Integrand size = 15, antiderivative size = 184 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=\frac {2 b^2}{7 a^3 \left (a+b \sqrt {x}\right )^7}+\frac {b^2}{a^4 \left (a+b \sqrt {x}\right )^6}+\frac {12 b^2}{5 a^5 \left (a+b \sqrt {x}\right )^5}+\frac {5 b^2}{a^6 \left (a+b \sqrt {x}\right )^4}+\frac {10 b^2}{a^7 \left (a+b \sqrt {x}\right )^3}+\frac {21 b^2}{a^8 \left (a+b \sqrt {x}\right )^2}+\frac {56 b^2}{a^9 \left (a+b \sqrt {x}\right )}-\frac {1}{a^8 x}+\frac {16 b}{a^9 \sqrt {x}}-\frac {72 b^2 \log \left (a+b \sqrt {x}\right )}{a^{10}}+\frac {36 b^2 \log (x)}{a^{10}} \] Output:
2/7*b^2/a^3/(a+b*x^(1/2))^7+b^2/a^4/(a+b*x^(1/2))^6+12/5*b^2/a^5/(a+b*x^(1 /2))^5+5*b^2/a^6/(a+b*x^(1/2))^4+10*b^2/a^7/(a+b*x^(1/2))^3+21*b^2/a^8/(a+ b*x^(1/2))^2+56*b^2/a^9/(a+b*x^(1/2))-1/a^8/x+16*b/a^9/x^(1/2)-72*b^2*ln(a +b*x^(1/2))/a^10+36*b^2*ln(x)/a^10
Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=\frac {\frac {a \left (-35 a^8+315 a^7 b \sqrt {x}+6534 a^6 b^2 x+28098 a^5 b^3 x^{3/2}+57834 a^4 b^4 x^2+66990 a^3 b^5 x^{5/2}+44940 a^2 b^6 x^3+16380 a b^7 x^{7/2}+2520 b^8 x^4\right )}{\left (a+b \sqrt {x}\right )^7 x}-2520 b^2 \log \left (a+b \sqrt {x}\right )+1260 b^2 \log (x)}{35 a^{10}} \] Input:
Integrate[1/((a + b*Sqrt[x])^8*x^2),x]
Output:
((a*(-35*a^8 + 315*a^7*b*Sqrt[x] + 6534*a^6*b^2*x + 28098*a^5*b^3*x^(3/2) + 57834*a^4*b^4*x^2 + 66990*a^3*b^5*x^(5/2) + 44940*a^2*b^6*x^3 + 16380*a* b^7*x^(7/2) + 2520*b^8*x^4))/((a + b*Sqrt[x])^7*x) - 2520*b^2*Log[a + b*Sq rt[x]] + 1260*b^2*Log[x])/(35*a^10)
Time = 0.59 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b \sqrt {x}\right )^8} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 2 \int \left (-\frac {36 b^3}{a^{10} \left (a+b \sqrt {x}\right )}-\frac {28 b^3}{a^9 \left (a+b \sqrt {x}\right )^2}-\frac {21 b^3}{a^8 \left (a+b \sqrt {x}\right )^3}-\frac {15 b^3}{a^7 \left (a+b \sqrt {x}\right )^4}-\frac {10 b^3}{a^6 \left (a+b \sqrt {x}\right )^5}-\frac {6 b^3}{a^5 \left (a+b \sqrt {x}\right )^6}-\frac {3 b^3}{a^4 \left (a+b \sqrt {x}\right )^7}-\frac {b^3}{a^3 \left (a+b \sqrt {x}\right )^8}+\frac {36 b^2}{a^{10} \sqrt {x}}-\frac {8 b}{a^9 x}+\frac {1}{a^8 x^{3/2}}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {36 b^2 \log \left (a+b \sqrt {x}\right )}{a^{10}}+\frac {36 b^2 \log \left (\sqrt {x}\right )}{a^{10}}+\frac {28 b^2}{a^9 \left (a+b \sqrt {x}\right )}+\frac {8 b}{a^9 \sqrt {x}}+\frac {21 