Integrand size = 15, antiderivative size = 148 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{7 a x^7}+\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{28 a^2 x^{20/3}}-\frac {b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{133 a^3 x^{19/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{798 a^4 x^6}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{6783 a^5 x^{17/3}}+\frac {b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{108528 a^6 x^{16/3}} \]
-1/7*(a+b*x^(1/3))^16/a/x^7+1/28*b*(a+b*x^(1/3))^16/a^2/x^(20/3)-1/133*b^2 *(a+b*x^(1/3))^16/a^3/x^(19/3)+1/798*b^3*(a+b*x^(1/3))^16/a^4/x^6-1/6783*b ^4*(a+b*x^(1/3))^16/a^5/x^(17/3)+1/108528*b^5*(a+b*x^(1/3))^16/a^6/x^(16/3 )
Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=\frac {-15504 a^{15}-244188 a^{14} b \sqrt [3]{x}-1799280 a^{13} b^2 x^{2/3}-8230040 a^{12} b^3 x-26142480 a^{11} b^4 x^{4/3}-61108047 a^{10} b^5 x^{5/3}-108636528 a^9 b^6 x^2-149652360 a^8 b^7 x^{7/3}-161164080 a^7 b^8 x^{8/3}-135795660 a^6 b^9 x^3-88884432 a^5 b^{10} x^{10/3}-44442216 a^4 b^{11} x^{11/3}-16460080 a^3 b^{12} x^4-4273290 a^2 b^{13} x^{13/3}-697680 a b^{14} x^{14/3}-54264 b^{15} x^5}{108528 x^7} \]
(-15504*a^15 - 244188*a^14*b*x^(1/3) - 1799280*a^13*b^2*x^(2/3) - 8230040* a^12*b^3*x - 26142480*a^11*b^4*x^(4/3) - 61108047*a^10*b^5*x^(5/3) - 10863 6528*a^9*b^6*x^2 - 149652360*a^8*b^7*x^(7/3) - 161164080*a^7*b^8*x^(8/3) - 135795660*a^6*b^9*x^3 - 88884432*a^5*b^10*x^(10/3) - 44442216*a^4*b^11*x^ (11/3) - 16460080*a^3*b^12*x^4 - 4273290*a^2*b^13*x^(13/3) - 697680*a*b^14 *x^(14/3) - 54264*b^15*x^5)/(108528*x^7)
Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {798, 55, 55, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{22/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 3 \left (-\frac {5 b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^7}d\sqrt [3]{x}}{21 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{21 a x^7}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 3 \left (-\frac {5 b \left (-\frac {b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{20/3}}d\sqrt [3]{x}}{5 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{20 a x^{20/3}}\right )}{21 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{21 a x^7}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 3 \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{19/3}}d\sqrt [3]{x}}{19 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{19 a x^{19/3}}\right )}{5 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{20 a x^{20/3}}\right )}{21 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{21 a x^7}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 3 \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^6}d\sqrt [3]{x}}{9 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{18 a x^6}\right )}{19 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{19 a x^{19/3}}\right )}{5 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{20 a x^{20/3}}\right )}{21 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{21 a x^7}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 3 \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{17/3}}d\sqrt [3]{x}}{17 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{17 a x^{17/3}}\right )}{9 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{18 a x^6}\right )}{19 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{19 a x^{19/3}}\right )}{5 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{20 a x^{20/3}}\right )}{21 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{21 a x^7}\right )\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle 3 \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (\frac {b \left (a+b \sqrt [3]{x}\right )^{16}}{272 a^2 x^{16/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{17 a x^{17/3}}\right )}{9 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{18 a x^6}\right )}{19 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{19 a x^{19/3}}\right )}{5 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{20 a x^{20/3}}\right )}{21 a}-\frac {\left (a+b \sqrt [3]{x}\right )^{16}}{21 a x^7}\right )\) |
3*((-5*b*(-1/5*(b*((-3*b*(-1/9*(b*(-1/17*(a + b*x^(1/3))^16/(a*x^(17/3)) + (b*(a + b*x^(1/3))^16)/(272*a^2*x^(16/3))))/a - (a + b*x^(1/3))^16/(18*a* x^6)))/(19*a) - (a + b*x^(1/3))^16/(19*a*x^(19/3))))/a - (a + b*x^(1/3))^1 6/(20*a*x^(20/3))))/(21*a) - (a + b*x^(1/3))^16/(21*a*x^7))
3.24.51.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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 = 3.74 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(-\frac {b^{15}}{2 x^{2}}-\frac {9009 a^{10} b^{5}}{16 x^{\frac {16}{3}}}-\frac {455 a^{3} b^{12}}{3 x^{3}}-\frac {1001 a^{9} b^{6}}{x^{5}}-\frac {45 a \,b^{14}}{7 x^{\frac {7}{3}}}-\frac {819 a^{5} b^{10}}{x^{\frac {11}{3}}}-\frac {455 a^{12} b^{3}}{6 x^{6}}-\frac {a^{15}}{7 x^{7}}-\frac {819 a^{4} b^{11}}{2 x^{\frac {10}{3}}}-\frac {9 a^{14} b}{4 x^{\frac {20}{3}}}-\frac {5005 a^{6} b^{9}}{4 x^{4}}-\frac {19305 a^{8} b^{7}}{14 x^{\frac {14}{3}}}-\frac {315 a^{2} b^{13}}{8 x^{\frac {8}{3}}}-\frac {4095 a^{11} b^{4}}{17 x^{\frac {17}{3}}}-\frac {315 a^{13} b^{2}}{19 x^{\frac {19}{3}}}-\frac {1485 a^{7} b^{8}}{x^{\frac {13}{3}}}\) | \(168\) |
| default | \(-\frac {b^{15}}{2 x^{2}}-\frac {9009 a^{10} b^{5}}{16 x^{\frac {16}{3}}}-\frac {455 a^{3} b^{12}}{3 x^{3}}-\frac {1001 a^{9} b^{6}}{x^{5}}-\frac {45 a \,b^{14}}{7 x^{\frac {7}{3}}}-\frac {819 a^{5} b^{10}}{x^{\frac {11}{3}}}-\frac {455 a^{12} b^{3}}{6 x^{6}}-\frac {a^{15}}{7 x^{7}}-\frac {819 a^{4} b^{11}}{2 x^{\frac {10}{3}}}-\frac {9 a^{14} b}{4 x^{\frac {20}{3}}}-\frac {5005 a^{6} b^{9}}{4 x^{4}}-\frac {19305 a^{8} b^{7}}{14 x^{\frac {14}{3}}}-\frac {315 a^{2} b^{13}}{8 x^{\frac {8}{3}}}-\frac {4095 a^{11} b^{4}}{17 x^{\frac {17}{3}}}-\frac {315 a^{13} b^{2}}{19 x^{\frac {19}{3}}}-\frac {1485 a^{7} b^{8}}{x^{\frac {13}{3}}}\) | \(168\) |
