3.2231 \(\int \frac {1}{(a+b \sqrt {x})^8 x^3} \, dx\)
Optimal. Leaf size=217 \[ -\frac {660 b^4 \log \left (a+b \sqrt {x}\right )}{a^{12}}+\frac {330 b^4 \log (x)}{a^{12}}+\frac {420 b^4}{a^{11} \left (a+b \sqrt {x}\right )}+\frac {240 b^3}{a^{11} \sqrt {x}}+\frac {126 b^4}{a^{10} \left (a+b \sqrt {x}\right )^2}-\frac {36 b^2}{a^{10} x}+\frac {140 b^4}{3 a^9 \left (a+b \sqrt {x}\right )^3}+\frac {16 b}{3 a^9 x^{3/2}}+\frac {35 b^4}{2 a^8 \left (a+b \sqrt {x}\right )^4}-\frac {1}{2 a^8 x^2}+\frac {6 b^4}{a^7 \left (a+b \sqrt {x}\right )^5}+\frac {5 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^6}+\frac {2 b^4}{7 a^5 \left (a+b \sqrt {x}\right )^7} \]
[Out]
-1/2/a^8/x^2+16/3*b/a^9/x^(3/2)-36*b^2/a^10/x+330*b^4*ln(x)/a^12-660*b^4*ln(a+b*x^(1/2))/a^12+240*b^3/a^11/x^(
1/2)+2/7*b^4/a^5/(a+b*x^(1/2))^7+5/3*b^4/a^6/(a+b*x^(1/2))^6+6*b^4/a^7/(a+b*x^(1/2))^5+35/2*b^4/a^8/(a+b*x^(1/
2))^4+140/3*b^4/a^9/(a+b*x^(1/2))^3+126*b^4/a^10/(a+b*x^(1/2))^2+420*b^4/a^11/(a+b*x^(1/2))
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Rubi [A] time = 0.19, antiderivative size = 217, 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} \[ \frac {420 b^4}{a^{11} \left (a+b \sqrt {x}\right )}+\frac {126 b^4}{a^{10} \left (a+b \sqrt {x}\right )^2}+\frac {140 b^4}{3 a^9 \left (a+b \sqrt {x}\right )^3}+\frac {35 b^4}{2 a^8 \left (a+b \sqrt {x}\right )^4}+\frac {6 b^4}{a^7 \left (a+b \sqrt {x}\right )^5}+\frac {5 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^6}+\frac {2 b^4}{7 a^5 \left (a+b \sqrt {x}\right )^7}+\frac {240 b^3}{a^{11} \sqrt {x}}-\frac {36 b^2}{a^{10} x}-\frac {660 b^4 \log \left (a+b \sqrt {x}\right )}{a^{12}}+\frac {330 b^4 \log (x)}{a^{12}}+\frac {16 b}{3 a^9 x^{3/2}}-\frac {1}{2 a^8 x^2} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*Sqrt[x])^8*x^3),x]
[Out]
(2*b^4)/(7*a^5*(a + b*Sqrt[x])^7) + (5*b^4)/(3*a^6*(a + b*Sqrt[x])^6) + (6*b^4)/(a^7*(a + b*Sqrt[x])^5) + (35*
b^4)/(2*a^8*(a + b*Sqrt[x])^4) + (140*b^4)/(3*a^9*(a + b*Sqrt[x])^3) + (126*b^4)/(a^10*(a + b*Sqrt[x])^2) + (4
20*b^4)/(a^11*(a + b*Sqrt[x])) - 1/(2*a^8*x^2) + (16*b)/(3*a^9*x^(3/2)) - (36*b^2)/(a^10*x) + (240*b^3)/(a^11*
Sqrt[x]) - (660*b^4*Log[a + b*Sqrt[x]])/a^12 + (330*b^4*Log[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])
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^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^5 (a+b x)^8} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a^8 x^5}-\frac {8 b}{a^9 x^4}+\frac {36 b^2}{a^{10} x^3}-\frac {120 b^3}{a^{11} x^2}+\frac {330 b^4}{a^{12} x}-\frac {b^5}{a^5 (a+b x)^8}-\frac {5 b^5}{a^6 (a+b x)^7}-\frac {15 b^5}{a^7 (a+b x)^6}-\frac {35 b^5}{a^8 (a+b x)^5}-\frac {70 b^5}{a^9 (a+b x)^4}-\frac {126 b^5}{a^{10} (a+b x)^3}-\frac {210 b^5}{a^{11} (a+b x)^2}-\frac {330 b^5}{a^{12} (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 b^4}{7 a^5 \left (a+b \sqrt {x}\right )^7}+\frac {5 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^6}+\frac {6 b^4}{a^7 \left (a+b \sqrt {x}\right )^5}+\frac {35 b^4}{2 a^8 \left (a+b \sqrt {x}\right )^4}+\frac {140 b^4}{3 a^9 \left (a+b \sqrt {x}\right )^3}+\frac {126 b^4}{a^{10} \left (a+b \sqrt {x}\right )^2}+\frac {420 b^4}{a^{11} \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^8 x^2}+\frac {16 b}{3 a^9 x^{3/2}}-\frac {36 b^2}{a^{10} x}+\frac {240 b^3}{a^{11} \sqrt {x}}-\frac {660 b^4 \log \left (a+b \sqrt {x}\right )}{a^{12}}+\frac {330 b^4 \log (x)}{a^{12}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 163, normalized size = 0.75 \[ \frac {\frac {a \left (-21 a^{10}+77 a^9 b \sqrt {x}-385 a^8 b^2 x+3465 a^7 b^3 x^{3/2}+71874 a^6 b^4 x^2+309078 a^5 b^5 x^{5/2}+636174 a^4 b^6 x^3+736890 a^3 b^7 x^{7/2}+494340 a^2 b^8 x^4+180180 a b^9 x^{9/2}+27720 b^{10} x^5\right )}{x^2 \left (a+b \sqrt {x}\right )^7}-27720 b^4 \log \left (a+b \sqrt {x}\right )+13860 b^4 \log (x)}{42 a^{12}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*Sqrt[x])^8*x^3),x]
[Out]
((a*(-21*a^10 + 77*a^9*b*Sqrt[x] - 385*a^8*b^2*x + 3465*a^7*b^3*x^(3/2) + 71874*a^6*b^4*x^2 + 309078*a^5*b^5*x
^(5/2) + 636174*a^4*b^6*x^3 + 736890*a^3*b^7*x^(7/2) + 494340*a^2*b^8*x^4 + 180180*a*b^9*x^(9/2) + 27720*b^10*
x^5))/((a + b*Sqrt[x])^7*x^2) - 27720*b^4*Log[a + b*Sqrt[x]] + 13860*b^4*Log[x])/(42*a^12)
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fricas [B] time = 1.02, size = 463, normalized size = 2.13 \[ -\frac {13860 \, a^{2} b^{16} x^{8} - 90090 \, a^{4} b^{14} x^{7} + 247170 \, a^{6} b^{12} x^{6} - 368445 \, a^{8} b^{10} x^{5} + 318087 \, a^{10} b^{8} x^{4} - 154532 \, a^{12} b^{6} x^{3} + 36104 \, a^{14} b^{4} x^{2} - 1365 \, a^{16} b^{2} x - 21 \, a^{18} + 27720 \, {\left (b^{18} x^{9} - 7 \, a^{2} b^{16} x^{8} + 21 \, a^{4} b^{14} x^{7} - 35 \, a^{6} b^{12} x^{6} + 35 \, a^{8} b^{10} x^{5} - 21 \, a^{10} b^{8} x^{4} + 7 \, a^{12} b^{6} x^{3} - a^{14} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 27720 \, {\left (b^{18} x^{9} - 7 \, a^{2} b^{16} x^{8} + 21 \, a^{4} b^{14} x^{7} - 35 \, a^{6} b^{12} x^{6} + 35 \, a^{8} b^{10} x^{5} - 21 \, a^{10} b^{8} x^{4} + 7 \, a^{12} b^{6} x^{3} - a^{14} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 8 \, {\left (3465 \, a b^{17} x^{8} - 23100 \, a^{3} b^{15} x^{7} + 65373 \, a^{5} b^{13} x^{6} - 101376 \, a^{7} b^{11} x^{5} + 92323 \, a^{9} b^{9} x^{4} - 48580 \, a^{11} b^{7} x^{3} + 13083 \, a^{13} b^{5} x^{2} - 1064 \, a^{15} b^{3} x - 28 \, a^{17} b\right )} \sqrt {x}}{42 \, {\left (a^{12} b^{14} x^{9} - 7 \, a^{14} b^{12} x^{8} + 21 \, a^{16} b^{10} x^{7} - 35 \, a^{18} b^{8} x^{6} + 35 \, a^{20} b^{6} x^{5} - 21 \, a^{22} b^{4} x^{4} + 7 \, a^{24} b^{2} x^{3} - a^{26} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(a+b*x^(1/2))^8,x, algorithm="fricas")
[Out]
-1/42*(13860*a^2*b^16*x^8 - 90090*a^4*b^14*x^7 + 247170*a^6*b^12*x^6 - 368445*a^8*b^10*x^5 + 318087*a^10*b^8*x
^4 - 154532*a^12*b^6*x^3 + 36104*a^14*b^4*x^2 - 1365*a^16*b^2*x - 21*a^18 + 27720*(b^18*x^9 - 7*a^2*b^16*x^8 +
21*a^4*b^14*x^7 - 35*a^6*b^12*x^6 + 35*a^8*b^10*x^5 - 21*a^10*b^8*x^4 + 7*a^12*b^6*x^3 - a^14*b^4*x^2)*log(b*
sqrt(x) + a) - 27720*(b^18*x^9 - 7*a^2*b^16*x^8 + 21*a^4*b^14*x^7 - 35*a^6*b^12*x^6 + 35*a^8*b^10*x^5 - 21*a^1
0*b^8*x^4 + 7*a^12*b^6*x^3 - a^14*b^4*x^2)*log(sqrt(x)) - 8*(3465*a*b^17*x^8 - 23100*a^3*b^15*x^7 + 65373*a^5*
b^13*x^6 - 101376*a^7*b^11*x^5 + 92323*a^9*b^9*x^4 - 48580*a^11*b^7*x^3 + 13083*a^13*b^5*x^2 - 1064*a^15*b^3*x
- 28*a^17*b)*sqrt(x))/(a^12*b^14*x^9 - 7*a^14*b^12*x^8 + 21*a^16*b^10*x^7 - 35*a^18*b^8*x^6 + 35*a^20*b^6*x^5
- 21*a^22*b^4*x^4 + 7*a^24*b^2*x^3 - a^26*x^2)