b^2}{2 a^8 \left (a+b \sqrt {x}\right )^2}-\frac {1}{2 a^8 x}+\frac {5 b^2}{a^7 \left (a+b \sqrt {x}\right )^3}+\frac {5 b^2}{2 a^6 \left (a+b \sqrt {x}\right )^4}+\frac {6 b^2}{5 a^5 \left (a+b \sqrt {x}\right )^5}+\frac {b^2}{2 a^4 \left (a+b \sqrt {x}\right )^6}+\frac {b^2}{7 a^3 \left (a+b \sqrt {x}\right )^7}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^8*x^2),x]
Output:
2*(b^2/(7*a^3*(a + b*Sqrt[x])^7) + b^2/(2*a^4*(a + b*Sqrt[x])^6) + (6*b^2) /(5*a^5*(a + b*Sqrt[x])^5) + (5*b^2)/(2*a^6*(a + b*Sqrt[x])^4) + (5*b^2)/( a^7*(a + b*Sqrt[x])^3) + (21*b^2)/(2*a^8*(a + b*Sqrt[x])^2) + (28*b^2)/(a^ 9*(a + b*Sqrt[x])) - 1/(2*a^8*x) + (8*b)/(a^9*Sqrt[x]) - (36*b^2*Log[a + b *Sqrt[x]])/a^10 + (36*b^2*Log[Sqrt[x]])/a^10)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.44 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {2 b^{2}}{7 a^{3} \left (a +b \sqrt {x}\right )^{7}}+\frac {b^{2}}{a^{4} \left (a +b \sqrt {x}\right )^{6}}+\frac {12 b^{2}}{5 a^{5} \left (a +b \sqrt {x}\right )^{5}}+\frac {5 b^{2}}{a^{6} \left (a +b \sqrt {x}\right )^{4}}+\frac {10 b^{2}}{a^{7} \left (a +b \sqrt {x}\right )^{3}}+\frac {21 b^{2}}{a^{8} \left (a +b \sqrt {x}\right )^{2}}+\frac {56 b^{2}}{a^{9} \left (a +b \sqrt {x}\right )}-\frac {1}{a^{8} x}+\frac {16 b}{a^{9} \sqrt {x}}-\frac {72 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{10}}+\frac {36 b^{2} \ln \left (x \right )}{a^{10}}\) | \(163\) |
| default | \(\frac {2 b^{2}}{7 a^{3} \left (a +b \sqrt {x}\right )^{7}}+\frac {b^{2}}{a^{4} \left (a +b \sqrt {x}\right )^{6}}+\frac {12 b^{2}}{5 a^{5} \left (a +b \sqrt {x}\right )^{5}}+\frac {5 b^{2}}{a^{6} \left (a +b \sqrt {x}\right )^{4}}+\frac {10 b^{2}}{a^{7} \left (a +b \sqrt {x}\right )^{3}}+\frac {21 b^{2}}{a^{8} \left (a +b \sqrt {x}\right )^{2}}+\frac {56 b^{2}}{a^{9} \left (a +b \sqrt {x}\right )}-\frac {1}{a^{8} x}+\frac {16 b}{a^{9} \sqrt {x}}-\frac {72 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{10}}+\frac {36 b^{2} \ln \left (x \right )}{a^{10}}\) | \(163\) |
Input:
int(1/(a+b*x^(1/2))^8/x^2,x,method=_RETURNVERBOSE)
Output:
2/7*b^2/a^3/(a+b*x^(1/2))^7+b^2/a^4/(a+b*x^(1/2))^6+12/5*b^2/a^5/(a+b*x^(1 /2))^5+5*b^2/a^6/(a+b*x^(1/2))^4+10*b^2/a^7/(a+b*x^(1/2))^3+21*b^2/a^8/(a+ b*x^(1/2))^2+56*b^2/a^9/(a+b*x^(1/2))-1/a^8/x+16*b/a^9/x^(1/2)-72*b^2*ln(a +b*x^(1/2))/a^10+36*b^2*ln(x)/a^10
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (162) = 324\).
