\(\int \frac {(a+b \sqrt {x})^{15}}{x^7} \, dx\) [70]

Optimal result

Integrand size = 15, antiderivative size = 196 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=-\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2}+455 a^3 b^{12} \log (x) \] Output:

-1/6*a^15/x^6-30/11*a^14*b/x^(11/2)-21*a^13*b^2/x^5-910/9*a^12*b^3/x^(9/2) 
-1365/4*a^11*b^4/x^4-858*a^10*b^5/x^(7/2)-5005/3*a^9*b^6/x^3-2574*a^8*b^7/ 
x^(5/2)-6435/2*a^7*b^8/x^2-10010/3*a^6*b^9/x^(3/2)-3003*a^5*b^10/x-2730*a^ 
4*b^11/x^(1/2)+210*a^2*b^13*x^(1/2)+15*a*b^14*x+2/3*b^15*x^(3/2)+455*a^3*b 
^12*ln(x)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=\frac {-66 a^{15}-1080 a^{14} b \sqrt {x}-8316 a^{13} b^2 x-40040 a^{12} b^3 x^{3/2}-135135 a^{11} b^4 x^2-339768 a^{10} b^5 x^{5/2}-660660 a^9 b^6 x^3-1019304 a^8 b^7 x^{7/2}-1274130 a^7 b^8 x^4-1321320 a^6 b^9 x^{9/2}-1189188 a^5 b^{10} x^5-1081080 a^4 b^{11} x^{11/2}+83160 a^2 b^{13} x^{13/2}+5940 a b^{14} x^7+264 b^{15} x^{15/2}}{396 x^6}+910 a^3 b^{12} \log \left (\sqrt {x}\right ) \] Input:

Integrate[(a + b*Sqrt[x])^15/x^7,x]
 

Output:

(-66*a^15 - 1080*a^14*b*Sqrt[x] - 8316*a^13*b^2*x - 40040*a^12*b^3*x^(3/2) 
 - 135135*a^11*b^4*x^2 - 339768*a^10*b^5*x^(5/2) - 660660*a^9*b^6*x^3 - 10 
19304*a^8*b^7*x^(7/2) - 1274130*a^7*b^8*x^4 - 1321320*a^6*b^9*x^(9/2) - 11 
89188*a^5*b^10*x^5 - 1081080*a^4*b^11*x^(11/2) + 83160*a^2*b^13*x^(13/2) + 
 5940*a*b^14*x^7 + 264*b^15*x^(15/2))/(396*x^6) + 910*a^3*b^12*Log[Sqrt[x] 
]
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 49, 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.

\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13/2}}d\sqrt {x}\)

\(\Big \downarrow \) 49

\(\displaystyle 2 \int \left (\frac {a^{15}}{x^{13/2}}+\frac {15 b a^{14}}{x^6}+\frac {105 b^2 a^{13}}{x^{11/2}}+\frac {455 b^3 a^{12}}{x^5}+\frac {1365 b^4 a^{11}}{x^{9/2}}+\frac {3003 b^5 a^{10}}{x^4}+\frac {5005 b^6 a^9}{x^{7/2}}+\frac {6435 b^7 a^8}{x^3}+\frac {6435 b^8 a^7}{x^{5/2}}+\frac {5005 b^9 a^6}{x^2}+\frac {3003 b^{10} a^5}{x^{3/2}}+\frac {1365 b^{11} a^4}{x}+\frac {455 b^{12} a^3}{\sqrt {x}}+105 b^{13} a^2+15 b^{14} \sqrt {x} a+b^{15} x\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^{15}}{12 x^6}-\frac {15 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{2 x^5}-\frac {455 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{8 x^4}-\frac {429 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{6 x^3}-\frac {1287 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{4 x^2}-\frac {5005 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{2 x}-\frac {1365 a^4 b^{11}}{\sqrt {x}}+455 a^3 b^{12} \log \left (\sqrt {x}\right )+105 a^2 b^{13} \sqrt {x}+\frac {15}{2} a b^{14} x+\frac {1}{3} b^{15} x^{3/2}\right )\)

