\(\int \frac {(a+b x^2)^{5/2}}{x^{18}} \, dx\) [418]

Optimal result

Integrand size = 15, antiderivative size = 140 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7} \] Output:

-1/17*(b*x^2+a)^(7/2)/a/x^17+2/51*b*(b*x^2+a)^(7/2)/a^2/x^15-16/663*b^2*(b 
*x^2+a)^(7/2)/a^3/x^13+32/2431*b^3*(b*x^2+a)^(7/2)/a^4/x^11-128/21879*b^4* 
(b*x^2+a)^(7/2)/a^5/x^9+256/153153*b^5*(b*x^2+a)^(7/2)/a^6/x^7
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (-9009 a^5+6006 a^4 b x^2-3696 a^3 b^2 x^4+2016 a^2 b^3 x^6-896 a b^4 x^8+256 b^5 x^{10}\right )}{153153 a^6 x^{17}} \] Input:

Integrate[(a + b*x^2)^(5/2)/x^18,x]
 

Output:

((a + b*x^2)^(7/2)*(-9009*a^5 + 6006*a^4*b*x^2 - 3696*a^3*b^2*x^4 + 2016*a 
^2*b^3*x^6 - 896*a*b^4*x^8 + 256*b^5*x^10))/(153153*a^6*x^17)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {245, 245, 245, 245, 245, 242}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {10 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{16}}dx}{17 a}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {10 b \left (-\frac {8 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{14}}dx}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{10}}dx}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^8}dx}{9 a}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}\)

Input:

Int[(a + b*x^2)^(5/2)/x^18,x]
 

Output:

-1/17*(a + b*x^2)^(7/2)/(a*x^17) - (10*b*(-1/15*(a + b*x^2)^(7/2)/(a*x^15) 
 - (8*b*(-1/13*(a + b*x^2)^(7/2)/(a*x^13) - (6*b*(-1/11*(a + b*x^2)^(7/2)/ 
(a*x^11) - (4*b*(-1/9*(a + b*x^2)^(7/2)/(a*x^9) + (2*b*(a + b*x^2)^(7/2))/ 
(63*a^2*x^7)))/(11*a)))/(13*a)))/(15*a)))/(17*a)
 

Defintions of rubi rules used

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.51

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-256 b^{5} x^{10}+896 a \,b^{4} x^{8}-2016 a^{2} b^{3} x^{6}+3696 a^{3} b^{2} x^{4}-6006 a^{4} b \,x^{2}+9009 a^{5}\right )}{153153 x^{17} a^{6}}\)	\(72\)
pseudoelliptic	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-256 b^{5} x^{10}+896 a \,b^{4} x^{8}-2016 a^{2} b^{3} x^{6}+3696 a^{3} b^{2} x^{4}-6006 a^{4} b \,x^{2}+9009 a^{5}\right )}{153153 x^{17} a^{6}}\)	\(72\)
orering	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-256 b^{5} x^{10}+896 a \,b^{4} x^{8}-2016 a^{2} b^{3} x^{6}+3696 a^{3} b^{2} x^{4}-6006 a^{4} b \,x^{2}+9009 a^{5}\right )}{153153 x^{17} a^{6}}\)	\(72\)
trager	\(-\frac {\left (-256 b^{8} x^{16}+128 a \,b^{7} x^{14}-96 a^{2} b^{6} x^{12}+80 a^{3} b^{5} x^{10}-70 a^{4} b^{4} x^{8}+63 a^{5} b^{3} x^{6}+12705 a^{6} b^{2} x^{4}+21021 a^{7} b \,x^{2}+9009 a^{8}\right ) \sqrt {b \,x^{2}+a}}{153153 x^{17} a^{6}}\)	\(105\)
risch	\(-\frac {\left (-256 b^{8} x^{16}+128 a \,b^{7} x^{14}-96 a^{2} b^{6} x^{12}+80 a^{3} b^{5} x^{10}-70 a^{4} b^{4} x^{8}+63 a^{5} b^{3} x^{6}+12705 a^{6} b^{2} x^{4}+21021 a^{7} b \,x^{2}+9009 a^{8}\right ) \sqrt {b \,x^{2}+a}}{153153 x^{17} a^{6}}\)	\(105\)
default	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{17 a \,x^{17}}-\frac {10 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{15 a \,x^{15}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\)	\(133\)