Time = 0.17 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=-\frac {1260 \, a^{2} b^{14} x^{7} - 8190 \, a^{4} b^{12} x^{6} + 22470 \, a^{6} b^{10} x^{5} - 33495 \, a^{8} b^{8} x^{4} + 28924 \, a^{10} b^{6} x^{3} - 13888 \, a^{12} b^{4} x^{2} + 3594 \, a^{14} b^{2} x - 35 \, a^{16} + 2520 \, {\left (b^{16} x^{8} - 7 \, a^{2} b^{14} x^{7} + 21 \, a^{4} b^{12} x^{6} - 35 \, a^{6} b^{10} x^{5} + 35 \, a^{8} b^{8} x^{4} - 21 \, a^{10} b^{6} x^{3} + 7 \, a^{12} b^{4} x^{2} - a^{14} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 2520 \, {\left (b^{16} x^{8} - 7 \, a^{2} b^{14} x^{7} + 21 \, a^{4} b^{12} x^{6} - 35 \, a^{6} b^{10} x^{5} + 35 \, a^{8} b^{8} x^{4} - 21 \, a^{10} b^{6} x^{3} + 7 \, a^{12} b^{4} x^{2} - a^{14} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 8 \, {\left (315 \, a b^{15} x^{7} - 2100 \, a^{3} b^{13} x^{6} + 5943 \, a^{5} b^{11} x^{5} - 9216 \, a^{7} b^{9} x^{4} + 8393 \, a^{9} b^{7} x^{3} - 4410 \, a^{11} b^{5} x^{2} + 1225 \, a^{13} b^{3} x - 70 \, a^{15} b\right )} \sqrt {x}}{35 \, {\left (a^{10} b^{14} x^{8} - 7 \, a^{12} b^{12} x^{7} + 21 \, a^{14} b^{10} x^{6} - 35 \, a^{16} b^{8} x^{5} + 35 \, a^{18} b^{6} x^{4} - 21 \, a^{20} b^{4} x^{3} + 7 \, a^{22} b^{2} x^{2} - a^{24} x\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^8/x^2,x, algorithm="fricas")
Output:
-1/35*(1260*a^2*b^14*x^7 - 8190*a^4*b^12*x^6 + 22470*a^6*b^10*x^5 - 33495* a^8*b^8*x^4 + 28924*a^10*b^6*x^3 - 13888*a^12*b^4*x^2 + 3594*a^14*b^2*x - 35*a^16 + 2520*(b^16*x^8 - 7*a^2*b^14*x^7 + 21*a^4*b^12*x^6 - 35*a^6*b^10* x^5 + 35*a^8*b^8*x^4 - 21*a^10*b^6*x^3 + 7*a^12*b^4*x^2 - a^14*b^2*x)*log( b*sqrt(x) + a) - 2520*(b^16*x^8 - 7*a^2*b^14*x^7 + 21*a^4*b^12*x^6 - 35*a^ 6*b^10*x^5 + 35*a^8*b^8*x^4 - 21*a^10*b^6*x^3 + 7*a^12*b^4*x^2 - a^14*b^2* x)*log(sqrt(x)) - 8*(315*a*b^15*x^7 - 2100*a^3*b^13*x^6 + 5943*a^5*b^11*x^ 5 - 9216*a^7*b^9*x^4 + 8393*a^9*b^7*x^3 - 4410*a^11*b^5*x^2 + 1225*a^13*b^ 3*x - 70*a^15*b)*sqrt(x))/(a^10*b^14*x^8 - 7*a^12*b^12*x^7 + 21*a^14*b^10* x^6 - 35*a^16*b^8*x^5 + 35*a^18*b^6*x^4 - 21*a^20*b^4*x^3 + 7*a^22*b^2*x^2 - a^24*x)
Leaf count of result is larger than twice the leaf count of optimal. 2854 vs. \(2 (178) = 356\).
