3.2230 \(\int \frac {1}{(a+b \sqrt {x})^8 x^2} \, dx\)
Optimal. Leaf size=184 \[ -\frac {72 b^2 \log \left (a+b \sqrt {x}\right )}{a^{10}}+\frac {36 b^2 \log (x)}{a^{10}}+\frac {56 b^2}{a^9 \left (a+b \sqrt {x}\right )}+\frac {16 b}{a^9 \sqrt {x}}+\frac {21 b^2}{a^8 \left (a+b \sqrt {x}\right )^2}-\frac {1}{a^8 x}+\frac {10 b^2}{a^7 \left (a+b \sqrt {x}\right )^3}+\frac {5 b^2}{a^6 \left (a+b \sqrt {x}\right )^4}+\frac {12 b^2}{5 a^5 \left (a+b \sqrt {x}\right )^5}+\frac {b^2}{a^4 \left (a+b \sqrt {x}\right )^6}+\frac {2 b^2}{7 a^3 \left (a+b \sqrt {x}\right )^7} \]
[Out]
-1/a^8/x+36*b^2*ln(x)/a^10-72*b^2*ln(a+b*x^(1/2))/a^10+16*b/a^9/x^(1/2)+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))
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Rubi [A] time = 0.16, antiderivative size = 184, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used =
{266, 44} \[ \frac {56 b^2}{a^9 \left (a+b \sqrt {x}\right )}+\frac {21 b^2}{a^8 \left (a+b \sqrt {x}\right )^2}+\frac {10 b^2}{a^7 \left (a+b \sqrt {x}\right )^3}+\frac {5 b^2}{a^6 \left (a+b \sqrt {x}\right )^4}+\frac {12 b^2}{5 a^5 \left (a+b \sqrt {x}\right )^5}+\frac {b^2}{a^4 \left (a+b \sqrt {x}\right )^6}+\frac {2 b^2}{7 a^3 \left (a+b \sqrt {x}\right )^7}-\frac {72 b^2 \log \left (a+b \sqrt {x}\right )}{a^{10}}+\frac {36 b^2 \log (x)}{a^{10}}+\frac {16 b}{a^9 \sqrt {x}}-\frac {1}{a^8 x} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*Sqrt[x])^8*x^2),x]
[Out]
(2*b^2)/(7*a^3*(a + b*Sqrt[x])^7) + b^2/(a^4*(a + b*Sqrt[x])^6) + (12*b^2)/(5*a^5*(a + b*Sqrt[x])^5) + (5*b^2)
/(a^6*(a + b*Sqrt[x])^4) + (10*b^2)/(a^7*(a + b*Sqrt[x])^3) + (21*b^2)/(a^8*(a + b*Sqrt[x])^2) + (56*b^2)/(a^9
*(a + b*Sqrt[x])) - 1/(a^8*x) + (16*b)/(a^9*Sqrt[x]) - (72*b^2*Log[a + b*Sqrt[x]])/a^10 + (36*b^2*Log[x])/a^10
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])
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^8} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a^8 x^3}-\frac {8 b}{a^9 x^2}+\frac {36 b^2}{a^{10} x}-\frac {b^3}{a^3 (a+b x)^8}-\frac {3 b^3}{a^4 (a+b x)^7}-\frac {6 b^3}{a^5 (a+b x)^6}-\frac {10 b^3}{a^6 (a+b x)^5}-\frac {15 b^3}{a^7 (a+b x)^4}-\frac {21 b^3}{a^8 (a+b x)^3}-\frac {28 b^3}{a^9 (a+b x)^2}-\frac {36 b^3}{a^{10} (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\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}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 139, normalized size = 0.76 \[ \frac {\frac {a \left (-35 a^8+315 a^7 b \sqrt {x}+6534 a^6 b^2 x+28098 a^5 b^3 x^{3/2}+57834 a^4 b^4 x^2+66990 a^3 b^5 x^{5/2}+44940 a^2 b^6 x^3+16380 a b^7 x^{7/2}+2520 b^8 x^4\right )}{x \left (a+b \sqrt {x}\right )^7}-2520 b^2 \log \left (a+b \sqrt {x}\right )+1260 b^2 \log (x)}{35 a^{10}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*Sqrt[x])^8*x^2),x]
[Out]
((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*
Sqrt[x]] + 1260*b^2*Log[x])/(35*a^10)