Input:

int((b*x^2+a)^(5/2)/x^18,x,method=_RETURNVERBOSE)
 

Output:

-1/153153*(b*x^2+a)^(7/2)*(-256*b^5*x^10+896*a*b^4*x^8-2016*a^2*b^3*x^6+36 
96*a^3*b^2*x^4-6006*a^4*b*x^2+9009*a^5)/x^17/a^6
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\frac {{\left (256 \, b^{8} x^{16} - 128 \, a b^{7} x^{14} + 96 \, a^{2} b^{6} x^{12} - 80 \, a^{3} b^{5} x^{10} + 70 \, a^{4} b^{4} x^{8} - 63 \, a^{5} b^{3} x^{6} - 12705 \, a^{6} b^{2} x^{4} - 21021 \, a^{7} b x^{2} - 9009 \, a^{8}\right )} \sqrt {b x^{2} + a}}{153153 \, a^{6} x^{17}} \] Input:

integrate((b*x^2+a)^(5/2)/x^18,x, algorithm="fricas")
 

Output:

1/153153*(256*b^8*x^16 - 128*a*b^7*x^14 + 96*a^2*b^6*x^12 - 80*a^3*b^5*x^1 
0 + 70*a^4*b^4*x^8 - 63*a^5*b^3*x^6 - 12705*a^6*b^2*x^4 - 21021*a^7*b*x^2 
- 9009*a^8)*sqrt(b*x^2 + a)/(a^6*x^17)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1346 vs. \(2 (133) = 266\).

Time = 2.12 (sec) , antiderivative size = 1346, normalized size of antiderivative = 9.61 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**(5/2)/x**18,x)
 

Output:

-9009*a**13*b**(51/2)*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765 
765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**2 
2 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 66066*a**12*b**(5 
3/2)*x**2*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b* 
*26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a 
**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 207900*a**11*b**(55/2)*x**4*s 
qrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 
 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x 
**24 + 153153*a**6*b**30*x**26) - 363888*a**10*b**(57/2)*x**6*sqrt(a/(b*x* 
*2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a* 
*9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 1531 
53*a**6*b**30*x**26) - 382550*a**9*b**(59/2)*x**8*sqrt(a/(b*x**2) + 1)/(15 
3153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x** 
20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**3 
0*x**26) - 241524*a**8*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(153153*a**11* 
b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 153153 
0*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 
84780*a**7*b**(63/2)*x**12*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 
+ 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28 
*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 12768*a**...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\frac {256 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}}{153153 \, a^{6} x^{7}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}{21879 \, a^{5} x^{9}} + \frac {32 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{2431 \, a^{4} x^{11}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{663 \, a^{3} x^{13}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{51 \, a^{2} x^{15}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{17 \, a x^{17}} \] Input:

integrate((b*x^2+a)^(5/2)/x^18,x, algorithm="maxima")
 

Output:

256/153153*(b*x^2 + a)^(7/2)*b^5/(a^6*x^7) - 128/21879*(b*x^2 + a)^(7/2)*b 
^4/(a^5*x^9) + 32/2431*(b*x^2 + a)^(7/2)*b^3/(a^4*x^11) - 16/663*(b*x^2 + 
a)^(7/2)*b^2/(a^3*x^13) + 2/51*(b*x^2 + a)^(7/2)*b/(a^2*x^15) - 1/17*(b*x^ 
2 + a)^(7/2)/(a*x^17)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (116) = 232\).

Time = 0.14 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\frac {512 \, {\left (102102 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} b^{\frac {17}{2}} + 364650 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a b^{\frac {17}{2}} + 692835 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{2} b^{\frac {17}{2}} + 668525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{3} b^{\frac {17}{2}} + 384098 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{4} b^{\frac {17}{2}} + 89726 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{5} b^{\frac {17}{2}} + 6188 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{6} b^{\frac {17}{2}} - 2380 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{7} b^{\frac {17}{2}} + 680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{8} b^{\frac {17}{2}} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{9} b^{\frac {17}{2}} + 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{10} b^{\frac {17}{2}} - a^{11} b^{\frac {17}{2}}\right )}}{153153 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{17}} \] Input:

integrate((b*x^2+a)^(5/2)/x^18,x, algorithm="giac")
 

Output:

512/153153*(102102*(sqrt(b)*x - sqrt(b*x^2 + a))^22*b^(17/2) + 364650*(sqr 
t(b)*x - sqrt(b*x^2 + a))^20*a*b^(17/2) + 692835*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^18*a^2*b^(17/2) + 668525*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^3*b^(17/2 
) + 384098*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^4*b^(17/2) + 89726*(sqrt(b)* 
x - sqrt(b*x^2 + a))^12*a^5*b^(17/2) + 6188*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
10*a^6*b^(17/2) - 2380*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^7*b^(17/2) + 680* 
(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^8*b^(17/2) - 136*(sqrt(b)*x - sqrt(b*x^2 
 + a))^4*a^9*b^(17/2) + 17*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^10*b^(17/2) - 
 a^11*b^(17/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^17
 

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\frac {10\,b^4\,\sqrt {b\,x^2+a}}{21879\,a^2\,x^9}-\frac {55\,b^2\,\sqrt {b\,x^2+a}}{663\,x^{13}}-\frac {b^3\,\sqrt {b\,x^2+a}}{2431\,a\,x^{11}}-\frac {a^2\,\sqrt {b\,x^2+a}}{17\,x^{17}}-\frac {80\,b^5\,\sqrt {b\,x^2+a}}{153153\,a^3\,x^7}+\frac {32\,b^6\,\sqrt {b\,x^2+a}}{51051\,a^4\,x^5}-\frac {128\,b^7\,\sqrt {b\,x^2+a}}{153153\,a^5\,x^3}+\frac {256\,b^8\,\sqrt {b\,x^2+a}}{153153\,a^6\,x}-\frac {7\,a\,b\,\sqrt {b\,x^2+a}}{51\,x^{15}} \] Input:

int((a + b*x^2)^(5/2)/x^18,x)
 

Output:

(10*b^4*(a + b*x^2)^(1/2))/(21879*a^2*x^9) - (55*b^2*(a + b*x^2)^(1/2))/(6 
63*x^13) - (b^3*(a + b*x^2)^(1/2))/(2431*a*x^11) - (a^2*(a + b*x^2)^(1/2)) 
/(17*x^17) - (80*b^5*(a + b*x^2)^(1/2))/(153153*a^3*x^7) + (32*b^6*(a + b* 
x^2)^(1/2))/(51051*a^4*x^5) - (128*b^7*(a + b*x^2)^(1/2))/(153153*a^5*x^3) 
 + (256*b^8*(a + b*x^2)^(1/2))/(153153*a^6*x) - (7*a*b*(a + b*x^2)^(1/2))/ 
(51*x^15)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx=\frac {-9009 \sqrt {b \,x^{2}+a}\, a^{8}-21021 \sqrt {b \,x^{2}+a}\, a^{7} b \,x^{2}-12705 \sqrt {b \,x^{2}+a}\, a^{6} b^{2} x^{4}-63 \sqrt {b \,x^{2}+a}\, a^{5} b^{3} x^{6}+70 \sqrt {b \,x^{2}+a}\, a^{4} b^{4} x^{8}-80 \sqrt {b \,x^{2}+a}\, a^{3} b^{5} x^{10}+96 \sqrt {b \,x^{2}+a}\, a^{2} b^{6} x^{12}-128 \sqrt {b \,x^{2}+a}\, a \,b^{7} x^{14}+256 \sqrt {b \,x^{2}+a}\, b^{8} x^{16}-256 \sqrt {b}\, b^{8} x^{17}}{153153 a^{6} x^{17}} \] Input:

int((b*x^2+a)^(5/2)/x^18,x)
 

Output:

( - 9009*sqrt(a + b*x**2)*a**8 - 21021*sqrt(a + b*x**2)*a**7*b*x**2 - 1270 
5*sqrt(a + b*x**2)*a**6*b**2*x**4 - 63*sqrt(a + b*x**2)*a**5*b**3*x**6 + 7 
0*sqrt(a + b*x**2)*a**4*b**4*x**8 - 80*sqrt(a + b*x**2)*a**3*b**5*x**10 + 
96*sqrt(a + b*x**2)*a**2*b**6*x**12 - 128*sqrt(a + b*x**2)*a*b**7*x**14 + 
256*sqrt(a + b*x**2)*b**8*x**16 - 256*sqrt(b)*b**8*x**17)/(153153*a**6*x** 
17)