| trager | \(\frac {\left (-1+x \right ) \left (12 a^{15} x^{6}+6370 a^{12} b^{3} x^{6}+84084 a^{9} b^{6} x^{6}+105105 a^{6} b^{9} x^{6}+12740 a^{3} b^{12} x^{6}+42 b^{15} x^{6}+12 a^{15} x^{5}+6370 a^{12} b^{3} x^{5}+84084 a^{9} b^{6} x^{5}+105105 a^{6} b^{9} x^{5}+12740 a^{3} b^{12} x^{5}+42 b^{15} x^{5}+12 x^{4} a^{15}+6370 a^{12} b^{3} x^{4}+84084 a^{9} b^{6} x^{4}+105105 a^{6} b^{9} x^{4}+12740 a^{3} b^{12} x^{4}+12 x^{3} a^{15}+6370 a^{12} b^{3} x^{3}+84084 a^{9} b^{6} x^{3}+105105 a^{6} b^{9} x^{3}+12 x^{2} a^{15}+6370 a^{12} b^{3} x^{2}+84084 a^{9} b^{6} x^{2}+12 x \,a^{15}+6370 a^{12} b^{3} x +12 a^{15}\right )}{84 x^{7}}-\frac {9 \left (4165 b^{12} x^{4}+86632 a^{3} b^{9} x^{3}+145860 a^{6} b^{6} x^{2}+25480 a^{9} b^{3} x +238 a^{12}\right ) a^{2} b}{952 x^{\frac {20}{3}}}-\frac {9 \left (1520 b^{12} x^{4}+96824 a^{3} b^{9} x^{3}+351120 a^{6} b^{6} x^{2}+133133 a^{9} b^{3} x +3920 a^{12}\right ) a \,b^{2}}{2128 x^{\frac {19}{3}}}\) | \(382\) |
-1/2/x^2*b^15-9009/16*a^10*b^5/x^(16/3)-455/3*a^3*b^12/x^3-1001*a^9*b^6/x^ 5-45/7*a*b^14/x^(7/3)-819*a^5*b^10/x^(11/3)-455/6/x^6*a^12*b^3-1/7*a^15/x^ 7-819/2*a^4*b^11/x^(10/3)-9/4*a^14*b/x^(20/3)-5005/4/x^4*a^6*b^9-19305/14* a^8*b^7/x^(14/3)-315/8*a^2*b^13/x^(8/3)-4095/17*a^11*b^4/x^(17/3)-315/19*a ^13*b^2/x^(19/3)-1485*a^7*b^8/x^(13/3)
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=-\frac {54264 \, b^{15} x^{5} + 16460080 \, a^{3} b^{12} x^{4} + 135795660 \, a^{6} b^{9} x^{3} + 108636528 \, a^{9} b^{6} x^{2} + 8230040 \, a^{12} b^{3} x + 15504 \, a^{15} + 459 \, {\left (1520 \, a b^{14} x^{4} + 96824 \, a^{4} b^{11} x^{3} + 351120 \, a^{7} b^{8} x^{2} + 133133 \, a^{10} b^{5} x + 3920 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 1026 \, {\left (4165 \, a^{2} b^{13} x^{4} + 86632 \, a^{5} b^{10} x^{3} + 145860 \, a^{8} b^{7} x^{2} + 25480 \, a^{11} b^{4} x + 238 \, a^{14} b\right )} x^{\frac {1}{3}}}{108528 \, x^{7}} \]
-1/108528*(54264*b^15*x^5 + 16460080*a^3*b^12*x^4 + 135795660*a^6*b^9*x^3 + 108636528*a^9*b^6*x^2 + 8230040*a^12*b^3*x + 15504*a^15 + 459*(1520*a*b^ 14*x^4 + 96824*a^4*b^11*x^3 + 351120*a^7*b^8*x^2 + 133133*a^10*b^5*x + 392 0*a^13*b^2)*x^(2/3) + 1026*(4165*a^2*b^13*x^4 + 86632*a^5*b^10*x^3 + 14586 0*a^8*b^7*x^2 + 25480*a^11*b^4*x + 238*a^14*b)*x^(1/3))/x^7
Time = 1.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=- \frac {a^{15}}{7 x^{7}} - \frac {9 a^{14} b}{4 x^{\frac {20}{3}}} - \frac {315 a^{13} b^{2}}{19 x^{\frac {19}{3}}} - \frac {455 a^{12} b^{3}}{6 x^{6}} - \frac {4095 a^{11} b^{4}}{17 x^{\frac {17}{3}}} - \frac {9009 a^{10} b^{5}}{16 x^{\frac {16}{3}}} - \frac {1001 a^{9} b^{6}}{x^{5}} - \frac {19305 a^{8} b^{7}}{14 x^{\frac {14}{3}}} - \frac {1485 a^{7} b^{8}}{x^{\frac {13}{3}}} - \frac {5005 a^{6} b^{9}}{4 x^{4}} - \frac {819 a^{5} b^{10}}{x^{\frac {11}{3}}} - \frac {819 a^{4} b^{11}}{2 x^{\frac {10}{3}}} - \frac {455 a^{3} b^{12}}{3 x^{3}} - \frac {315 a^{2} b^{13}}{8 x^{\frac {8}{3}}} - \frac {45 a b^{14}}{7 x^{\frac {7}{3}}} - \frac {b^{15}}{2 x^{2}} \]
-a**15/(7*x**7) - 9*a**14*b/(4*x**(20/3)) - 315*a**13*b**2/(19*x**(19/3)) - 455*a**12*b**3/(6*x**6) - 4095*a**11*b**4/(17*x**(17/3)) - 9009*a**10*b* *5/(16*x**(16/3)) - 1001*a**9*b**6/x**5 - 19305*a**8*b**7/(14*x**(14/3)) - 1485*a**7*b**8/x**(13/3) - 5005*a**6*b**9/(4*x**4) - 819*a**5*b**10/x**(1 1/3) - 819*a**4*b**11/(2*x**(10/3)) - 455*a**3*b**12/(3*x**3) - 315*a**2*b **13/(8*x**(8/3)) - 45*a*b**14/(7*x**(7/3)) - b**15/(2*x**2)
