Integrand size = 30, antiderivative size = 328 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^{10}}{10 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}+\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{16/3}}-\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac {\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{16/3}} \]
(-4*a^3*f+3*a^2*b*e-2*a*b^2*d+b^3*c)*x/b^5+1/4*(3*a^2*f-2*a*b*e+b^2*d)*x^4 /b^4+1/7*(-2*a*f+b*e)*x^7/b^3+1/10*f*x^10/b^2+1/3*a*(-a^3*f+a^2*b*e-a*b^2* d+b^3*c)*x/b^5/(b*x^3+a)-1/9*a^(1/3)*(-13*a^3*f+10*a^2*b*e-7*a*b^2*d+4*b^3 *c)*ln(a^(1/3)+b^(1/3)*x)/b^(16/3)+1/18*a^(1/3)*(-13*a^3*f+10*a^2*b*e-7*a* b^2*d+4*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(16/3)+1/9*a^(1 /3)*(-13*a^3*f+10*a^2*b*e-7*a*b^2*d+4*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3) *x)/a^(1/3)*3^(1/2))/b^(16/3)*3^(1/2)
Time = 0.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.96 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {1260 \sqrt [3]{b} \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x+315 b^{4/3} \left (b^2 d-2 a b e+3 a^2 f\right ) x^4+180 b^{7/3} (b e-2 a f) x^7+126 b^{10/3} f x^{10}+\frac {420 a \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{a+b x^3}-140 \sqrt {3} \sqrt [3]{a} \left (-4 b^3 c+7 a b^2 d-10 a^2 b e+13 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+140 \sqrt [3]{a} \left (-4 b^3 c+7 a b^2 d-10 a^2 b e+13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-70 \sqrt [3]{a} \left (-4 b^3 c+7 a b^2 d-10 a^2 b e+13 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1260 b^{16/3}} \]
(1260*b^(1/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x + 315*b^(4/3)*(b ^2*d - 2*a*b*e + 3*a^2*f)*x^4 + 180*b^(7/3)*(b*e - 2*a*f)*x^7 + 126*b^(10/ 3)*f*x^10 + (420*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x ^3) - 140*Sqrt[3]*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*A rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-4*b^3*c + 7*a*b ^2*d - 10*a^2*b*e + 13*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-4*b^ 3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1260*b^(16/3))
Time = 0.70 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2367, 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.
| \(\displaystyle \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {-3 a b^4 f x^{12}-3 a b^3 (b e-a f) x^9-3 a b^2 \left (f a^2-b e a+b^2 d\right ) x^6-3 a b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{b x^3+a}dx}{3 a b^5}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac {\int \left (-3 a b^3 f x^9-3 a b^2 (b e-2 a f) x^6-3 a b \left (3 f a^2-2 b e a+b^2 d\right ) x^3-3 a \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right )+\frac {-13 f a^5+10 b e a^4-7 b^2 d a^3+4 b^3 c a^2}{b x^3+a}\right )dx}{3 a b^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac {-\frac {3}{4} a b x^4 \left (3 a^2 f-2 a b e+b^2 d\right )-3 a x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )-\frac {a^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \sqrt [3]{b}}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{6 \sqrt [3]{b}}-\frac {3}{10} a b^3 f x^{10}-\frac {3}{7} a b^2 x^7 (b e-2 a f)}{3 a b^5}\) |