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giac [A] time = 0.17, size = 156, normalized size = 0.72 \[ -\frac {660 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{12}} + \frac {330 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac {27720 \, a b^{10} x^{5} + 180180 \, a^{2} b^{9} x^{\frac {9}{2}} + 494340 \, a^{3} b^{8} x^{4} + 736890 \, a^{4} b^{7} x^{\frac {7}{2}} + 636174 \, a^{5} b^{6} x^{3} + 309078 \, a^{6} b^{5} x^{\frac {5}{2}} + 71874 \, a^{7} b^{4} x^{2} + 3465 \, a^{8} b^{3} x^{\frac {3}{2}} - 385 \, a^{9} b^{2} x + 77 \, a^{10} b \sqrt {x} - 21 \, a^{11}}{42 \, {\left (b \sqrt {x} + a\right )}^{7} a^{12} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(a+b*x^(1/2))^8,x, algorithm="giac")
[Out]
-660*b^4*log(abs(b*sqrt(x) + a))/a^12 + 330*b^4*log(abs(x))/a^12 + 1/42*(27720*a*b^10*x^5 + 180180*a^2*b^9*x^(
9/2) + 494340*a^3*b^8*x^4 + 736890*a^4*b^7*x^(7/2) + 636174*a^5*b^6*x^3 + 309078*a^6*b^5*x^(5/2) + 71874*a^7*b
^4*x^2 + 3465*a^8*b^3*x^(3/2) - 385*a^9*b^2*x + 77*a^10*b*sqrt(x) - 21*a^11)/((b*sqrt(x) + a)^7*a^12*x^2)
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maple [A] time = 0.02, size = 186, normalized size = 0.86 \[ \frac {2 b^{4}}{7 \left (b \sqrt {x}+a \right )^{7} a^{5}}+\frac {5 b^{4}}{3 \left (b \sqrt {x}+a \right )^{6} a^{6}}+\frac {6 b^{4}}{\left (b \sqrt {x}+a \right )^{5} a^{7}}+\frac {35 b^{4}}{2 \left (b \sqrt {x}+a \right )^{4} a^{8}}+\frac {140 b^{4}}{3 \left (b \sqrt {x}+a \right )^{3} a^{9}}+\frac {126 b^{4}}{\left (b \sqrt {x}+a \right )^{2} a^{10}}+\frac {420 b^{4}}{\left (b \sqrt {x}+a \right ) a^{11}}+\frac {330 b^{4} \ln \relax (x )}{a^{12}}-\frac {660 b^{4} \ln \left (b \sqrt {x}+a \right )}{a^{12}}+\frac {240 b^{3}}{a^{11} \sqrt {x}}-\frac {36 b^{2}}{a^{10} x}+\frac {16 b}{3 a^{9} x^{\frac {3}{2}}}-\frac {1}{2 a^{8} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(b*x^(1/2)+a)^8,x)
[Out]
-1/2/a^8/x^2+16/3*b/a^9/x^(3/2)-36*b^2/a^10/x+330*b^4*ln(x)/a^12-660*b^4*ln(b*x^(1/2)+a)/a^12+240*b^3/a^11/x^(
1/2)+2/7*b^4/a^5/(b*x^(1/2)+a)^7+5/3*b^4/a^6/(b*x^(1/2)+a)^6+6*b^4/a^7/(b*x^(1/2)+a)^5+35/2*b^4/a^8/(b*x^(1/2)
+a)^4+140/3*b^4/a^9/(b*x^(1/2)+a)^3+126*b^4/a^10/(b*x^(1/2)+a)^2+420*b^4/a^11/(b*x^(1/2)+a)
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maxima [A] time = 1.11, size = 220, normalized size = 1.01 \[ \frac {27720 \, b^{10} x^{5} + 180180 \, a b^{9} x^{\frac {9}{2}} + 494340 \, a^{2} b^{8} x^{4} + 736890 \, a^{3} b^{7} x^{\frac {7}{2}} + 636174 \, a^{4} b^{6} x^{3} + 309078 \, a^{5} b^{5} x^{\frac {5}{2}} + 71874 \, a^{6} b^{4} x^{2} + 3465 \, a^{7} b^{3} x^{\frac {3}{2}} - 385 \, a^{8} b^{2} x + 77 \, a^{9} b \sqrt {x} - 21 \, a^{10}}{42 \, {\left (a^{11} b^{7} x^{\frac {11}{2}} + 7 \, a^{12} b^{6} x^{5} + 21 \, a^{13} b^{5} x^{\frac {9}{2}} + 35 \, a^{14} b^{4} x^{4} + 35 \, a^{15} b^{3} x^{\frac {7}{2}} + 21 \, a^{16} b^{2} x^{3} + 7 \, a^{17} b x^{\frac {5}{2}} + a^{18} x^{2}\right )}} - \frac {660 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{12}} + \frac {330 \, b^{4} \log \relax (x)}{a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(a+b*x^(1/2))^8,x, algorithm="maxima")
[Out]
1/42*(27720*b^10*x^5 + 180180*a*b^9*x^(9/2) + 494340*a^2*b^8*x^4 + 736890*a^3*b^7*x^(7/2) + 636174*a^4*b^6*x^3
