Integrand size = 30, antiderivative size = 316 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac {(b e-a f) x^{11}}{11 b^2}+\frac {f x^{14}}{14 b}-\frac {a^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{17/3}}-\frac {a^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{17/3}}+\frac {a^{5/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{17/3}} \]
-1/2*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/b^5+1/5*(-a^3*f+a^2*b*e-a*b^2*d+ b^3*c)*x^5/b^4+1/8*(a^2*f-a*b*e+b^2*d)*x^8/b^3+1/11*(-a*f+b*e)*x^11/b^2+1/ 14*f*x^14/b-1/3*a^(5/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)* x)/b^(17/3)+1/6*a^(5/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1/3)* b^(1/3)*x+b^(2/3)*x^2)/b^(17/3)-1/3*a^(5/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c) *arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/b^(17/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.98 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac {(b e-a f) x^{11}}{11 b^2}+\frac {f x^{14}}{14 b}+\frac {a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{17/3}}+\frac {a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{17/3}}-\frac {a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{17/3}} \]
(a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(2*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^8)/(8*b^3) + ((b*e - a*f)*x^11)/(11*b^2) + (f*x^14)/(14*b) + (a^(5/3)*(-(b^3*c) + a*b^ 2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[ 3]*b^(17/3)) + (a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(17/3)) - (a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3 *f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(17/3))
Time = 0.52 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\int \frac {14 x^7 \left ((b e-a f) x^6+b d x^3+b c\right )}{b x^3+a}dx}{14 b}+\frac {f x^{14}}{14 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^7 \left ((b e-a f) x^6+b d x^3+b c\right )}{b x^3+a}dx}{b}+\frac {f x^{14}}{14 b}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\int \left (\frac {(b e-a f) x^{10}}{b}+\frac {\left (f a^2-b e a+b^2 d\right ) x^7}{b^2}+\frac {\left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^4}{b^3}-\frac {a \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x}{b^4}-\frac {\left (f a^5-b e a^4+b^2 d a^3-b^3 c a^2\right ) x}{b^4 \left (b x^3+a\right )}\right )dx}{b}+\frac {f x^{14}}{14 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^2}+\frac {x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^3}-\frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^4}-\frac {a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} b^{14/3}}+\frac {a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{14/3}}-\frac {a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac {x^{11} (b e-a f)}{11 b}}{b}+\frac {f x^{14}}{14 b}\) |
(f*x^14)/(14*b) + (-1/2*(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/b^4 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^3) + ((b^2*d - a*b*e + a^2* f)*x^8)/(8*b^2) + ((b*e - a*f)*x^11)/(11*b) - (a^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[ 3]*b^(14/3)) - (a^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(14/3)) + (a^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Lo g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(14/3)))/b
3.3.34.3.1 Defintions of rubi rules used
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_)*((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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(\frac {f \,x^{14}}{14 b}-\frac {x^{11} a f}{11 b^{2}}+\frac {e \,x^{11}}{11 b}+\frac {x^{8} a^{2} f}{8 b^{3}}-\frac {a e \,x^{8}}{8 b^{2}}+\frac {x^{8} d}{8 b}-\frac {x^{5} a^{3} f}{5 b^{4}}+\frac {a^{2} e \,x^{5}}{5 b^{3}}-\frac {x^{5} a d}{5 b^{2}}+\frac {x^{5} c}{5 b}+\frac {x^{2} a^{4} f}{2 b^{5}}-\frac {a^{3} e \,x^{2}}{2 b^{4}}+\frac {x^{2} a^{2} d}{2 b^{3}}-\frac {a c \,x^{2}}{2 b^{2}}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-f \,a^{3}+a^{2} b e -a \,b^{2} d +b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{3 b^{6}}\) | \(204\) |