Input:

Int[(a + b*Sqrt[x])^15/x^7,x]
 

Output:

2*(-1/12*a^15/x^6 - (15*a^14*b)/(11*x^(11/2)) - (21*a^13*b^2)/(2*x^5) - (4 
55*a^12*b^3)/(9*x^(9/2)) - (1365*a^11*b^4)/(8*x^4) - (429*a^10*b^5)/x^(7/2 
) - (5005*a^9*b^6)/(6*x^3) - (1287*a^8*b^7)/x^(5/2) - (6435*a^7*b^8)/(4*x^ 
2) - (5005*a^6*b^9)/(3*x^(3/2)) - (3003*a^5*b^10)/(2*x) - (1365*a^4*b^11)/ 
Sqrt[x] + 105*a^2*b^13*Sqrt[x] + (15*a*b^14*x)/2 + (b^15*x^(3/2))/3 + 455* 
a^3*b^12*Log[Sqrt[x]])
 

Defintions of rubi rules used

rule 49

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] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 798

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]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 23.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(-\frac {a^{15}}{6 x^{6}}-\frac {30 a^{14} b}{11 x^{\frac {11}{2}}}-\frac {21 a^{13} b^{2}}{x^{5}}-\frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}}-\frac {1365 a^{11} b^{4}}{4 x^{4}}-\frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}}-\frac {5005 a^{9} b^{6}}{3 x^{3}}-\frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}}-\frac {6435 a^{7} b^{8}}{2 x^{2}}-\frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}}-\frac {3003 a^{5} b^{10}}{x}-\frac {2730 a^{4} b^{11}}{\sqrt {x}}+210 a^{2} b^{13} \sqrt {x}+15 a \,b^{14} x +\frac {2 b^{15} x^{\frac {3}{2}}}{3}+455 a^{3} b^{12} \ln \left (x \right )\)	\(165\)
default	\(-\frac {a^{15}}{6 x^{6}}-\frac {30 a^{14} b}{11 x^{\frac {11}{2}}}-\frac {21 a^{13} b^{2}}{x^{5}}-\frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}}-\frac {1365 a^{11} b^{4}}{4 x^{4}}-\frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}}-\frac {5005 a^{9} b^{6}}{3 x^{3}}-\frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}}-\frac {6435 a^{7} b^{8}}{2 x^{2}}-\frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}}-\frac {3003 a^{5} b^{10}}{x}-\frac {2730 a^{4} b^{11}}{\sqrt {x}}+210 a^{2} b^{13} \sqrt {x}+15 a \,b^{14} x +\frac {2 b^{15} x^{\frac {3}{2}}}{3}+455 a^{3} b^{12} \ln \left (x \right )\)	\(165\)
trager	\(\frac {\left (-1+x \right ) \left (180 b^{14} x^{6}+2 a^{14} x^{5}+252 a^{12} b^{2} x^{5}+4095 a^{10} b^{4} x^{5}+20020 a^{8} b^{6} x^{5}+38610 a^{6} b^{8} x^{5}+36036 a^{4} b^{10} x^{5}+2 a^{14} x^{4}+252 a^{12} b^{2} x^{4}+4095 b^{4} x^{4} a^{10}+20020 x^{4} b^{6} a^{8}+38610 a^{6} b^{8} x^{4}+2 a^{14} x^{3}+252 a^{12} b^{2} x^{3}+4095 a^{10} b^{4} x^{3}+20020 a^{8} b^{6} x^{3}+2 a^{14} x^{2}+252 a^{12} b^{2} x^{2}+4095 a^{10} b^{4} x^{2}+2 a^{14} x +252 a^{12} b^{2} x +2 a^{14}\right ) a}{12 x^{6}}-\frac {2 \left (-33 x^{7} b^{14}-10395 a^{2} b^{12} x^{6}+135135 a^{4} b^{10} x^{5}+165165 a^{6} b^{8} x^{4}+127413 a^{8} b^{6} x^{3}+42471 a^{10} b^{4} x^{2}+5005 a^{12} b^{2} x +135 a^{14}\right ) b}{99 x^{\frac {11}{2}}}+455 a^{3} b^{12} \ln \left (x \right )\)	\(320\)

