Integrand size = 30, antiderivative size = 369 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{19/3}}+\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}} \]
-a*(-5*a^3*f+4*a^2*b*e-3*a*b^2*d+2*b^3*c)*x/b^6+1/4*(-4*a^3*f+3*a^2*b*e-2* a*b^2*d+b^3*c)*x^4/b^5+1/7*(3*a^2*f-2*a*b*e+b^2*d)*x^7/b^4+1/10*(-2*a*f+b* e)*x^10/b^3+1/13*f*x^13/b^2-1/3*a^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^6/( b*x^3+a)+1/9*a^(4/3)*(-16*a^3*f+13*a^2*b*e-10*a*b^2*d+7*b^3*c)*ln(a^(1/3)+ b^(1/3)*x)/b^(19/3)-1/18*a^(4/3)*(-16*a^3*f+13*a^2*b*e-10*a*b^2*d+7*b^3*c) *ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(19/3)-1/9*a^(4/3)*(-16*a^3*f +13*a^2*b*e-10*a*b^2*d+7*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3 ^(1/2))/b^(19/3)*3^(1/2)
Time = 0.43 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.99 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}+\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac {a^{4/3} \left (-7 b^3 c+10 a b^2 d-13 a^2 b e+16 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{19/3}}-\frac {a^{4/3} \left (-7 b^3 c+10 a b^2 d-13 a^2 b e+16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}+\frac {a^{4/3} \left (-7 b^3 c+10 a b^2 d-13 a^2 b e+16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}} \]
(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2 *d + 3*a^2*b*e - 4*a^3*f)*x^4)/(4*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7) /(7*b^4) + ((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) + (a^2*(-(b^3 *c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(3*b^6*(a + b*x^3)) + (a^(4/3)*(-7*b^3 *c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3) )/Sqrt[3]])/(3*Sqrt[3]*b^(19/3)) - (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^ 2*b*e + 16*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(19/3)) + (a^(4/3)*(-7*b^ 3*c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))
Time = 0.95 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2367, 25, 2426, 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.
| \(\displaystyle \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle -\frac {\int -\frac {3 a b^5 f x^{15}+3 a b^4 (b e-a f) x^{12}+3 a b^3 \left (f a^2-b e a+b^2 d\right ) x^9+3 a b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6-3 a^2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{b x^3+a}dx}{3 a b^6}-\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a b^5 f x^{15}+3 a b^4 (b e-a f) x^{12}+3 a b^3 \left (f a^2-b e a+b^2 d\right ) x^9+3 a b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6-3 a^2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{b x^3+a}dx}{3 a b^6}-\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {\int \left (3 a b^4 f x^{12}+3 a b^3 (b e-2 a f) x^9+3 a b^2 \left (3 f a^2-2 b e a+b^2 d\right ) x^6+3 a b \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3-3 a^2 \left (-5 f a^3+4 b e a^2-3 b^2 d a+2 b^3 c\right )+\frac {-16 f a^6+13 b e a^5-10 b^2 d a^4+7 b^3 c a^3}{b x^3+a}\right )dx}{3 a b^6}-\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{7} a b^2 x^7 \left (3 a^2 f-2 a b e+b^2 d\right )+\frac {3}{4} a b x^4 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )-3 a^2 x \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )-\frac {a^{7/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \sqrt [3]{b}}-\frac {a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{6 \sqrt [3]{b}}+\frac {3}{13} a b^4 f x^{13}+\frac {3}{10} a b^3 x^{10} (b e-2 a f)}{3 a b^6}-\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}\) |
-1/3*(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(b^6*(a + b*x^3)) + (-3*a ^2*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x + (3*a*b*(b^3*c - 2*a*b^2 *d + 3*a^2*b*e - 4*a^3*f)*x^4)/4 + (3*a*b^2*(b^2*d - 2*a*b*e + 3*a^2*f)*x^ 7)/7 + (3*a*b^3*(b*e - 2*a*f)*x^10)/10 + (3*a*b^4*f*x^13)/13 - (a^(7/3)*(7 *b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x )/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(1/3)) + (a^(7/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(1/3)) - (a^(7/3)*( 7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3 )*x + b^(2/3)*x^2])/(6*b^(1/3)))/(3*a*b^6)