| default | \(\frac {\frac {f \,x^{14} b^{4}}{14}+\frac {\left (-a \,b^{3} f +b^{4} e \right ) x^{11}}{11}+\frac {\left (a^{2} b^{2} f -a \,b^{3} e +b^{4} d \right ) x^{8}}{8}+\frac {\left (-a^{3} b f +a^{2} e \,b^{2}-a \,b^{3} d +b^{4} c \right ) x^{5}}{5}+\frac {\left (a^{4} f -a^{3} b e +a^{2} b^{2} d -a \,b^{3} c \right ) x^{2}}{2}}{b^{5}}-\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{b^{5}}\) | \(249\) |
1/14*f*x^14/b-1/11/b^2*x^11*a*f+1/11/b*e*x^11+1/8/b^3*x^8*a^2*f-1/8/b^2*a* e*x^8+1/8/b*x^8*d-1/5/b^4*x^5*a^3*f+1/5/b^3*a^2*e*x^5-1/5/b^2*x^5*a*d+1/5/ b*x^5*c+1/2/b^5*x^2*a^4*f-1/2/b^4*a^3*e*x^2+1/2/b^3*x^2*a^2*d-1/2/b^2*a*c* x^2+1/3/b^6*a^2*sum((-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/_R*ln(x-_R),_R=RootOf(_ Z^3*b+a))
Time = 0.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.02 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {660 \, b^{4} f x^{14} + 840 \, {\left (b^{4} e - a b^{3} f\right )} x^{11} + 1155 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{8} + 1848 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{5} - 4620 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{2} + 3080 \, \sqrt {3} {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 1540 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 3080 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{9240 \, b^{5}} \]
1/9240*(660*b^4*f*x^14 + 840*(b^4*e - a*b^3*f)*x^11 + 1155*(b^4*d - a*b^3* e + a^2*b^2*f)*x^8 + 1848*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^5 - 46 20*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^2 + 3080*sqrt(3)*(a*b^3*c - a ^2*b^2*d + a^3*b*e - a^4*f)*(a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a^2 /b^2)^(1/3) - sqrt(3)*a)/a) + 1540*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f) *(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)) - 30 80*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(a^2/b^2)^(1/3)*log(a*x + b*(a^ 2/b^2)^(2/3)))/b^5
Time = 0.85 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.62 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=x^{11} \left (- \frac {a f}{11 b^{2}} + \frac {e}{11 b}\right ) + x^{8} \left (\frac {a^{2} f}{8 b^{3}} - \frac {a e}{8 b^{2}} + \frac {d}{8 b}\right ) + x^{5} \left (- \frac {a^{3} f}{5 b^{4}} + \frac {a^{2} e}{5 b^{3}} - \frac {a d}{5 b^{2}} + \frac {c}{5 b}\right ) + x^{2} \left (\frac {a^{4} f}{2 b^{5}} - \frac {a^{3} e}{2 b^{4}} + \frac {a^{2} d}{2 b^{3}} - \frac {a c}{2 b^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{17} - a^{14} f^{3} + 3 a^{13} b e f^{2} - 3 a^{12} b^{2} d f^{2} - 3 a^{12} b^{2} e^{2} f + 3 a^{11} b^{3} c f^{2} + 6 a^{11} b^{3} d e f + a^{11} b^{3} e^{3} - 6 a^{10} b^{4} c e f - 3 a^{10} b^{4} d^{2} f - 3 a^{10} b^{4} d e^{2} + 6 a^{9} b^{5} c d f + 3 a^{9} b^{5} c e^{2} + 3 a^{9} b^{5} d^{2} e - 3 a^{8} b^{6} c^{2} f - 6 a^{8} b^{6} c d e - a^{8} b^{6} d^{3} + 3 a^{7} b^{7} c^{2} e + 3 a^{7} b^{7} c d^{2} - 3 a^{6} b^{8} c^{2} d + a^{5} b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} b^{11}}{a^{9} f^{2} - 2 a^{8} b e f + 2 a^{7} b^{2} d f + a^{7} b^{2} e^{2} - 2 a^{6} b^{3} c f - 2 a^{6} b^{3} d e + 2 a^{5} b^{4} c e + a^{5} b^{4} d^{2} - 2 a^{4} b^{5} c d + a^{3} b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{14}}{14 b} \]
x**11*(-a*f/(11*b**2) + e/(11*b)) + x**8*(a**2*f/(8*b**3) - a*e/(8*b**2) + d/(8*b)) + x**5*(-a**3*f/(5*b**4) + a**2*e/(5*b**3) - a*d/(5*b**2) + c/(5 *b)) + x**2*(a**4*f/(2*b**5) - a**3*e/(2*b**4) + a**2*d/(2*b**3) - a*c/(2* b**2)) + RootSum(27*_t**3*b**17 - a**14*f**3 + 3*a**13*b*e*f**2 - 3*a**12* b**2*d*f**2 - 3*a**12*b**2*e**2*f + 3*a**11*b**3*c*f**2 + 6*a**11*b**3*d*e *f + a**11*b**3*e**3 - 6*a**10*b**4*c*e*f - 3*a**10*b**4*d**2*f - 3*a**10* b**4*d*e**2 + 6*a**9*b**5*c*d*f + 3*a**9*b**5*c*e**2 + 3*a**9*b**5*d**2*e - 3*a**8*b**6*c**2*f - 6*a**8*b**6*c*d*e - a**8*b**6*d**3 + 3*a**7*b**7*c* *2*e + 3*a**7*b**7*c*d**2 - 3*a**6*b**8*c**2*d + a**5*b**9*c**3, Lambda(_t , _t*log(9*_t**2*b**11/(a**9*f**2 - 2*a**8*b*e*f + 2*a**7*b**2*d*f + a**7* b**2*e**2 - 2*a**6*b**3*c*f - 2*a**6*b**3*d*e + 2*a**5*b**4*c*e + a**5*b** 4*d**2 - 2*a**4*b**5*c*d + a**3*b**6*c**2) + x))) + f*x**14/(14*b)
Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.99 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {\sqrt {3} {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {220 \, b^{4} f x^{14} + 280 \, {\left (b^{4} e - a b^{3} f\right )} x^{11} + 385 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{8} + 616 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{5} - 1540 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{2}}{3080 \, b^{5}} + \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/3*sqrt(3)*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*arctan(1/3*sqrt(3)*( 2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(1/3)) + 1/3080*(220*b^4*f*x^14 + 280*(b^4*e - a*b^3*f)*x^11 + 385*(b^4*d - a*b^3*e + a^2*b^2*f)*x^8 + 61 6*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^5 - 1540*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^2)/b^5 + 1/6*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f )*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^6*(a/b)^(1/3)) - 1/3*(a^2*b^3* c - a^3*b^2*d + a^4*b*e - a^5*f)*log(x + (a/b)^(1/3))/(b^6*(a/b)^(1/3))
Time = 0.28 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.37 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b^{2} d + \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b e - \left (-a b^{2}\right )^{\frac {2}{3}} a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{7}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b^{2} d + \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b e - \left (-a b^{2}\right )^{\frac {2}{3}} a^{4} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{7}} - \frac {{\left (a^{2} b^{12} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b^{11} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{4} b^{10} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{5} b^{9} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{14}} + \frac {220 \, b^{13} f x^{14} + 280 \, b^{13} e x^{11} - 280 \, a b^{12} f x^{11} + 385 \, b^{13} d x^{8} - 385 \, a b^{12} e x^{8} + 385 \, a^{2} b^{11} f x^{8} + 616 \, b^{13} c x^{5} - 616 \, a b^{12} d x^{5} + 616 \, a^{2} b^{11} e x^{5} - 616 \, a^{3} b^{10} f x^{5} - 1540 \, a b^{12} c x^{2} + 1540 \, a^{2} b^{11} d x^{2} - 1540 \, a^{3} b^{10} e x^{2} + 1540 \, a^{4} b^{9} f x^{2}}{3080 \, b^{14}} \]
-1/3*sqrt(3)*((-a*b^2)^(2/3)*a*b^3*c - (-a*b^2)^(2/3)*a^2*b^2*d + (-a*b^2) ^(2/3)*a^3*b*e - (-a*b^2)^(2/3)*a^4*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1 /3))/(-a/b)^(1/3))/b^7 + 1/6*((-a*b^2)^(2/3)*a*b^3*c - (-a*b^2)^(2/3)*a^2* b^2*d + (-a*b^2)^(2/3)*a^3*b*e - (-a*b^2)^(2/3)*a^4*f)*log(x^2 + x*(-a/b)^ (1/3) + (-a/b)^(2/3))/b^7 - 1/3*(a^2*b^12*c*(-a/b)^(1/3) - a^3*b^11*d*(-a/ b)^(1/3) + a^4*b^10*e*(-a/b)^(1/3) - a^5*b^9*f*(-a/b)^(1/3))*(-a/b)^(1/3)* log(abs(x - (-a/b)^(1/3)))/(a*b^14) + 1/3080*(220*b^13*f*x^14 + 280*b^13*e *x^11 - 280*a*b^12*f*x^11 + 385*b^13*d*x^8 - 385*a*b^12*e*x^8 + 385*a^2*b^ 11*f*x^8 + 616*b^13*c*x^5 - 616*a*b^12*d*x^5 + 616*a^2*b^11*e*x^5 - 616*a^ 3*b^10*f*x^5 - 1540*a*b^12*c*x^2 + 1540*a^2*b^11*d*x^2 - 1540*a^3*b^10*e*x ^2 + 1540*a^4*b^9*f*x^2)/b^14
Time = 9.86 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.99 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=x^{11}\,\left (\frac {e}{11\,b}-\frac {a\,f}{11\,b^2}\right )+x^8\,\left (\frac {d}{8\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{8\,b}\right )+x^5\,\left (\frac {c}{5\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{5\,b}\right )+\frac {f\,x^{14}}{14\,b}-\frac {a^{5/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{17/3}}-\frac {a\,x^2\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{2\,b}+\frac {a^{5/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{17/3}}-\frac {a^{5/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{17/3}} \]
x^11*(e/(11*b) - (a*f)/(11*b^2)) + x^8*(d/(8*b) - (a*(e/b - (a*f)/b^2))/(8 *b)) + x^5*(c/(5*b) - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/(5*b)) + (f*x^14 )/(14*b) - (a^(5/3)*log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^ 2*b*e))/(3*b^(17/3)) - (a*x^2*(c/b - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/b ))/(2*b) + (a^(5/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1 /2)*1i)/2 + 1/2)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^(17/3)) - (a^(5 /3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2) *(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^(17/3))