+ 309078*a^5*b^5*x^(5/2) + 71874*a^6*b^4*x^2 + 3465*a^7*b^3*x^(3/2) - 385*a^8*b^2*x + 77*a^9*b*sqrt(x) - 21*a
^10)/(a^11*b^7*x^(11/2) + 7*a^12*b^6*x^5 + 21*a^13*b^5*x^(9/2) + 35*a^14*b^4*x^4 + 35*a^15*b^3*x^(7/2) + 21*a^
16*b^2*x^3 + 7*a^17*b*x^(5/2) + a^18*x^2) - 660*b^4*log(b*sqrt(x) + a)/a^12 + 330*b^4*log(x)/a^12
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mupad [B] time = 1.60, size = 213, normalized size = 0.98 \[ \frac {\frac {11\,b\,\sqrt {x}}{6\,a^2}-\frac {1}{2\,a}-\frac {55\,b^2\,x}{6\,a^3}+\frac {11979\,b^4\,x^2}{7\,a^5}+\frac {165\,b^3\,x^{3/2}}{2\,a^4}+\frac {15147\,b^6\,x^3}{a^7}+\frac {7359\,b^5\,x^{5/2}}{a^6}+\frac {11770\,b^8\,x^4}{a^9}+\frac {17545\,b^7\,x^{7/2}}{a^8}+\frac {660\,b^{10}\,x^5}{a^{11}}+\frac {4290\,b^9\,x^{9/2}}{a^{10}}}{a^7\,x^2+b^7\,x^{11/2}+7\,a\,b^6\,x^5+7\,a^6\,b\,x^{5/2}+21\,a^5\,b^2\,x^3+35\,a^3\,b^4\,x^4+35\,a^4\,b^3\,x^{7/2}+21\,a^2\,b^5\,x^{9/2}}-\frac {1320\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3*(a + b*x^(1/2))^8),x)
[Out]
((11*b*x^(1/2))/(6*a^2) - 1/(2*a) - (55*b^2*x)/(6*a^3) + (11979*b^4*x^2)/(7*a^5) + (165*b^3*x^(3/2))/(2*a^4) +
(15147*b^6*x^3)/a^7 + (7359*b^5*x^(5/2))/a^6 + (11770*b^8*x^4)/a^9 + (17545*b^7*x^(7/2))/a^8 + (660*b^10*x^5)
/a^11 + (4290*b^9*x^(9/2))/a^10)/(a^7*x^2 + b^7*x^(11/2) + 7*a*b^6*x^5 + 7*a^6*b*x^(5/2) + 21*a^5*b^2*x^3 + 35
*a^3*b^4*x^4 + 35*a^4*b^3*x^(7/2) + 21*a^2*b^5*x^(9/2)) - (1320*b^4*atanh((2*b*x^(1/2))/a + 1))/a^12
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sympy [A] time = 30.48, size = 3077, normalized size = 14.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(a+b*x**(1/2))**8,x)
[Out]
Piecewise((zoo/x**6, Eq(a, 0) & Eq(b, 0)), (-1/(2*a**8*x**2), Eq(b, 0)), (-1/(6*b**8*x**6), Eq(a, 0)), (-21*a*
*11*sqrt(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**1
5*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 77*a**10*b*x/(42*a**1
9*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 88
2*a**14*b**5*x**5 + 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) - 385*a**9*b**2*x**(3/2)/(42*a**19*x**(5/2)
+ 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b*
*5*x**5 + 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 3465*a**8*b**3*x**2/(42*a**19*x**(5/2) + 294*a**18*
b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294
*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 13860*a**7*b**4*x**(5/2)*log(x)/(42*a**19*x**(5/2) + 294*a**18*b
*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*
a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) - 27720*a**7*b**4*x**(5/2)*log(a/b + sqrt(x))/(42*a**19*x**(5/2) +
294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*
x**5 + 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 71874*a**7*b**4*x**(5/2)/(42*a**19*x**(5/2) + 294*a**1
8*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 2
94*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 97020*a**6*b**5*x**3*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x
**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a*
*13*b**6*x**(11/2) + 42*a**12*b**7*x**6) - 194040*a**6*b**5*x**3*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*a
**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5
+ 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 309078*a**6*b**5*x**3/(42*a**19*x**(5/2) + 294*a**18*b*x**3
