Integrand size = 30, antiderivative size = 384 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{20/3}}+\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{20/3}} \] Output:
1/2*(-10*a^3*f+6*a^2*b*e-3*a*b^2*d+b^3*c)*x^2/b^6+1/5*(6*a^2*f-3*a*b*e+b^2 *d)*x^5/b^5+1/8*(-3*a*f+b*e)*x^8/b^4+1/11*f*x^11/b^3-1/6*a^2*(-a^3*f+a^2*b *e-a*b^2*d+b^3*c)*x^2/b^6/(b*x^3+a)^2+1/9*a*(-16*a^3*f+13*a^2*b*e-10*a*b^2 *d+7*b^3*c)*x^2/b^6/(b*x^3+a)+1/27*a^(2/3)*(-119*a^3*f+77*a^2*b*e-44*a*b^2 *d+20*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b^( 20/3)+1/27*a^(2/3)*(-119*a^3*f+77*a^2*b*e-44*a*b^2*d+20*b^3*c)*ln(a^(1/3)+ b^(1/3)*x)/b^(20/3)-1/54*a^(2/3)*(-119*a^3*f+77*a^2*b*e-44*a*b^2*d+20*b^3* c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(20/3)
Time = 0.29 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.99 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}+\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}-\frac {a^{2/3} \left (-20 b^3 c+44 a b^2 d-77 a^2 b e+119 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{20/3}}-\frac {a^{2/3} \left (-20 b^3 c+44 a b^2 d-77 a^2 b e+119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}+\frac {a^{2/3} \left (-20 b^3 c+44 a b^2 d-77 a^2 b e+119 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{20/3}} \] Input:
Integrate[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
Output:
((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b *e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^ 3) + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2 ) + (a*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x ^3)) - (a^(2/3)*(-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*ArcTan[( 1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(20/3)) - (a^(2/3)*(-20* b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27 *b^(20/3)) + (a^(2/3)*(-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*Lo g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(20/3))
Time = 3.38 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2367, 27, 2390, 2367, 2390, 2375, 27, 2375, 27, 1812, 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.
| \(\displaystyle \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle -\frac {\int -\frac {2 \left (3 a b^6 f x^{16}+3 a b^5 (b e-a f) x^{13}+3 a b^4 \left (f a^2-b e a+b^2 d\right ) x^{10}+3 a b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^7-3 a^2 b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^4+a^3 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x\right )}{\left (b x^3+a\right )^2}dx}{6 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a b^6 f x^{16}+3 a b^5 (b e-a f) x^{13}+3 a b^4 \left (f a^2-b e a+b^2 d\right ) x^{10}+3 a b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^7-3 a^2 b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^4+a^3 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x}{\left (b x^3+a\right )^2}dx}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2390 |