(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*b^5*(a + b*x^3)) - (-3*a*(b^3 *c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x - (3*a*b*(b^2*d - 2*a*b*e + 3*a^2* f)*x^4)/4 - (3*a*b^2*(b*e - 2*a*f)*x^7)/7 - (3*a*b^3*f*x^10)/10 - (a^(4/3) *(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3) *x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(1/3)) + (a^(4/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(1/3)) - (a^(4/3)* (4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*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^5)
3.3.62.3.1 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 = 1.54 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(\frac {f \,x^{10}}{10 b^{2}}-\frac {2 x^{7} f a}{7 b^{3}}+\frac {x^{7} e}{7 b^{2}}+\frac {3 x^{4} f \,a^{2}}{4 b^{4}}-\frac {x^{4} a e}{2 b^{3}}+\frac {d \,x^{4}}{4 b^{2}}-\frac {4 x f \,a^{3}}{b^{5}}+\frac {3 x \,a^{2} e}{b^{4}}-\frac {2 x a d}{b^{3}}+\frac {x c}{b^{2}}+\frac {\left (-\frac {1}{3} a^{4} f +\frac {1}{3} a^{3} b e -\frac {1}{3} a^{2} b^{2} d +\frac {1}{3} a \,b^{3} c \right ) x}{b^{5} \left (b \,x^{3}+a \right )}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (13 f \,a^{3}-10 a^{2} b e +7 a \,b^{2} d -4 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{6}}\) | \(193\) |
| default | \(-\frac {-\frac {1}{10} b^{3} f \,x^{10}+\frac {2}{7} x^{7} a \,b^{2} f -\frac {1}{7} x^{7} b^{3} e -\frac {3}{4} a^{2} b f \,x^{4}+\frac {1}{2} a \,b^{2} e \,x^{4}-\frac {1}{4} d \,x^{4} b^{3}+4 f \,a^{3} x -3 a^{2} b e x +2 a \,b^{2} d x -b^{3} c x}{b^{5}}+\frac {a \left (\frac {\left (-\frac {1}{3} f \,a^{3}+\frac {1}{3} a^{2} b e -\frac {1}{3} a \,b^{2} d +\frac {1}{3} b^{3} c \right ) x}{b \,x^{3}+a}+\frac {\left (13 f \,a^{3}-10 a^{2} b e +7 a \,b^{2} d -4 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}\right )}{b^{5}}\) | \(258\) |
1/10*f*x^10/b^2-2/7/b^3*x^7*f*a+1/7/b^2*x^7*e+3/4/b^4*x^4*f*a^2-1/2/b^3*x^ 4*a*e+1/4/b^2*d*x^4-4/b^5*x*f*a^3+3/b^4*x*a^2*e-2/b^3*x*a*d+1/b^2*x*c+(-1/ 3*a^4*f+1/3*a^3*b*e-1/3*a^2*b^2*d+1/3*a*b^3*c)*x/b^5/(b*x^3+a)+1/9/b^6*a*s um((13*a^3*f-10*a^2*b*e+7*a*b^2*d-4*b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+ a))
Time = 0.32 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.29 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {126 \, b^{4} f x^{13} + 18 \, {\left (10 \, b^{4} e - 13 \, a b^{3} f\right )} x^{10} + 45 \, {\left (7 \, b^{4} d - 10 \, a b^{3} e + 13 \, a^{2} b^{2} f\right )} x^{7} + 315 \, {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{4} - 140 \, \sqrt {3} {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f + {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 70 \, {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f + {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 140 \, {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f + {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 420 \, {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} x}{1260 \, {\left (b^{6} x^{3} + a b^{5}\right )}} \]