Time = 6.46 (sec) , antiderivative size = 2854, normalized size of antiderivative = 15.51 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*x**(1/2))**8/x**2,x)
Output:
Piecewise((zoo/x**5, Eq(a, 0) & Eq(b, 0)), (-1/(a**8*x), Eq(b, 0)), (-1/(5 *b**8*x**5), Eq(a, 0)), (-35*a**9*sqrt(x)/(35*a**17*x**(3/2) + 245*a**16*b *x**2 + 735*a**15*b**2*x**(5/2) + 1225*a**14*b**3*x**3 + 1225*a**13*b**4*x **(7/2) + 735*a**12*b**5*x**4 + 245*a**11*b**6*x**(9/2) + 35*a**10*b**7*x* *5) + 315*a**8*b*x/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b**2* x**(5/2) + 1225*a**14*b**3*x**3 + 1225*a**13*b**4*x**(7/2) + 735*a**12*b** 5*x**4 + 245*a**11*b**6*x**(9/2) + 35*a**10*b**7*x**5) + 1260*a**7*b**2*x* *(3/2)*log(x)/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b**2*x**(5 /2) + 1225*a**14*b**3*x**3 + 1225*a**13*b**4*x**(7/2) + 735*a**12*b**5*x** 4 + 245*a**11*b**6*x**(9/2) + 35*a**10*b**7*x**5) - 2520*a**7*b**2*x**(3/2 )*log(a/b + sqrt(x))/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b** 2*x**(5/2) + 1225*a**14*b**3*x**3 + 1225*a**13*b**4*x**(7/2) + 735*a**12*b **5*x**4 + 245*a**11*b**6*x**(9/2) + 35*a**10*b**7*x**5) + 6534*a**7*b**2* x**(3/2)/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b**2*x**(5/2) + 1225*a**14*b**3*x**3 + 1225*a**13*b**4*x**(7/2) + 735*a**12*b**5*x**4 + 2 45*a**11*b**6*x**(9/2) + 35*a**10*b**7*x**5) + 8820*a**6*b**3*x**2*log(x)/ (35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b**2*x**(5/2) + 1225*a** 14*b**3*x**3 + 1225*a**13*b**4*x**(7/2) + 735*a**12*b**5*x**4 + 245*a**11* b**6*x**(9/2) + 35*a**10*b**7*x**5) - 17640*a**6*b**3*x**2*log(a/b + sqrt( x))/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b**2*x**(5/2) + 1...
Time = 0.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=\frac {2520 \, b^{8} x^{4} + 16380 \, a b^{7} x^{\frac {7}{2}} + 44940 \, a^{2} b^{6} x^{3} + 66990 \, a^{3} b^{5} x^{\frac {5}{2}} + 57834 \, a^{4} b^{4} x^{2} + 28098 \, a^{5} b^{3} x^{\frac {3}{2}} + 6534 \, a^{6} b^{2} x + 315 \, a^{7} b \sqrt {x} - 35 \, a^{8}}{35 \, {\left (a^{9} b^{7} x^{\frac {9}{2}} + 7 \, a^{10} b^{6} x^{4} + 21 \, a^{11} b^{5} x^{\frac {7}{2}} + 35 \, a^{12} b^{4} x^{3} + 35 \, a^{13} b^{3} x^{\frac {5}{2}} + 21 \, a^{14} b^{2} x^{2} + 7 \, a^{15} b x^{\frac {3}{2}} + a^{16} x\right )}} - \frac {72 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{10}} + \frac {36 \, b^{2} \log \left (x\right )}{a^{10}} \] Input:
integrate(1/(a+b*x^(1/2))^8/x^2,x, algorithm="maxima")
Output:
1/35*(2520*b^8*x^4 + 16380*a*b^7*x^(7/2) + 44940*a^2*b^6*x^3 + 66990*a^3*b ^5*x^(5/2) + 57834*a^4*b^4*x^2 + 28098*a^5*b^3*x^(3/2) + 6534*a^6*b^2*x + 315*a^7*b*sqrt(x) - 35*a^8)/(a^9*b^7*x^(9/2) + 7*a^10*b^6*x^4 + 21*a^11*b^ 5*x^(7/2) + 35*a^12*b^4*x^3 + 35*a^13*b^3*x^(5/2) + 21*a^14*b^2*x^2 + 7*a^ 15*b*x^(3/2) + a^16*x) - 72*b^2*log(b*sqrt(x) + a)/a^10 + 36*b^2*log(x)/a^ 10
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=-\frac {72 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{10}} + \frac {36 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{10}} + \frac {2520 \, a b^{8} x^{4} + 16380 \, a^{2} b^{7} x^{\frac {7}{2}} + 44940 \, a^{3} b^{6} x^{3} + 66990 \, a^{4} b^{5} x^{\frac {5}{2}} + 57834 \, a^{5} b^{4} x^{2} + 28098 \, a^{6} b^{3} x^{\frac {3}{2}} + 6534 \, a^{7} b^{2} x + 315 \, a^{8} b \sqrt {x} - 35 \, a^{9}}{35 \, {\left (b \sqrt {x} + a\right )}^{7} a^{10} x} \] Input:
integrate(1/(a+b*x^(1/2))^8/x^2,x, algorithm="giac")
Output:
-72*b^2*log(abs(b*sqrt(x) + a))/a^10 + 36*b^2*log(abs(x))/a^10 + 1/35*(252 0*a*b^8*x^4 + 16380*a^2*b^7*x^(7/2) + 44940*a^3*b^6*x^3 + 66990*a^4*b^5*x^ (5/2) + 57834*a^5*b^4*x^2 + 28098*a^6*b^3*x^(3/2) + 6534*a^7*b^2*x + 315*a ^8*b*sqrt(x) - 35*a^9)/((b*sqrt(x) + a)^7*a^10*x)
Time = 0.55 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=\frac {\frac {9\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {6534\,b^2\,x}{35\,a^3}+\frac {8262\,b^4\,x^2}{5\,a^5}+\frac {4014\,b^3\,x^{3/2}}{5\,a^4}+\frac {1284\,b^6\,x^3}{a^7}+\frac {1914\,b^5\,x^{5/2}}{a^6}+\frac {72\,b^8\,x^4}{a^9}+\frac {468\,b^7\,x^{7/2}}{a^8}}{a^7\,x+b^7\,x^{9/2}+7\,a\,b^6\,x^4+7\,a^6\,b\,x^{3/2}+21\,a^5\,b^2\,x^2+35\,a^3\,b^4\,x^3+35\,a^4\,b^3\,x^{5/2}+21\,a^2\,b^5\,x^{7/2}}-\frac {144\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^{10}} \] Input:
int(1/(x^2*(a + b*x^(1/2))^8),x)
Output:
((9*b*x^(1/2))/a^2 - 1/a + (6534*b^2*x)/(35*a^3) + (8262*b^4*x^2)/(5*a^5) + (4014*b^3*x^(3/2))/(5*a^4) + (1284*b^6*x^3)/a^7 + (1914*b^5*x^(5/2))/a^6 + (72*b^8*x^4)/a^9 + (468*b^7*x^(7/2))/a^8)/(a^7*x + b^7*x^(9/2) + 7*a*b^ 6*x^4 + 7*a^6*b*x^(3/2) + 21*a^5*b^2*x^2 + 35*a^3*b^4*x^3 + 35*a^4*b^3*x^( 5/2) + 21*a^2*b^5*x^(7/2)) - (144*b^2*atanh((2*b*x^(1/2))/a + 1))/a^10