| \(\displaystyle \frac {\int \frac {x \left (3 a b^6 f x^{15}+3 a b^5 (b e-a f) x^{12}+3 a b^4 \left (f a^2-b e a+b^2 d\right ) x^9+3 a b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6-3 a^2 b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^3 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )\right )}{\left (b x^3+a\right )^2}dx}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {-9 a^2 b^{11} f x^{13}-9 a^2 b^{10} (b e-2 a f) x^{10}-9 a^2 b^9 \left (3 f a^2-2 b e a+b^2 d\right ) x^7-9 a^2 b^8 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^4+a^3 b^7 \left (-29 f a^3+23 b e a^2-17 b^2 d a+11 b^3 c\right ) x}{b x^3+a}dx}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2390 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {x \left (-9 a^2 b^{11} f x^{12}-9 a^2 b^{10} (b e-2 a f) x^9-9 a^2 b^9 \left (3 f a^2-2 b e a+b^2 d\right ) x^6-9 a^2 b^8 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3+a^3 b^7 \left (-29 f a^3+23 b e a^2-17 b^2 d a+11 b^3 c\right )\right )}{b x^3+a}dx}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\int \frac {11 x \left (-9 a^2 (b e-3 a f) x^9 b^{11}-9 a^2 \left (3 f a^2-2 b e a+b^2 d\right ) x^6 b^{10}-9 a^2 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3 b^9+a^3 \left (-29 f a^3+23 b e a^2-17 b^2 d a+11 b^3 c\right ) b^8\right )}{b x^3+a}dx}{11 b}-\frac {9}{11} a^2 b^{10} f x^{11}}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\int \frac {x \left (-9 a^2 (b e-3 a f) x^9 b^{11}-9 a^2 \left (3 f a^2-2 b e a+b^2 d\right ) x^6 b^{10}-9 a^2 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3 b^9+a^3 \left (-29 f a^3+23 b e a^2-17 b^2 d a+11 b^3 c\right ) b^8\right )}{b x^3+a}dx}{b}-\frac {9}{11} a^2 b^{10} f x^{11}}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\frac {\int \frac {8 x \left (-9 a^2 \left (6 f a^2-3 b e a+b^2 d\right ) x^6 b^{11}-9 a^2 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3 b^{10}+a^3 \left (-29 f a^3+23 b e a^2-17 b^2 d a+11 b^3 c\right ) b^9\right )}{b x^3+a}dx}{8 b}-\frac {9}{8} a^2 b^{10} x^8 (b e-3 a f)}{b}-\frac {9}{11} a^2 b^{10} f x^{11}}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\frac {\int \frac {x \left (-9 a^2 \left (6 f a^2-3 b e a+b^2 d\right ) x^6 b^{11}-9 a^2 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3 b^{10}+a^3 \left (-29 f a^3+23 b e a^2-17 b^2 d a+11 b^3 c\right ) b^9\right )}{b x^3+a}dx}{b}-\frac {9}{8} a^2 b^{10} x^8 (b e-3 a f)}{b}-\frac {9}{11} a^2 b^{10} f x^{11}}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\frac {\int \left (-9 a^2 \left (6 f a^2-3 b e a+b^2 d\right ) x^4 b^{10}-9 a^2 \left (-10 f a^3+6 b e a^2-3 b^2 d a+b^3 c\right ) x b^9+\frac {\left (20 a^3 c b^{12}-44 a^4 d b^{11}+77 a^5 e b^{10}-119 a^6 f b^9\right ) x}{b x^3+a}\right )dx}{b}-\frac {9}{8} a^2 b^{10} x^8 (b e-3 a f)}{b}-\frac {9}{11} a^2 b^{10} f x^{11}}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^2 b x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\frac {-\frac {9}{5} a^2 b^{10} x^5 \left (6 a^2 f-3 a b e+b^2 d\right )-\frac {9}{2} a^2 b^9 x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )-\frac {a^{8/3} b^{25/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{\sqrt {3}}+\frac {1}{6} a^{8/3} b^{25/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )-\frac {1}{3} a^{8/3} b^{25/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{b}-\frac {9}{8} a^2 b^{10} x^8 (b e-3 a f)}{b}-\frac {9}{11} a^2 b^{10} f x^{11}}{3 a b^6}}{3 a b^7}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}\) |