1/1260*(126*b^4*f*x^13 + 18*(10*b^4*e - 13*a*b^3*f)*x^10 + 45*(7*b^4*d - 1 0*a*b^3*e + 13*a^2*b^2*f)*x^7 + 315*(4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^4 - 140*sqrt(3)*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a ^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)* arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3)*a)/a) + 70*(4*a*b^3*c - 7* a^2*b^2*d + 10*a^3*b*e - 13*a^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140* (4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f + (4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 420*(4* a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*x)/(b^6*x^3 + a*b^5)
Time = 76.44 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.37 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^{7} \left (- \frac {2 a f}{7 b^{3}} + \frac {e}{7 b^{2}}\right ) + x^{4} \cdot \left (\frac {3 a^{2} f}{4 b^{4}} - \frac {a e}{2 b^{3}} + \frac {d}{4 b^{2}}\right ) + x \left (- \frac {4 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{b^{4}} - \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{16} - 2197 a^{10} f^{3} + 5070 a^{9} b e f^{2} - 3549 a^{8} b^{2} d f^{2} - 3900 a^{8} b^{2} e^{2} f + 2028 a^{7} b^{3} c f^{2} + 5460 a^{7} b^{3} d e f + 1000 a^{7} b^{3} e^{3} - 3120 a^{6} b^{4} c e f - 1911 a^{6} b^{4} d^{2} f - 2100 a^{6} b^{4} d e^{2} + 2184 a^{5} b^{5} c d f + 1200 a^{5} b^{5} c e^{2} + 1470 a^{5} b^{5} d^{2} e - 624 a^{4} b^{6} c^{2} f - 1680 a^{4} b^{6} c d e - 343 a^{4} b^{6} d^{3} + 480 a^{3} b^{7} c^{2} e + 588 a^{3} b^{7} c d^{2} - 336 a^{2} b^{8} c^{2} d + 64 a b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t b^{5}}{13 a^{3} f - 10 a^{2} b e + 7 a b^{2} d - 4 b^{3} c} + x \right )} \right )\right )} + \frac {f x^{10}}{10 b^{2}} \]
x**7*(-2*a*f/(7*b**3) + e/(7*b**2)) + x**4*(3*a**2*f/(4*b**4) - a*e/(2*b** 3) + d/(4*b**2)) + x*(-4*a**3*f/b**5 + 3*a**2*e/b**4 - 2*a*d/b**3 + c/b**2 ) + x*(-a**4*f + a**3*b*e - a**2*b**2*d + a*b**3*c)/(3*a*b**5 + 3*b**6*x** 3) + RootSum(729*_t**3*b**16 - 2197*a**10*f**3 + 5070*a**9*b*e*f**2 - 3549 *a**8*b**2*d*f**2 - 3900*a**8*b**2*e**2*f + 2028*a**7*b**3*c*f**2 + 5460*a **7*b**3*d*e*f + 1000*a**7*b**3*e**3 - 3120*a**6*b**4*c*e*f - 1911*a**6*b* *4*d**2*f - 2100*a**6*b**4*d*e**2 + 2184*a**5*b**5*c*d*f + 1200*a**5*b**5* c*e**2 + 1470*a**5*b**5*d**2*e - 624*a**4*b**6*c**2*f - 1680*a**4*b**6*c*d *e - 343*a**4*b**6*d**3 + 480*a**3*b**7*c**2*e + 588*a**3*b**7*c*d**2 - 33 6*a**2*b**8*c**2*d + 64*a*b**9*c**3, Lambda(_t, _t*log(9*_t*b**5/(13*a**3* f - 10*a**2*b*e + 7*a*b**2*d - 4*b**3*c) + x))) + f*x**10/(10*b**2)
Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.98 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{3 \, {\left (b^{6} x^{3} + a b^{5}\right )}} + \frac {14 \, b^{3} f x^{10} + 20 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{7} + 35 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{4} + 140 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x}{140 \, b^{5}} - \frac {\sqrt {3} {\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x/(b^6*x^3 + a*b^5) + 1/140*(1 4*b^3*f*x^10 + 20*(b^3*e - 2*a*b^2*f)*x^7 + 35*(b^3*d - 2*a*b^2*e + 3*a^2* b*f)*x^4 + 140*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 - 1/9*sqrt (3)*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*arctan(1/3*sqrt(3)*( 2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(2/3)) + 