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fricas [B] time = 1.48, size = 435, normalized size = 2.36 \[ -\frac {1260 \, a^{2} b^{14} x^{7} - 8190 \, a^{4} b^{12} x^{6} + 22470 \, a^{6} b^{10} x^{5} - 33495 \, a^{8} b^{8} x^{4} + 28924 \, a^{10} b^{6} x^{3} - 13888 \, a^{12} b^{4} x^{2} + 3594 \, a^{14} b^{2} x - 35 \, a^{16} + 2520 \, {\left (b^{16} x^{8} - 7 \, a^{2} b^{14} x^{7} + 21 \, a^{4} b^{12} x^{6} - 35 \, a^{6} b^{10} x^{5} + 35 \, a^{8} b^{8} x^{4} - 21 \, a^{10} b^{6} x^{3} + 7 \, a^{12} b^{4} x^{2} - a^{14} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 2520 \, {\left (b^{16} x^{8} - 7 \, a^{2} b^{14} x^{7} + 21 \, a^{4} b^{12} x^{6} - 35 \, a^{6} b^{10} x^{5} + 35 \, a^{8} b^{8} x^{4} - 21 \, a^{10} b^{6} x^{3} + 7 \, a^{12} b^{4} x^{2} - a^{14} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 8 \, {\left (315 \, a b^{15} x^{7} - 2100 \, a^{3} b^{13} x^{6} + 5943 \, a^{5} b^{11} x^{5} - 9216 \, a^{7} b^{9} x^{4} + 8393 \, a^{9} b^{7} x^{3} - 4410 \, a^{11} b^{5} x^{2} + 1225 \, a^{13} b^{3} x - 70 \, a^{15} b\right )} \sqrt {x}}{35 \, {\left (a^{10} b^{14} x^{8} - 7 \, a^{12} b^{12} x^{7} + 21 \, a^{14} b^{10} x^{6} - 35 \, a^{16} b^{8} x^{5} + 35 \, a^{18} b^{6} x^{4} - 21 \, a^{20} b^{4} x^{3} + 7 \, a^{22} b^{2} x^{2} - a^{24} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*x^(1/2))^8,x, algorithm="fricas")
[Out]
-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 - 1
3888*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^1
0*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)
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giac [A] time = 0.21, size = 134, normalized size = 0.73 \[ -\frac {72 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{10}} + \frac {36 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{10}} + \frac {2520 \, a b^{8} x^{4} + 16380 \, a^{2} b^{7} x^{\frac {7}{2}} + 44940 \, a^{3} b^{6} x^{3} + 66990 \, a^{4} b^{5} x^{\frac {5}{2}} + 57834 \, a^{5} b^{4} x^{2} + 28098 \, a^{6} b^{3} x^{\frac {3}{2}} + 6534 \, a^{7} b^{2} x + 315 \, a^{8} b \sqrt {x} - 35 \, a^{9}}{35 \, {\left (b \sqrt {x} + a\right )}^{7} a^{10} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*x^(1/2))^8,x, algorithm="giac")
[Out]
-72*b^2*log(abs(b*sqrt(x) + a))/a^10 + 36*b^2*log(abs(x))/a^10 + 1/35*(2520*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)
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maple [A] time = 0.01, size = 163, normalized size = 0.89 \[ \frac {2 b^{2}}{7 \left (b \sqrt {x}+a \right )^{7} a^{3}}+\frac {b^{2}}{\left (b \sqrt {x}+a \right )^{6} a^{4}}+\frac {12 b^{2}}{5 \left (b \sqrt {x}+a \right )^{5} a^{5}}+\frac {5 b^{2}}{\left (b \sqrt {x}+a \right )^{4} a^{6}}+\frac {10 b^{2}}{\left (b \sqrt {x}+a \right )^{3} a^{7}}+\frac {21 b^{2}}{\left (b \sqrt {x}+a \right )^{2} a^{8}}+\frac {56 b^{2}}{\left (b \sqrt {x}+a \right ) a^{9}}+\frac {36 b^{2} \ln \relax (x )}{a^{10}}-\frac {72 b^{2} \ln \left (b \sqrt {x}+a \right )}{a^{10}}+\frac {16 b}{a^{9} \sqrt {x}}-\frac {1}{a^{8} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/(b*x^(1/2)+a)^8,x)
[Out]
-1/a^8/x+36*b^2*ln(x)/a^10-72*b^2*ln(b*x^(1/2)+a)/a^10+16*b/a^9/x^(1/2)+2/7*b^2/a^3/(b*x^(1/2)+a)^7+b^2/a^4/(b
*x^(1/2)+a)^6+12/5*b^2/a^5/(b*x^(1/2)+a)^5+5*b^2/a^6/(b*x^(1/2)+a)^4+10*b^2/a^7/(b*x^(1/2)+a)^3+21*b^2/a^8/(b*
x^(1/2)+a)^2+56*b^2/a^9/(b*x^(1/2)+a)
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maxima [A] time = 0.98, size = 196, normalized size = 1.07 \[ \frac {2520 \, b^{8} x^{4} + 16380 \, a b^{7} x^{\frac {7}{2}} + 44940 \, a^{2} b^{6} x^{3} + 66990 \, a^{3} b^{5} x^{\frac {5}{2}} + 57834 \, a^{4} b^{4} x^{2} + 28098 \, a^{5} b^{3} x^{\frac {3}{2}} + 6534 \, a^{6} b^{2} x + 315 \, a^{7} b \sqrt {x} - 35 \, a^{8}}{35 \, {\left (a^{9} b^{7} x^{\frac {9}{2}} + 7 \, a^{10} b^{6} x^{4} + 21 \, a^{11} b^{5} x^{\frac {7}{2}} + 35 \, a^{12} b^{4} x^{3} + 35 \, a^{13} b^{3} x^{\frac {5}{2}} + 21 \, a^{14} b^{2} x^{2} + 7 \, a^{15} b x^{\frac {3}{2}} + a^{16} x\right )}} - \frac {72 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{10}} + \frac {36 \, b^{2} \log \relax (x)}{a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*x^(1/2))^8,x, algorithm="maxima")