Input:
Int[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
Output:
-1/6*(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(b^6*(a + b*x^3)^2) + ( (a^2*b*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(3*(a + b*x^3)) - ((-9*a^2*b^10*f*x^11)/11 + ((-9*a^2*b^10*(b*e - 3*a*f)*x^8)/8 + ((-9*a^ 2*b^9*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/2 - (9*a^2*b^10*(b^2 *d - 3*a*b*e + 6*a^2*f)*x^5)/5 - (a^(8/3)*b^(25/3)*(20*b^3*c - 44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)) ])/Sqrt[3] - (a^(8/3)*b^(25/3)*(20*b^3*c - 44*a*b^2*d + 77*a^2*b*e - 119*a ^3*f)*Log[a^(1/3) + b^(1/3)*x])/3 + (a^(8/3)*b^(25/3)*(20*b^3*c - 44*a*b^2 *d + 77*a^2*b*e - 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2 ])/6)/b)/b)/(3*a*b^6))/(3*a*b^7)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( (d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) *x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) I nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 ] && LtQ[p, -1] && IGtQ[m, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Int[x*PolynomialQuot ient[Pq, x, x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {f \,x^{11}}{11 b^{3}}-\frac {3 x^{8} f a}{8 b^{4}}+\frac {x^{8} e}{8 b^{3}}+\frac {6 x^{5} f \,a^{2}}{5 b^{5}}-\frac {3 x^{5} e a}{5 b^{4}}+\frac {x^{5} d}{5 b^{3}}-\frac {5 x^{2} f \,a^{3}}{b^{6}}+\frac {3 x^{2} e \,a^{2}}{b^{5}}-\frac {3 a d \,x^{2}}{2 b^{4}}+\frac {x^{2} c}{2 b^{3}}+\frac {\left (-\frac {16}{9} a^{4} b f +\frac {13}{9} a^{3} b^{2} e -\frac {10}{9} a^{2} b^{3} d +\frac {7}{9} a \,b^{4} c \right ) x^{5}-\frac {a^{2} \left (29 f \,a^{3}-23 e \,a^{2} b +17 d a \,b^{2}-11 b^{3} c \right ) x^{2}}{18}}{b^{6} \left (b \,x^{3}+a \right )^{2}}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (119 f \,a^{3}-77 e \,a^{2} b +44 d a \,b^{2}-20 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{27 b^{7}}\) | \(244\) |
| default | \(-\frac {-\frac {b^{3} f \,x^{11}}{11}+\frac {\left (3 f a \,b^{2}-e \,b^{3}\right ) x^{8}}{8}+\frac {\left (-6 f \,a^{2} b +3 e a \,b^{2}-b^{3} d \right ) x^{5}}{5}+\frac {\left (10 f \,a^{3}-6 e \,a^{2} b +3 d a \,b^{2}-b^{3} c \right ) x^{2}}{2}}{b^{6}}+\frac {a \left (\frac {\left (-\frac {16}{9} a^{3} b f +\frac {13}{9} a^{2} b^{2} e -\frac {10}{9} a \,b^{3} d +\frac {7}{9} b^{4} c \right ) x^{5}-\frac {a \left (29 f \,a^{3}-23 e \,a^{2} b +17 d a \,b^{2}-11 b^{3} c \right ) x^{2}}{18}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {119}{9} f \,a^{3}-\frac {77}{9} e \,a^{2} b +\frac {44}{9} d a \,b^{2}-\frac {20}{9} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{b^{6}}\) | \(296\) |
Input:
int(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/11*f*x^11/b^3-3/8/b^4*x^8*f*a+1/8/b^3*x^8*e+6/5/b^5*x^5*f*a^2-3/5/b^4*x^ 5*e*a+1/5/b^3*x^5*d-5/b^6*x^2*f*a^3+3/b^5*x^2*e*a^2-3/2/b^4*a*d*x^2+1/2/b^ 3*x^2*c+((-16/9*a^4*b*f+13/9*a^3*b^2*e-10/9*a^2*b^3*d+7/9*a*b^4*c)*x^5-1/1 8*a^2*(29*a^3*f-23*a^2*b*e+17*a*b^2*d-11*b^3*c)*x^2)/b^6/(b*x^3+a)^2+1/27/ b^7*a*sum((119*a^3*f-77*a^2*b*e+44*a*b^2*d-20*b^3*c)/_R*ln(x-_R),_R=RootOf (_Z^3*b+a))