+ 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13
*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 291060*a**5*b**6*x**(7/2)*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**3
+ 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13
*b**6*x**(11/2) + 42*a**12*b**7*x**6) - 582120*a**5*b**6*x**(7/2)*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*
a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5
+ 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 636174*a**5*b**6*x**(7/2)/(42*a**19*x**(5/2) + 294*a**18*b
*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*
a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 485100*a**4*b**7*x**4*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**
3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**1
3*b**6*x**(11/2) + 42*a**12*b**7*x**6) - 970200*a**4*b**7*x**4*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*a**
18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 +
294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 736890*a**4*b**7*x**4/(42*a**19*x**(5/2) + 294*a**18*b*x**3 +
882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b
**6*x**(11/2) + 42*a**12*b**7*x**6) + 485100*a**3*b**8*x**(9/2)*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**3 +
882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b
**6*x**(11/2) + 42*a**12*b**7*x**6) - 970200*a**3*b**8*x**(9/2)*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*a*
*18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 +
294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 494340*a**3*b**8*x**(9/2)/(42*a**19*x**(5/2) + 294*a**18*b*x
**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a*
*13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 291060*a**2*b**9*x**5*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**3
+ 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*
b**6*x**(11/2) + 42*a**12*b**7*x**6) - 582120*a**2*b**9*x**5*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*a**18
*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 29
4*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 180180*a**2*b**9*x**5/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 8
82*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**
6*x**(11/2) + 42*a**12*b**7*x**6) + 97020*a*b**10*x**(11/2)*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882
*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**6*
x**(11/2) + 42*a**12*b**7*x**6) - 194040*a*b**10*x**(11/2)*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*a**18*b
*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*
a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 27720*a*b**10*x**(11/2)/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 8
82*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**
6*x**(11/2) + 42*a**12*b**7*x**6) + 13860*b**11*x**6*log(x)/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*
b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**6*x**(11/
2) + 42*a**12*b**7*x**6) - 27720*b**11*x**6*log(a/b + sqrt(x))/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**
17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**6*x**(
11/2) + 42*a**12*b**7*x**6), True))
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