 84*a^19*b^3*x^6 + 36*a^20*b^2*x^5 + 9*a^21*b*x^4 +
a^22*x^3)

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Sympy [A]  time = 4.14182, size = 258, normalized size = 1.3 \begin{align*} - \frac{42 a^{11} - 252 a^{10} b x + 2772 a^{9} b^{2} x^{2} + 78419 a^{8} b^{3} x^{3} + 456291 a^{7} b^{4} x^{4} + 1326204 a^{6} b^{5} x^{5} + 2318316 a^{5} b^{6} x^{6} + 2604294 a^{4} b^{7} x^{7} + 1905750 a^{3} b^{8} x^{8} + 882420 a^{2} b^{9} x^{9} + 235620 a b^{10} x^{10} + 27720 b^{11} x^{11}}{126 a^{21} x^{3} + 1134 a^{20} b x^{4} + 4536 a^{19} b^{2} x^{5} + 10584 a^{18} b^{3} x^{6} + 15876 a^{17} b^{4} x^{7} + 15876 a^{16} b^{5} x^{8} + 10584 a^{15} b^{6} x^{9} + 4536 a^{14} b^{7} x^{10} + 1134 a^{13} b^{8} x^{11} + 126 a^{12} b^{9} x^{12}} + \frac{220 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**10,x)

[Out]

-(42*a**11 - 252*a**10*b*x + 2772*a**9*b**2*x**2 + 78419*a**8*b**3*x**3 + 456291*a**7*b**4*x**4 + 1326204*a**6
*b**5*x**5 + 2318316*a**5*b**6*x**6 + 2604294*a**4*b**7*x**7 + 1905750*a**3*b**8*x**8 + 882420*a**2*b**9*x**9
+ 235620*a*b**10*x**10 + 27720*b**11*x**11)/(126*a**21*x**3 + 1134*a**20*b*x**4 + 4536*a**19*b**2*x**5 + 10584
*a**18*b**3*x**6 + 15876*a**17*b**4*x**7 + 15876*a**16*b**5*x**8 + 10584*a**15*b**6*x**9 + 4536*a**14*b**7*x**
10 + 1134*a**13*b**8*x**11 + 126*a**12*b**9*x**12) + 220*b**3*(-log(x) + log(a/b + x))/a**13

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Giac [A]  time = 1.18127, size = 220, normalized size = 1.11 \begin{align*} \frac{220 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{13}} - \frac{220 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{13}} - \frac{27720 \, a b^{11} x^{11} + 235620 \, a^{2} b^{10} x^{10} + 882420 \, a^{3} b^{9} x^{9} + 1905750 \, a^{4} b^{8} x^{8} + 2604294 \, a^{5} b^{7} x^{7} + 2318316 \, a^{6} b^{6} x^{6} + 1326204 \, a^{7} b^{5} x^{5} + 456291 \, a^{8} b^{4} x^{4} + 78419 \, a^{9} b^{3} x^{3} + 2772 \, a^{10} b^{2} x^{2} - 252 \, a^{11} b x + 42 \, a^{12}}{126 \,{\left (b x + a\right )}^{9} a^{13} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^10,x, algorithm="giac")

[Out]

220*b^3*log(abs(b*x + a))/a^13 - 220*b^3*log(abs(x))/a^13 - 1/126*(27720*a*b^11*x^11 + 235620*a^2*b^10*x^10 +
882420*a^3*b^9*x^9 + 1905750*a^4*b^8*x^8 + 2604294*a^5*b^7*x^7 + 2318316*a^6*b^6*x^6 + 1326204*a^7*b^5*x^5 + 4
56291*a^8*b^4*x^4 + 78419*a^9*b^3*x^3 + 2772*a^10*b^2*x^2 - 252*a^11*b*x + 42*a^12)/((b*x + a)^9*a^13*x^3)