Time = 0.09 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.65 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {1080 \, b^{5} f x^{17} + 135 \, {\left (11 \, b^{5} e - 17 \, a b^{4} f\right )} x^{14} + 54 \, {\left (44 \, b^{5} d - 77 \, a b^{4} e + 119 \, a^{2} b^{3} f\right )} x^{11} + 297 \, {\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{8} + 1056 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{5} + 660 \, {\left (20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f\right )} x^{2} - 440 \, \sqrt {3} {\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) + 220 \, {\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 440 \, {\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{11880 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \] Input:
integrate(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
1/11880*(1080*b^5*f*x^17 + 135*(11*b^5*e - 17*a*b^4*f)*x^14 + 54*(44*b^5*d - 77*a*b^4*e + 119*a^2*b^3*f)*x^11 + 297*(20*b^5*c - 44*a*b^4*d + 77*a^2* b^3*e - 119*a^3*b^2*f)*x^8 + 1056*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2* e - 119*a^4*b*f)*x^5 + 660*(20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119 *a^5*f)*x^2 - 440*sqrt(3)*((20*b^5*c - 44*a*b^4*d + 77*a^2*b^3*e - 119*a^3 *b^2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2*(20 *a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^2)^(1/3 )*arctan(1/3*(2*sqrt(3)*b*x*(-a^2/b^2)^(1/3) + sqrt(3)*a)/a) + 220*((20*b^ 5*c - 44*a*b^4*d + 77*a^2*b^3*e - 119*a^3*b^2*f)*x^6 + 20*a^2*b^3*c - 44*a ^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3* b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^2)^(1/3)*log(a*x^2 - b*x*(-a^2/b^2)^(2/3 ) - a*(-a^2/b^2)^(1/3)) - 440*((20*b^5*c - 44*a*b^4*d + 77*a^2*b^3*e - 119 *a^3*b^2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2 *(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^2)^ (1/3)*log(a*x + b*(-a^2/b^2)^(2/3)))/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6)
Timed out. \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**10*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.99 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{5} + {\left (11 \, a^{2} b^{3} c - 17 \, a^{3} b^{2} d + 23 \, a^{4} b e - 29 \, a^{5} f\right )} x^{2}}{18 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} - \frac {\sqrt {3} {\left (20 \, a b^{3} c - 44 \, a^{2} b^{2} d + 77 \, a^{3} b e - 119 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {40 \, b^{3} f x^{11} + 55 \, {\left (b^{3} e - 3 \, a b^{2} f\right )} x^{8} + 88 \, {\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{5} + 220 \, {\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x^{2}}{440 \, b^{6}} - \frac {{\left (20 \, a b^{3} c - 44 \, a^{2} b^{2} d + 77 \, a^{3} b e - 119 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (20 \, a b^{3} c - 44 \, a^{2} b^{2} d + 77 \, a^{3} b e - 119 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/18*(2*(7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^5 + (11*a ^2*b^3*c - 17*a^3*b^2*d + 23*a^4*b*e - 29*a^5*f)*x^2)/(b^8*x^6 + 2*a*b^7*x ^3 + a^2*b^6) - 1/27*sqrt(3)*(20*a*b^3*c - 44*a^2*b^2*d + 77*a^3*b*e - 119 *a^4*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^7*(a/b)^(1/ 3)) + 1/440*(40*b^3*f*x^11 + 55*(b^3*e - 3*a*b^2*f)*x^8 + 88*(b^3*d - 