[Out]
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 + 280
98*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*lo
g(b*sqrt(x) + a)/a^10 + 36*b^2*log(x)/a^10
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mupad [B] time = 1.26, size = 189, normalized size = 1.03 \[ \frac {\frac {9\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {6534\,b^2\,x}{35\,a^3}+\frac {8262\,b^4\,x^2}{5\,a^5}+\frac {4014\,b^3\,x^{3/2}}{5\,a^4}+\frac {1284\,b^6\,x^3}{a^7}+\frac {1914\,b^5\,x^{5/2}}{a^6}+\frac {72\,b^8\,x^4}{a^9}+\frac {468\,b^7\,x^{7/2}}{a^8}}{a^7\,x+b^7\,x^{9/2}+7\,a\,b^6\,x^4+7\,a^6\,b\,x^{3/2}+21\,a^5\,b^2\,x^2+35\,a^3\,b^4\,x^3+35\,a^4\,b^3\,x^{5/2}+21\,a^2\,b^5\,x^{7/2}}-\frac {144\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2*(a + b*x^(1/2))^8),x)
[Out]
((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
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sympy [A] time = 20.42, size = 2854, normalized size = 15.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**2/(a+b*x**(1/2))**8,x)
[Out]
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*sq
rt(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 + 245*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) + 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) + 28098*a**6*b**3*x**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 + 245*a
**11*b**6*x**(9/2) + 35*a**10*b**7*x**5) + 26460*a**5*b**4*x**(5/2)*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) - 52920*a**5*b**4*x**(5/2)*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) + 57834*a**5*b**4*x**(5/2)/(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 + 245*a*
*11*b**6*x**(9/2) + 35*a**10*b**7*x**5) + 44100*a**4*b**5*x**3*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) - 88200*a**4*b**5*x**3*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) + 66990*a**4*b**5*x**3/(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) + 44100*a**3*b**6*x**(7/2)*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) - 88200*a**3*b**6*x**(7/2)*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) + 44940*a**3*b**6*x**(7/2)/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**1
5*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) + 26460*a**2*b**7*x**4*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) - 52920*a**2*b**7*x**4*log(a/b + sqrt(x))/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**1
5*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) + 16380*a**2*b**7*x**4/(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**1
0*b**7*x**5) + 8820*a*b**8*x**(9/2)*log(x)/(35*a**17*x**(3/2) + 245*a**16*b*x**2 + 735*a**15*b**2*x**(5/2) + 1
225*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*b**8*x**(9/2)*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) + 2520*a*b**8*x**(9/2)/(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 + 245*a**11*b**6*x**(9/2) + 35*a**10*b**7*x**5)
+ 1260*b**9*x**5*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*b**9
*x**5*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), True))
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