3.3.60.3.1 Defintions of rubi rules used
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_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {f \,x^{13}}{13 b^{2}}-\frac {x^{10} a f}{5 b^{3}}+\frac {x^{10} e}{10 b^{2}}+\frac {3 x^{7} a^{2} f}{7 b^{4}}-\frac {2 x^{7} a e}{7 b^{3}}+\frac {d \,x^{7}}{7 b^{2}}-\frac {a^{3} f \,x^{4}}{b^{5}}+\frac {3 a^{2} e \,x^{4}}{4 b^{4}}-\frac {a d \,x^{4}}{2 b^{3}}+\frac {c \,x^{4}}{4 b^{2}}+\frac {5 a^{4} f x}{b^{6}}-\frac {4 a^{3} e x}{b^{5}}+\frac {3 a^{2} d x}{b^{4}}-\frac {2 a c x}{b^{3}}+\frac {\left (\frac {1}{3} f \,a^{5}-\frac {1}{3} a^{4} e b +\frac {1}{3} a^{3} d \,b^{2}-\frac {1}{3} a^{2} c \,b^{3}\right ) x}{b^{6} \left (b \,x^{3}+a \right )}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-16 f \,a^{3}+13 a^{2} b e -10 a \,b^{2} d +7 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{7}}\) | \(244\) |
| default | \(\frac {\frac {1}{13} f \,x^{13} b^{4}-\frac {1}{5} x^{10} a \,b^{3} f +\frac {1}{10} x^{10} b^{4} e +\frac {3}{7} x^{7} a^{2} b^{2} f -\frac {2}{7} x^{7} a \,b^{3} e +\frac {1}{7} b^{4} d \,x^{7}-a^{3} b f \,x^{4}+\frac {3}{4} a^{2} b^{2} e \,x^{4}-\frac {1}{2} a \,b^{3} d \,x^{4}+\frac {1}{4} b^{4} c \,x^{4}+5 a^{4} f x -4 a^{3} b e x +3 a^{2} b^{2} d x -2 a \,b^{3} c x}{b^{6}}-\frac {a^{2} \left (\frac {\left (-\frac {1}{3} f \,a^{3}+\frac {1}{3} a^{2} b e -\frac {1}{3} a \,b^{2} d +\frac {1}{3} b^{3} c \right ) x}{b \,x^{3}+a}+\frac {\left (16 f \,a^{3}-13 a^{2} b e +10 a \,b^{2} d -7 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}\right )}{b^{6}}\) | \(306\) |
1/13*f*x^13/b^2-1/5/b^3*x^10*a*f+1/10/b^2*x^10*e+3/7/b^4*x^7*a^2*f-2/7/b^3 *x^7*a*e+1/7/b^2*d*x^7-1/b^5*a^3*f*x^4+3/4/b^4*a^2*e*x^4-1/2/b^3*a*d*x^4+1 /4/b^2*c*x^4+5/b^6*a^4*f*x-4/b^5*a^3*e*x+3/b^4*a^2*d*x-2/b^3*a*c*x+(1/3*f* a^5-1/3*a^4*e*b+1/3*a^3*d*b^2-1/3*a^2*c*b^3)*x/b^6/(b*x^3+a)+1/9/b^7*a^2*s um((-16*a^3*f+13*a^2*b*e-10*a*b^2*d+7*b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3* b+a))
Time = 0.30 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.32 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {1260 \, b^{5} f x^{16} + 126 \, {\left (13 \, b^{5} e - 16 \, a b^{4} f\right )} x^{13} + 234 \, {\left (10 \, b^{5} d - 13 \, a b^{4} e + 16 \, a^{2} b^{3} f\right )} x^{10} + 585 \, {\left (7 \, b^{5} c - 10 \, a b^{4} d + 13 \, a^{2} b^{3} e - 16 \, a^{3} b^{2} f\right )} x^{7} - 4095 \, {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{4} - 1820 \, \sqrt {3} {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f + {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 910 \, {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f + {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 1820 \, {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f + {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 5460 \, {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} x}{16380 \, {\left (b^{7} x^{3} + a b^{6}\right )}} \]
1/16380*(1260*b^5*f*x^16 + 126*(13*b^5*e - 16*a*b^4*f)*x^13 + 234*(10*b^5* d - 13*a*b^4*e + 16*a^2*b^3*f)*x^10 + 585*(7*b^5*c - 10*a*b^4*d + 13*a^2*b ^3*e - 16*a^3*b^2*f)*x^7 - 4095*(7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^4 - 1820*sqrt(3)*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f + (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3)*(- a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 910*(7 *a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f + (7*a*b^4*c - 10*a^2*b^ 3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3)*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3 ) + (-a/b)^(2/3)) - 1820*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5 *f + (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3)*(-a/b)^(1 /3)*log(x - (-a/b)^(1/3)) - 5460*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f)*x)/(b^7*x^3 + a*b^6)