Input:

int((a+b*x^(1/2))^15/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/6*a^15/x^6-30/11*a^14*b/x^(11/2)-21*a^13*b^2/x^5-910/9*a^12*b^3/x^(9/2) 
-1365/4*a^11*b^4/x^4-858*a^10*b^5/x^(7/2)-5005/3*a^9*b^6/x^3-2574*a^8*b^7/ 
x^(5/2)-6435/2*a^7*b^8/x^2-10010/3*a^6*b^9/x^(3/2)-3003*a^5*b^10/x-2730*a^ 
4*b^11/x^(1/2)+210*a^2*b^13*x^(1/2)+15*a*b^14*x+2/3*b^15*x^(3/2)+455*a^3*b 
^12*ln(x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=\frac {5940 \, a b^{14} x^{7} + 360360 \, a^{3} b^{12} x^{6} \log \left (\sqrt {x}\right ) - 1189188 \, a^{5} b^{10} x^{5} - 1274130 \, a^{7} b^{8} x^{4} - 660660 \, a^{9} b^{6} x^{3} - 135135 \, a^{11} b^{4} x^{2} - 8316 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \, {\left (33 \, b^{15} x^{7} + 10395 \, a^{2} b^{13} x^{6} - 135135 \, a^{4} b^{11} x^{5} - 165165 \, a^{6} b^{9} x^{4} - 127413 \, a^{8} b^{7} x^{3} - 42471 \, a^{10} b^{5} x^{2} - 5005 \, a^{12} b^{3} x - 135 \, a^{14} b\right )} \sqrt {x}}{396 \, x^{6}} \] Input:

integrate((a+b*x^(1/2))^15/x^7,x, algorithm="fricas")
 

Output:

1/396*(5940*a*b^14*x^7 + 360360*a^3*b^12*x^6*log(sqrt(x)) - 1189188*a^5*b^ 
10*x^5 - 1274130*a^7*b^8*x^4 - 660660*a^9*b^6*x^3 - 135135*a^11*b^4*x^2 - 
8316*a^13*b^2*x - 66*a^15 + 8*(33*b^15*x^7 + 10395*a^2*b^13*x^6 - 135135*a 
^4*b^11*x^5 - 165165*a^6*b^9*x^4 - 127413*a^8*b^7*x^3 - 42471*a^10*b^5*x^2 
 - 5005*a^12*b^3*x - 135*a^14*b)*sqrt(x))/x^6
 

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=- \frac {a^{15}}{6 x^{6}} - \frac {30 a^{14} b}{11 x^{\frac {11}{2}}} - \frac {21 a^{13} b^{2}}{x^{5}} - \frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}} - \frac {1365 a^{11} b^{4}}{4 x^{4}} - \frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}} - \frac {5005 a^{9} b^{6}}{3 x^{3}} - \frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}} - \frac {6435 a^{7} b^{8}}{2 x^{2}} - \frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}} - \frac {3003 a^{5} b^{10}}{x} - \frac {2730 a^{4} b^{11}}{\sqrt {x}} + 455 a^{3} b^{12} \log {\left (x \right )} + 210 a^{2} b^{13} \sqrt {x} + 15 a b^{14} x + \frac {2 b^{15} x^{\frac {3}{2}}}{3} \] Input:

integrate((a+b*x**(1/2))**15/x**7,x)
 

Output:

-a**15/(6*x**6) - 30*a**14*b/(11*x**(11/2)) - 21*a**13*b**2/x**5 - 910*a** 
12*b**3/(9*x**(9/2)) - 1365*a**11*b**4/(4*x**4) - 858*a**10*b**5/x**(7/2) 
- 5005*a**9*b**6/(3*x**3) - 2574*a**8*b**7/x**(5/2) - 6435*a**7*b**8/(2*x* 
*2) - 10010*a**6*b**9/(3*x**(3/2)) - 3003*a**5*b**10/x - 2730*a**4*b**11/s 
qrt(x) + 455*a**3*b**12*log(x) + 210*a**2*b**13*sqrt(x) + 15*a*b**14*x + 2 
*b**15*x**(3/2)/3
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=\frac {2}{3} \, b^{15} x^{\frac {3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \left (x\right ) + 210 \, a^{2} b^{13} \sqrt {x} - \frac {1081080 \, a^{4} b^{11} x^{\frac {11}{2}} + 1189188 \, a^{5} b^{10} x^{5} + 1321320 \, a^{6} b^{9} x^{\frac {9}{2}} + 1274130 \, a^{7} b^{8} x^{4} + 1019304 \, a^{8} b^{7} x^{\frac {7}{2}} + 660660 \, a^{9} b^{6} x^{3} + 339768 \, a^{10} b^{5} x^{\frac {5}{2}} + 135135 \, a^{11} b^{4} x^{2} + 40040 \, a^{12} b^{3} x^{\frac {3}{2}} + 8316 \, a^{13} b^{2} x + 1080 \, a^{14} b \sqrt {x} + 66 \, a^{15}}{396 \, x^{6}} \] Input:

integrate((a+b*x^(1/2))^15/x^7,x, algorithm="maxima")
 

Output:

2/3*b^15*x^(3/2) + 15*a*b^14*x + 455*a^3*b^12*log(x) + 210*a^2*b^13*sqrt(x 
) - 1/396*(1081080*a^4*b^11*x^(11/2) + 1189188*a^5*b^10*x^5 + 1321320*a^6* 
b^9*x^(9/2) + 1274130*a^7*b^8*x^4 + 1019304*a^8*b^7*x^(7/2) + 660660*a^9*b 
^6*x^3 + 339768*a^10*b^5*x^(5/2) + 135135*a^11*b^4*x^2 + 40040*a^12*b^3*x^ 
(3/2) + 8316*a^13*b^2*x + 1080*a^14*b*sqrt(x) + 66*a^15)/x^6
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=\frac {2}{3} \, b^{15} x^{\frac {3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \left ({\left | x \right |}\right ) + 210 \, a^{2} b^{13} \sqrt {x} - \frac {1081080 \, a^{4} b^{11} x^{\frac {11}{2}} + 1189188 \, a^{5} b^{10} x^{5} + 1321320 \, a^{6} b^{9} x^{\frac {9}{2}} + 1274130 \, a^{7} b^{8} x^{4} + 1019304 \, a^{8} b^{7} x^{\frac {7}{2}} + 660660 \, a^{9} b^{6} x^{3} + 339768 \, a^{10} b^{5} x^{\frac {5}{2}} + 135135 \, a^{11} b^{4} x^{2} + 40040 \, a^{12} b^{3} x^{\frac {3}{2}} + 8316 \, a^{13} b^{2} x + 1080 \, a^{14} b \sqrt {x} + 66 \, a^{15}}{396 \, x^{6}} \] Input:

integrate((a+b*x^(1/2))^15/x^7,x, algorithm="giac")
 

Output:

2/3*b^15*x^(3/2) + 15*a*b^14*x + 455*a^3*b^12*log(abs(x)) + 210*a^2*b^13*s 
qrt(x) - 1/396*(1081080*a^4*b^11*x^(11/2) + 1189188*a^5*b^10*x^5 + 1321320 
*a^6*b^9*x^(9/2) + 1274130*a^7*b^8*x^4 + 1019304*a^8*b^7*x^(7/2) + 660660* 
a^9*b^6*x^3 + 339768*a^10*b^5*x^(5/2) + 135135*a^11*b^4*x^2 + 40040*a^12*b 
^3*x^(3/2) + 8316*a^13*b^2*x + 1080*a^14*b*sqrt(x) + 66*a^15)/x^6
 