3*a* b^2*e + 6*a^2*b*f)*x^5 + 220*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^ 2)/b^6 - 1/54*(20*a*b^3*c - 44*a^2*b^2*d + 77*a^3*b*e - 119*a^4*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^7*(a/b)^(1/3)) + 1/27*(20*a*b^3*c - 44* a^2*b^2*d + 77*a^3*b*e - 119*a^4*f)*log(x + (a/b)^(1/3))/(b^7*(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.26 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (20 \, a b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 44 \, a^{2} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 77 \, a^{3} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 119 \, a^{4} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{6}} + \frac {\sqrt {3} {\left (20 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 77 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 119 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{8}} - \frac {{\left (20 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 77 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 119 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{8}} + \frac {14 \, a b^{4} c x^{5} - 20 \, a^{2} b^{3} d x^{5} + 26 \, a^{3} b^{2} e x^{5} - 32 \, a^{4} b f x^{5} + 11 \, a^{2} b^{3} c x^{2} - 17 \, a^{3} b^{2} d x^{2} + 23 \, a^{4} b e x^{2} - 29 \, a^{5} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{6}} + \frac {40 \, b^{30} f x^{11} + 55 \, b^{30} e x^{8} - 165 \, a b^{29} f x^{8} + 88 \, b^{30} d x^{5} - 264 \, a b^{29} e x^{5} + 528 \, a^{2} b^{28} f x^{5} + 220 \, b^{30} c x^{2} - 660 \, a b^{29} d x^{2} + 1320 \, a^{2} b^{28} e x^{2} - 2200 \, a^{3} b^{27} f x^{2}}{440 \, b^{33}} \] Input:
integrate(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")
Output:
1/27*(20*a*b^3*c*(-a/b)^(1/3) - 44*a^2*b^2*d*(-a/b)^(1/3) + 77*a^3*b*e*(-a /b)^(1/3) - 119*a^4*f*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)) )/(a*b^6) + 1/27*sqrt(3)*(20*(-a*b^2)^(2/3)*b^3*c - 44*(-a*b^2)^(2/3)*a*b^ 2*d + 77*(-a*b^2)^(2/3)*a^2*b*e - 119*(-a*b^2)^(2/3)*a^3*f)*arctan(1/3*sqr t(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^8 - 1/54*(20*(-a*b^2)^(2/3)*b^3* c - 44*(-a*b^2)^(2/3)*a*b^2*d + 77*(-a*b^2)^(2/3)*a^2*b*e - 119*(-a*b^2)^( 2/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^8 + 1/18*(14*a*b^4* c*x^5 - 20*a^2*b^3*d*x^5 + 26*a^3*b^2*e*x^5 - 32*a^4*b*f*x^5 + 11*a^2*b^3* c*x^2 - 17*a^3*b^2*d*x^2 + 23*a^4*b*e*x^2 - 29*a^5*f*x^2)/((b*x^3 + a)^2*b ^6) + 1/440*(40*b^30*f*x^11 + 55*b^30*e*x^8 - 165*a*b^29*f*x^8 + 88*b^30*d *x^5 - 264*a*b^29*e*x^5 + 528*a^2*b^28*f*x^5 + 220*b^30*c*x^2 - 660*a*b^29 *d*x^2 + 1320*a^2*b^28*e*x^2 - 2200*a^3*b^27*f*x^2)/b^33
Time = 6.59 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.11 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^8\,\left (\frac {e}{8\,b^3}-\frac {3\,a\,f}{8\,b^4}\right )+x^2\,\left (\frac {c}{2\,b^3}-\frac {a^3\,f}{2\,b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{2\,b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{2\,b}\right )-\frac {\left (\frac {16\,f\,a^4\,b}{9}-\frac {13\,e\,a^3\,b^2}{9}+\frac {10\,d\,a^2\,b^3}{9}-\frac {7\,c\,a\,b^4}{9}\right )\,x^5+\left (\frac {29\,f\,a^5}{18}-\frac {23\,e\,a^4\,b}{18}+\frac {17\,d\,a^3\,b^2}{18}-\frac {11\,c\,a^2\,b^3}{18}\right )\,x^2}{a^2\,b^6+2\,a\,b^7\,x^3+b^8\,x^6}-x^5\,\left (\frac {3\,a^2\,f}{5\,b^5}-\frac {d}{5\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{5\,b}\right )+\frac {f\,x^{11}}{11\,b^3}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-119\,f\,a^3+77\,e\,a^2\,b-44\,d\,a\,b^2+20\,c\,b^3\right )}{27\,b^{20/3}}-\frac {a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-119\,f\,a^3+77\,e\,a^2\,b-44\,d\,a\,b^2+20\,c\,b^3\right )}{27\,b^{20/3}}+\frac {a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-119\,f\,a^3+77\,e\,a^2\,b-44\,d\,a\,b^2+20\,c\,b^3\right )}{27\,b^{20/3}} \] Input:
int((x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x)
Output:
x^8*(e/(8*b^3) - (3*a*f)/(8*b^4)) + x^2*(c/(2*b^3) - (a^3*f)/(2*b^6) - (3* a^2*(e/b^3 - (3*a*f)/b^4))/(2*b^2) + (3*a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e /b^3 - (3*a*f)/b^4))/b))/(2*b)) - (x^2*((29*a^5*f)/18 - (11*a^2*b^3*c)/18 + (17*a^3*b^2*d)/18 - (23*a^4*b*e)/18) + x^5*((10*a^2*b^3*d)/9 - (13*a^3*b ^2*e)/9 - (7*a*b^4*c)/9 + (16*a^4*b*f)/9))/(a^2*b^6 + b^8*x^6 + 2*a*b^7*x^ 3) - x^5*((3*a^2*f)/(5*b^5) - d/(5*b^3) + (3*a*(e/b^3 - (3*a*f)/b^4))/(5*b )) + (f*x^11)/(11*b^3) + (a^(2/3)*log(b^(1/3)*x + a^(1/3))*(20*b^3*c - 119 *a^3*f - 44*a*b^2*d + 77*a^2*b*e))/(27*b^(20/3)) - (a^(2/3)*log(3^(1/2)*a^ (1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(20*b^3*c - 119*a ^3*f - 44*a*b^2*d + 77*a^2*b*e))/(27*b^(20/3)) + (a^(2/3)*log(3^(1/2)*a^(1 /3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(20*b^3*c - 119*a^3 *f - 44*a*b^2*d + 77*a^2*b*e))/(27*b^(20/3))
Time = 0.15 (sec) , antiderivative size = 1238, normalized size of antiderivative = 3.22 \[ \int \frac {x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)
Output:
( - 52360*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**6* f + 33880*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**5* b*e - 104720*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a* *5*b*f*x**3 - 19360*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt( 3)))*a**4*b**2*d + 67760*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)* sqrt(3)))*a**4*b**2*e*x**3 - 52360*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/ (a**(1/3)*sqrt(3)))*a**4*b**2*f*x**6 + 8800*sqrt(3)*atan((a**(1/3) - 2*b** (1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**3*c - 38720*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**3*d*x**3 + 33880*sqrt(3)*atan((a **(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**3*e*x**6 + 17600*sqrt( 3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**4*c*x**3 - 1 9360*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**4* d*x**6 + 8800*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a *b**5*c*x**6 - 78540*b**(2/3)*a**(1/3)*a**5*f*x**2 + 50820*b**(2/3)*a**(1/ 3)*a**4*b*e*x**2 - 125664*b**(2/3)*a**(1/3)*a**4*b*f*x**5 - 29040*b**(2/3) *a**(1/3)*a**3*b**2*d*x**2 + 81312*b**(2/3)*a**(1/3)*a**3*b**2*e*x**5 - 35 343*b**(2/3)*a**(1/3)*a**3*b**2*f*x**8 + 13200*b**(2/3)*a**(1/3)*a**2*b**3 *c*x**2 - 46464*b**(2/3)*a**(1/3)*a**2*b**3*d*x**5 + 22869*b**(2/3)*a**(1/ 3)*a**2*b**3*e*x**8 + 6426*b**(2/3)*a**(1/3)*a**2*b**3*f*x**11 + 21120*b** (2/3)*a**(1/3)*a*b**4*c*x**5 - 13068*b**(2/3)*a**(1/3)*a*b**4*d*x**8 - ...