Time = 85.66 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.36 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^{10} \left (- \frac {a f}{5 b^{3}} + \frac {e}{10 b^{2}}\right ) + x^{7} \cdot \left (\frac {3 a^{2} f}{7 b^{4}} - \frac {2 a e}{7 b^{3}} + \frac {d}{7 b^{2}}\right ) + x^{4} \left (- \frac {a^{3} f}{b^{5}} + \frac {3 a^{2} e}{4 b^{4}} - \frac {a d}{2 b^{3}} + \frac {c}{4 b^{2}}\right ) + x \left (\frac {5 a^{4} f}{b^{6}} - \frac {4 a^{3} e}{b^{5}} + \frac {3 a^{2} d}{b^{4}} - \frac {2 a c}{b^{3}}\right ) + \frac {x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{3 a b^{6} + 3 b^{7} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{19} + 4096 a^{13} f^{3} - 9984 a^{12} b e f^{2} + 7680 a^{11} b^{2} d f^{2} + 8112 a^{11} b^{2} e^{2} f - 5376 a^{10} b^{3} c f^{2} - 12480 a^{10} b^{3} d e f - 2197 a^{10} b^{3} e^{3} + 8736 a^{9} b^{4} c e f + 4800 a^{9} b^{4} d^{2} f + 5070 a^{9} b^{4} d e^{2} - 6720 a^{8} b^{5} c d f - 3549 a^{8} b^{5} c e^{2} - 3900 a^{8} b^{5} d^{2} e + 2352 a^{7} b^{6} c^{2} f + 5460 a^{7} b^{6} c d e + 1000 a^{7} b^{6} d^{3} - 1911 a^{6} b^{7} c^{2} e - 2100 a^{6} b^{7} c d^{2} + 1470 a^{5} b^{8} c^{2} d - 343 a^{4} b^{9} c^{3}, \left ( t \mapsto t \log {\left (- \frac {9 t b^{6}}{16 a^{4} f - 13 a^{3} b e + 10 a^{2} b^{2} d - 7 a b^{3} c} + x \right )} \right )\right )} + \frac {f x^{13}}{13 b^{2}} \]
x**10*(-a*f/(5*b**3) + e/(10*b**2)) + x**7*(3*a**2*f/(7*b**4) - 2*a*e/(7*b **3) + d/(7*b**2)) + x**4*(-a**3*f/b**5 + 3*a**2*e/(4*b**4) - a*d/(2*b**3) + c/(4*b**2)) + x*(5*a**4*f/b**6 - 4*a**3*e/b**5 + 3*a**2*d/b**4 - 2*a*c/ b**3) + x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/(3*a*b**6 + 3*b* *7*x**3) + RootSum(729*_t**3*b**19 + 4096*a**13*f**3 - 9984*a**12*b*e*f**2 + 7680*a**11*b**2*d*f**2 + 8112*a**11*b**2*e**2*f - 5376*a**10*b**3*c*f** 2 - 12480*a**10*b**3*d*e*f - 2197*a**10*b**3*e**3 + 8736*a**9*b**4*c*e*f + 4800*a**9*b**4*d**2*f + 5070*a**9*b**4*d*e**2 - 6720*a**8*b**5*c*d*f - 35 49*a**8*b**5*c*e**2 - 3900*a**8*b**5*d**2*e + 2352*a**7*b**6*c**2*f + 5460 *a**7*b**6*c*d*e + 1000*a**7*b**6*d**3 - 1911*a**6*b**7*c**2*e - 2100*a**6 *b**7*c*d**2 + 1470*a**5*b**8*c**2*d - 343*a**4*b**9*c**3, Lambda(_t, _t*l og(-9*_t*b**6/(16*a**4*f - 13*a**3*b*e + 10*a**2*b**2*d - 7*a*b**3*c) + x) )) + f*x**13/(13*b**2)
Time = 0.34 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3 \, {\left (b^{7} x^{3} + a b^{6}\right )}} + \frac {140 \, b^{4} f x^{13} + 182 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{10} + 260 \, {\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{7} + 455 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{4} - 1820 \, {\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x}{1820 \, b^{6}} + \frac {\sqrt {3} {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/3*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x/(b^7*x^3 + a*b^6) + 1/182 0*(140*b^4*f*x^13 + 182*(b^4*e - 2*a*b^3*f)*x^10 + 260*(b^4*d - 2*a*b^3*e + 3*a^2*b^2*f)*x^7 + 455*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^4 - 1820*(2*a*b^3*c - 3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*x)/b^6 + 1/9*sqrt( 3)*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f)*arctan(1/3*sqrt(3) *(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^7*(a/b)^(2/3)) - 1/18*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3 ))/(b^7*(a/b)^(2/3)) + 1/9*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a ^5*f)*log(x + (a/b)^(1/3))/(b^7*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.20 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} {\left (7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d + 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e - 16 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{7}} - \frac {{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{6}} + \frac {{\left (7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d + 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e - 16 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{7}} - \frac {a^{2} b^{3} c x - a^{3} b^{2} d x + a^{4} b e x - a^{5} f x}{3 \, {\left (b x^{3} + a\right )} b^{6}} + \frac {140 \, b^{24} f x^{13} + 182 \, b^{24} e x^{10} - 364 \, a b^{23} f x^{10} + 260 \, b^{24} d x^{7} - 520 \, a b^{23} e x^{7} + 780 \, a^{2} b^{22} f x^{7} + 455 \, b^{24} c x^{4} - 910 \, a b^{23} d x^{4} + 1365 \, a^{2} b^{22} e x^{4} - 1820 \, a^{3} b^{21} f x^{4} - 3640 \, a b^{23} c x + 5460 \, a^{2} b^{22} d x - 7280 \, a^{3} b^{21} e x + 9100 \, a^{4} b^{20} f x}{1820 \, b^{26}} \]
1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*a*b^3*c - 10*(-a*b^2)^(1/3)*a^2*b^2*d + 13*( -a*b^2)^(1/3)*a^3*b*e - 16*(-a*b^2)^(1/3)*a^4*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^7 - 1/9*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^ 4*b*e - 16*a^5*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^6) + 1/18*( 7*(-a*b^2)^(1/3)*a*b^3*c - 10*(-a*b^2)^(1/3)*a^2*b^2*d + 13*(-a*b^2)^(1/3) *a^3*b*e - 16*(-a*b^2)^(1/3)*a^4*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3 ))/b^7 - 1/3*(a^2*b^3*c*x - a^3*b^2*d*x + a^4*b*e*x - a^5*f*x)/((b*x^3 + a )*b^6) + 1/1820*(140*b^24*f*x^13 + 182*b^24*e*x^10 - 364*a*b^23*f*x^10 + 2 60*b^24*d*x^7 - 520*a*b^23*e*x^7 + 780*a^2*b^22*f*x^7 + 455*b^24*c*x^4 - 9 10*a*b^23*d*x^4 + 1365*a^2*b^22*e*x^4 - 1820*a^3*b^21*f*x^4 - 3640*a*b^23* c*x + 5460*a^2*b^22*d*x - 7280*a^3*b^21*e*x + 9100*a^4*b^20*f*x)/b^26
Time = 0.35 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.30 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^{10}\,\left (\frac {e}{10\,b^2}-\frac {a\,f}{5\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )}{b}-\frac {a^2\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b^2}\right )-x^7\,\left (\frac {a^2\,f}{7\,b^4}-\frac {d}{7\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{7\,b}\right )+x^4\,\left (\frac {c}{4\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{4\,b^2}+\frac {a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{2\,b}\right )+\frac {f\,x^{13}}{13\,b^2}+\frac {x\,\left (\frac {f\,a^5}{3}-\frac {e\,a^4\,b}{3}+\frac {d\,a^3\,b^2}{3}-\frac {c\,a^2\,b^3}{3}\right )}{b^7\,x^3+a\,b^6}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-16\,f\,a^3+13\,e\,a^2\,b-10\,d\,a\,b^2+7\,c\,b^3\right )}{9\,b^{19/3}}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-16\,f\,a^3+13\,e\,a^2\,b-10\,d\,a\,b^2+7\,c\,b^3\right )}{9\,b^{19/3}}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-16\,f\,a^3+13\,e\,a^2\,b-10\,d\,a\,b^2+7\,c\,b^3\right )}{9\,b^{19/3}} \]
x^10*(e/(10*b^2) - (a*f)/(5*b^3)) - x*((2*a*(c/b^2 - (a^2*(e/b^2 - (2*a*f) /b^3))/b^2 + (2*a*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/b ))/b - (a^2*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/b^2) - x^7*((a^2*f)/(7*b^4) - d/(7*b^2) + (2*a*(e/b^2 - (2*a*f)/b^3))/(7*b)) + x^ 4*(c/(4*b^2) - (a^2*(e/b^2 - (2*a*f)/b^3))/(4*b^2) + (a*((a^2*f)/b^4 - d/b ^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/(2*b)) + (f*x^13)/(13*b^2) + (x*((a^5 *f)/3 - (a^2*b^3*c)/3 + (a^3*b^2*d)/3 - (a^4*b*e)/3))/(a*b^6 + b^7*x^3) + (a^(4/3)*log(b^(1/3)*x + a^(1/3))*(7*b^3*c - 16*a^3*f - 10*a*b^2*d + 13*a^ 2*b*e))/(9*b^(19/3)) + (a^(4/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^( 1/3))*((3^(1/2)*1i)/2 - 1/2)*(7*b^3*c - 16*a^3*f - 10*a*b^2*d + 13*a^2*b*e ))/(9*b^(19/3)) - (a^(4/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3)) *((3^(1/2)*1i)/2 + 1/2)*(7*b^3*c - 16*a^3*f - 10*a*b^2*d + 13*a^2*b*e))/(9 *b^(19/3))