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=\frac {2\,b^{15}\,x^{3/2}}{3}-\frac {\frac {a^{15}}{6}+21\,a^{13}\,b^2\,x+\frac {30\,a^{14}\,b\,\sqrt {x}}{11}+\frac {1365\,a^{11}\,b^4\,x^2}{4}+\frac {5005\,a^9\,b^6\,x^3}{3}+\frac {6435\,a^7\,b^8\,x^4}{2}+3003\,a^5\,b^{10}\,x^5+\frac {910\,a^{12}\,b^3\,x^{3/2}}{9}+858\,a^{10}\,b^5\,x^{5/2}+2574\,a^8\,b^7\,x^{7/2}+\frac {10010\,a^6\,b^9\,x^{9/2}}{3}+2730\,a^4\,b^{11}\,x^{11/2}}{x^6}+910\,a^3\,b^{12}\,\ln \left (\sqrt {x}\right )+210\,a^2\,b^{13}\,\sqrt {x}+15\,a\,b^{14}\,x \] Input:

int((a + b*x^(1/2))^15/x^7,x)
 

Output:

(2*b^15*x^(3/2))/3 - (a^15/6 + 21*a^13*b^2*x + (30*a^14*b*x^(1/2))/11 + (1 
365*a^11*b^4*x^2)/4 + (5005*a^9*b^6*x^3)/3 + (6435*a^7*b^8*x^4)/2 + 3003*a 
^5*b^10*x^5 + (910*a^12*b^3*x^(3/2))/9 + 858*a^10*b^5*x^(5/2) + 2574*a^8*b 
^7*x^(7/2) + (10010*a^6*b^9*x^(9/2))/3 + 2730*a^4*b^11*x^(11/2))/x^6 + 910 
*a^3*b^12*log(x^(1/2)) + 210*a^2*b^13*x^(1/2) + 15*a*b^14*x
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^7} \, dx=\frac {180180 \sqrt {x}\, \mathrm {log}\left (x \right ) a^{3} b^{12} x^{6}-66 \sqrt {x}\, a^{15}-8316 \sqrt {x}\, a^{13} b^{2} x -135135 \sqrt {x}\, a^{11} b^{4} x^{2}-660660 \sqrt {x}\, a^{9} b^{6} x^{3}-1274130 \sqrt {x}\, a^{7} b^{8} x^{4}-1189188 \sqrt {x}\, a^{5} b^{10} x^{5}+5940 \sqrt {x}\, a \,b^{14} x^{7}-1080 a^{14} b x -40040 a^{12} b^{3} x^{2}-339768 a^{10} b^{5} x^{3}-1019304 a^{8} b^{7} x^{4}-1321320 a^{6} b^{9} x^{5}-1081080 a^{4} b^{11} x^{6}+83160 a^{2} b^{13} x^{7}+264 b^{15} x^{8}}{396 \sqrt {x}\, x^{6}} \] Input:

int((a+b*x^(1/2))^15/x^7,x)
 

Output:

(180180*sqrt(x)*log(x)*a**3*b**12*x**6 - 66*sqrt(x)*a**15 - 8316*sqrt(x)*a 
**13*b**2*x - 135135*sqrt(x)*a**11*b**4*x**2 - 660660*sqrt(x)*a**9*b**6*x* 
*3 - 1274130*sqrt(x)*a**7*b**8*x**4 - 1189188*sqrt(x)*a**5*b**10*x**5 + 59 
40*sqrt(x)*a*b**14*x**7 - 1080*a**14*b*x - 40040*a**12*b**3*x**2 - 339768* 
a**10*b**5*x**3 - 1019304*a**8*b**7*x**4 - 1321320*a**6*b**9*x**5 - 108108 
0*a**4*b**11*x**6 + 83160*a**2*b**13*x**7 + 264*b**15*x**8)/(396*sqrt(x)*x 
**6)