Time = 0.22 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.34 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx=\frac {-88200 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b^{5} x^{2}-52920 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{7} x^{3}-35 a^{9}+25578 \sqrt {x}\, a^{6} b^{3} x -88200 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{6} x^{3}-2520 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{9} x^{4}+2520 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) b^{9} x^{4}-17640 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{8} x^{4}+2520 \,\mathrm {log}\left (\sqrt {x}\right ) a^{7} b^{2} x +52920 \,\mathrm {log}\left (\sqrt {x}\right ) a^{5} b^{4} x^{2}+88200 \,\mathrm {log}\left (\sqrt {x}\right ) a^{3} b^{6} x^{3}+17640 \,\mathrm {log}\left (\sqrt {x}\right ) a \,b^{8} x^{4}+315 \sqrt {x}\, a^{8} b -360 \sqrt {x}\, b^{9} x^{4}+6174 a^{7} b^{2} x +50274 a^{5} b^{4} x^{2}+32340 a^{3} b^{6} x^{3}+17640 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{6} b^{3} x +88200 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{4} b^{5} x^{2}+52920 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{2} b^{7} x^{3}+54390 \sqrt {x}\, a^{4} b^{5} x^{2}+8820 \sqrt {x}\, a^{2} b^{7} x^{3}-2520 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7} b^{2} x -52920 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{4} x^{2}-17640 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b^{3} x}{35 a^{10} x \left (7 \sqrt {x}\, a^{6} b +35 \sqrt {x}\, a^{4} b^{3} x +21 \sqrt {x}\, a^{2} b^{5} x^{2}+\sqrt {x}\, b^{7} x^{3}+a^{7}+21 a^{5} b^{2} x +35 a^{3} b^{4} x^{2}+7 a \,b^{6} x^{3}\right )} \] Input:
int(1/(a+b*x^(1/2))^8/x^2,x)
Output:
( - 17640*sqrt(x)*log(sqrt(x)*b + a)*a**6*b**3*x - 88200*sqrt(x)*log(sqrt( x)*b + a)*a**4*b**5*x**2 - 52920*sqrt(x)*log(sqrt(x)*b + a)*a**2*b**7*x**3 - 2520*sqrt(x)*log(sqrt(x)*b + a)*b**9*x**4 + 17640*sqrt(x)*log(sqrt(x))* a**6*b**3*x + 88200*sqrt(x)*log(sqrt(x))*a**4*b**5*x**2 + 52920*sqrt(x)*lo g(sqrt(x))*a**2*b**7*x**3 + 2520*sqrt(x)*log(sqrt(x))*b**9*x**4 + 315*sqrt (x)*a**8*b + 25578*sqrt(x)*a**6*b**3*x + 54390*sqrt(x)*a**4*b**5*x**2 + 88 20*sqrt(x)*a**2*b**7*x**3 - 360*sqrt(x)*b**9*x**4 - 2520*log(sqrt(x)*b + a )*a**7*b**2*x - 52920*log(sqrt(x)*b + a)*a**5*b**4*x**2 - 88200*log(sqrt(x )*b + a)*a**3*b**6*x**3 - 17640*log(sqrt(x)*b + a)*a*b**8*x**4 + 2520*log( sqrt(x))*a**7*b**2*x + 52920*log(sqrt(x))*a**5*b**4*x**2 + 88200*log(sqrt( x))*a**3*b**6*x**3 + 17640*log(sqrt(x))*a*b**8*x**4 - 35*a**9 + 6174*a**7* b**2*x + 50274*a**5*b**4*x**2 + 32340*a**3*b**6*x**3)/(35*a**10*x*(7*sqrt( x)*a**6*b + 35*sqrt(x)*a**4*b**3*x + 21*sqrt(x)*a**2*b**5*x**2 + sqrt(x)*b **7*x**3 + a**7 + 21*a**5*b**2*x + 35*a**3*b**4*x**2 + 7*a*b**6*x**3))