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=-\frac {54264 \, b^{15} x^{5} + 697680 \, a b^{14} x^{\frac {14}{3}} + 4273290 \, a^{2} b^{13} x^{\frac {13}{3}} + 16460080 \, a^{3} b^{12} x^{4} + 44442216 \, a^{4} b^{11} x^{\frac {11}{3}} + 88884432 \, a^{5} b^{10} x^{\frac {10}{3}} + 135795660 \, a^{6} b^{9} x^{3} + 161164080 \, a^{7} b^{8} x^{\frac {8}{3}} + 149652360 \, a^{8} b^{7} x^{\frac {7}{3}} + 108636528 \, a^{9} b^{6} x^{2} + 61108047 \, a^{10} b^{5} x^{\frac {5}{3}} + 26142480 \, a^{11} b^{4} x^{\frac {4}{3}} + 8230040 \, a^{12} b^{3} x + 1799280 \, a^{13} b^{2} x^{\frac {2}{3}} + 244188 \, a^{14} b x^{\frac {1}{3}} + 15504 \, a^{15}}{108528 \, x^{7}} \]
-1/108528*(54264*b^15*x^5 + 697680*a*b^14*x^(14/3) + 4273290*a^2*b^13*x^(1 3/3) + 16460080*a^3*b^12*x^4 + 44442216*a^4*b^11*x^(11/3) + 88884432*a^5*b ^10*x^(10/3) + 135795660*a^6*b^9*x^3 + 161164080*a^7*b^8*x^(8/3) + 1496523 60*a^8*b^7*x^(7/3) + 108636528*a^9*b^6*x^2 + 61108047*a^10*b^5*x^(5/3) + 2 6142480*a^11*b^4*x^(4/3) + 8230040*a^12*b^3*x + 1799280*a^13*b^2*x^(2/3) + 244188*a^14*b*x^(1/3) + 15504*a^15)/x^7
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=-\frac {54264 \, b^{15} x^{5} + 697680 \, a b^{14} x^{\frac {14}{3}} + 4273290 \, a^{2} b^{13} x^{\frac {13}{3}} + 16460080 \, a^{3} b^{12} x^{4} + 44442216 \, a^{4} b^{11} x^{\frac {11}{3}} + 88884432 \, a^{5} b^{10} x^{\frac {10}{3}} + 135795660 \, a^{6} b^{9} x^{3} + 161164080 \, a^{7} b^{8} x^{\frac {8}{3}} + 149652360 \, a^{8} b^{7} x^{\frac {7}{3}} + 108636528 \, a^{9} b^{6} x^{2} + 61108047 \, a^{10} b^{5} x^{\frac {5}{3}} + 26142480 \, a^{11} b^{4} x^{\frac {4}{3}} + 8230040 \, a^{12} b^{3} x + 1799280 \, a^{13} b^{2} x^{\frac {2}{3}} + 244188 \, a^{14} b x^{\frac {1}{3}} + 15504 \, a^{15}}{108528 \, x^{7}} \]
-1/108528*(54264*b^15*x^5 + 697680*a*b^14*x^(14/3) + 4273290*a^2*b^13*x^(1 3/3) + 16460080*a^3*b^12*x^4 + 44442216*a^4*b^11*x^(11/3) + 88884432*a^5*b ^10*x^(10/3) + 135795660*a^6*b^9*x^3 + 161164080*a^7*b^8*x^(8/3) + 1496523 60*a^8*b^7*x^(7/3) + 108636528*a^9*b^6*x^2 + 61108047*a^10*b^5*x^(5/3) + 2 6142480*a^11*b^4*x^(4/3) + 8230040*a^12*b^3*x + 1799280*a^13*b^2*x^(2/3) + 244188*a^14*b*x^(1/3) + 15504*a^15)/x^7
Time = 6.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^8} \, dx=-\frac {\frac {a^{15}}{7}+\frac {b^{15}\,x^5}{2}+\frac {455\,a^{12}\,b^3\,x}{6}+\frac {9\,a^{14}\,b\,x^{1/3}}{4}+\frac {45\,a\,b^{14}\,x^{14/3}}{7}+1001\,a^9\,b^6\,x^2+\frac {5005\,a^6\,b^9\,x^3}{4}+\frac {455\,a^3\,b^{12}\,x^4}{3}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{19}+\frac {4095\,a^{11}\,b^4\,x^{4/3}}{17}+\frac {9009\,a^{10}\,b^5\,x^{5/3}}{16}+\frac {19305\,a^8\,b^7\,x^{7/3}}{14}+1485\,a^7\,b^8\,x^{8/3}+819\,a^5\,b^{10}\,x^{10/3}+\frac {819\,a^4\,b^{11}\,x^{11/3}}{2}+\frac {315\,a^2\,b^{13}\,x^{13/3}}{8}}{x^7} \]
-(a^15/7 + (b^15*x^5)/2 + (455*a^12*b^3*x)/6 + (9*a^14*b*x^(1/3))/4 + (45* a*b^14*x^(14/3))/7 + 1001*a^9*b^6*x^2 + (5005*a^6*b^9*x^3)/4 + (455*a^3*b^ 12*x^4)/3 + (315*a^13*b^2*x^(2/3))/19 + (4095*a^11*b^4*x^(4/3))/17 + (9009 *a^10*b^5*x^(5/3))/16 + (19305*a^8*b^7*x^(7/3))/14 + 1485*a^7*b^8*x^(8/3) + 819*a^5*b^10*x^(10/3) + (819*a^4*b^11*x^(11/3))/2 + (315*a^2*b^13*x^(13/ 3))/8)/x^7