1/18*(4*a*b^3*c - 7*a^ 2*b^2*d + 10*a^3*b*e - 13*a^4*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b ^6*(a/b)^(2/3)) - 1/9*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*lo g(x + (a/b)^(1/3))/(b^6*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.18 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{6}} + \frac {{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{5}} - \frac {{\left (4 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{6}} + \frac {a b^{3} c x - a^{2} b^{2} d x + a^{3} b e x - a^{4} f x}{3 \, {\left (b x^{3} + a\right )} b^{5}} + \frac {14 \, b^{18} f x^{10} + 20 \, b^{18} e x^{7} - 40 \, a b^{17} f x^{7} + 35 \, b^{18} d x^{4} - 70 \, a b^{17} e x^{4} + 105 \, a^{2} b^{16} f x^{4} + 140 \, b^{18} c x - 280 \, a b^{17} d x + 420 \, a^{2} b^{16} e x - 560 \, a^{3} b^{15} f x}{140 \, b^{20}} \]
-1/9*sqrt(3)*(4*(-a*b^2)^(1/3)*b^3*c - 7*(-a*b^2)^(1/3)*a*b^2*d + 10*(-a*b ^2)^(1/3)*a^2*b*e - 13*(-a*b^2)^(1/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a /b)^(1/3))/(-a/b)^(1/3))/b^6 + 1/9*(4*a*b^3*c - 7*a^2*b^2*d + 10*a^3*b*e - 13*a^4*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/18*(4*(-a*b ^2)^(1/3)*b^3*c - 7*(-a*b^2)^(1/3)*a*b^2*d + 10*(-a*b^2)^(1/3)*a^2*b*e - 1 3*(-a*b^2)^(1/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6 + 1/3 *(a*b^3*c*x - a^2*b^2*d*x + a^3*b*e*x - a^4*f*x)/((b*x^3 + a)*b^5) + 1/140 *(14*b^18*f*x^10 + 20*b^18*e*x^7 - 40*a*b^17*f*x^7 + 35*b^18*d*x^4 - 70*a* b^17*e*x^4 + 105*a^2*b^16*f*x^4 + 140*b^18*c*x - 280*a*b^17*d*x + 420*a^2* b^16*e*x - 560*a^3*b^15*f*x)/b^20
Time = 10.18 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.09 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^7\,\left (\frac {e}{7\,b^2}-\frac {2\,a\,f}{7\,b^3}\right )+x\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )-x^4\,\left (\frac {a^2\,f}{4\,b^4}-\frac {d}{4\,b^2}+\frac {a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{2\,b}\right )-\frac {x\,\left (\frac {f\,a^4}{3}-\frac {e\,a^3\,b}{3}+\frac {d\,a^2\,b^2}{3}-\frac {c\,a\,b^3}{3}\right )}{b^6\,x^3+a\,b^5}+\frac {f\,x^{10}}{10\,b^2}-\frac {a^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-13\,f\,a^3+10\,e\,a^2\,b-7\,d\,a\,b^2+4\,c\,b^3\right )}{9\,b^{16/3}}-\frac {a^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-13\,f\,a^3+10\,e\,a^2\,b-7\,d\,a\,b^2+4\,c\,b^3\right )}{9\,b^{16/3}}+\frac {a^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-13\,f\,a^3+10\,e\,a^2\,b-7\,d\,a\,b^2+4\,c\,b^3\right )}{9\,b^{16/3}} \]
x^7*(e/(7*b^2) - (2*a*f)/(7*b^3)) + x*(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) - x^ 4*((a^2*f)/(4*b^4) - d/(4*b^2) + (a*(e/b^2 - (2*a*f)/b^3))/(2*b)) - (x*((a ^4*f)/3 + (a^2*b^2*d)/3 - (a*b^3*c)/3 - (a^3*b*e)/3))/(a*b^5 + b^6*x^3) + (f*x^10)/(10*b^2) - (a^(1/3)*log(b^(1/3)*x + a^(1/3))*(4*b^3*c - 13*a^3*f - 7*a*b^2*d + 10*a^2*b*e))/(9*b^(16/3)) - (a^(1/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(4*b^3*c - 13*a^3*f - 7*a* b^2*d + 10*a^2*b*e))/(9*b^(16/3)) + (a^(1/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^ (1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(4*b^3*c - 13*a^3*f - 7*a*b^2*d + 10*a^2*b*e))/(9*b^(16/3))