Integrand size = 30, antiderivative size = 351 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=-\frac {c}{16 a x^{16}}+\frac {b c-a d}{13 a^2 x^{13}}-\frac {b^2 c-a b d+a^2 e}{10 a^3 x^{10}}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{7 a^4 x^7}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{4 a^5 x^4}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}-\frac {b^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{19/3}}-\frac {b^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{19/3}}+\frac {b^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{19/3}} \]
-1/16*c/a/x^16+1/13*(-a*d+b*c)/a^2/x^13+1/10*(-a^2*e+a*b*d-b^2*c)/a^3/x^10 +1/7*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^4/x^7-1/4*b*(-a^3*f+a^2*b*e-a*b^2*d+ b^3*c)/a^5/x^4+b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^6/x-1/3*b^(7/3)*(-a^3* f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(19/3)+1/6*b^(7/3)*(-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)/a^(19/3 )-1/3*b^(7/3)*(-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))/a^(19/3)*3^(1/2)
Time = 0.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=-\frac {c}{16 a x^{16}}+\frac {b c-a d}{13 a^2 x^{13}}-\frac {b^2 c-a b d+a^2 e}{10 a^3 x^{10}}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{7 a^4 x^7}+\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{4 a^5 x^4}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {b^{7/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{19/3}}+\frac {b^{7/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{19/3}}+\frac {b^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{19/3}} \]
-1/16*c/(a*x^16) + (b*c - a*d)/(13*a^2*x^13) - (b^2*c - a*b*d + a^2*e)/(10 *a^3*x^10) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(7*a^4*x^7) + (b*(-(b^3*c ) + a*b^2*d - a^2*b*e + a^3*f))/(4*a^5*x^4) + (b^2*(b^3*c - a*b^2*d + a^2* b*e - a^3*f))/(a^6*x) + (b^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Ar cTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(19/3)) + (b^(7/3)*( -(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(19/3 )) + (b^(7/3)*(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*a^(19/3))
Time = 0.52 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2373, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \int \left (\frac {a d-b c}{a^2 x^{14}}+\frac {a^2 e-a b d+b^2 c}{a^3 x^{11}}-\frac {b^3 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^6 \left (a+b x^3\right )}+\frac {b^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^6 x^2}-\frac {b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^5 x^5}+\frac {a^3 f-a^2 b e+a b^2 d-b^3 c}{a^4 x^8}+\frac {c}{a x^{17}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c-a d}{13 a^2 x^{13}}-\frac {a^2 e-a b d+b^2 c}{10 a^3 x^{10}}-\frac {b^{7/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{19/3}}+\frac {b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{19/3}}-\frac {b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{19/3}}+\frac {b^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6 x}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 x^4}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{7 a^4 x^7}-\frac {c}{16 a x^{16}}\) |
-1/16*c/(a*x^16) + (b*c - a*d)/(13*a^2*x^13) - (b^2*c - a*b*d + a^2*e)/(10 *a^3*x^10) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(7*a^4*x^7) - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(4*a^5*x^4) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a^6*x) - (b^(7/3)*(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]*a^(19/3)) - (b^(7/3)*( b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(19/3)) + (b^(7/3)*(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*a^(19/3))
3.3.50.3.1 Defintions of rubi rules used
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
Time = 1.66 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(-\frac {c}{16 a \,x^{16}}-\frac {a d -b c}{13 a^{2} x^{13}}-\frac {a^{2} e -a b d +b^{2} c}{10 a^{3} x^{10}}-\frac {f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c}{7 a^{4} x^{7}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) b^{2}}{a^{6} x}+\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) b}{4 a^{5} x^{4}}-\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{3} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{6}}\) | \(277\) |
| risch | \(\frac {-\frac {c}{16 a}-\frac {\left (a d -b c \right ) x^{3}}{13 a^{2}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{6}}{10 a^{3}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{9}}{7 a^{4}}+\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) b \,x^{12}}{4 a^{5}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) b^{2} x^{15}}{a^{6}}}{x^{16}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{19} \textit {\_Z}^{3}-a^{9} b^{7} f^{3}+3 a^{8} b^{8} e \,f^{2}-3 a^{7} b^{9} d \,f^{2}-3 a^{7} b^{9} e^{2} f +3 a^{6} b^{10} c \,f^{2}+6 a^{6} b^{10} d e f +a^{6} b^{10} e^{3}-6 a^{5} b^{11} c e f -3 a^{5} b^{11} d^{2} f -3 a^{5} b^{11} d \,e^{2}+6 a^{4} b^{12} c d f +3 a^{4} b^{12} c \,e^{2}+3 a^{4} b^{12} d^{2} e -3 a^{3} b^{13} c^{2} f -6 a^{3} b^{13} c d e -a^{3} b^{13} d^{3}+3 a^{2} b^{14} c^{2} e +3 a^{2} b^{14} c \,d^{2}-3 a \,b^{15} c^{2} d +b^{16} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{19}+3 a^{9} b^{7} f^{3}-9 a^{8} b^{8} e \,f^{2}+9 a^{7} b^{9} d \,f^{2}+9 a^{7} b^{9} e^{2} f -9 a^{6} b^{10} c \,f^{2}-18 a^{6} b^{10} d e f -3 a^{6} b^{10} e^{3}+18 a^{5} b^{11} c e f +9 a^{5} b^{11} d^{2} f +9 a^{5} b^{11} d \,e^{2}-18 a^{4} b^{12} c d f -9 a^{4} b^{12} c \,e^{2}-9 a^{4} b^{12} d^{2} e +9 a^{3} b^{13} c^{2} f +18 a^{3} b^{13} c d e +3 a^{3} b^{13} d^{3}-9 a^{2} b^{14} c^{2} e -9 a^{2} b^{14} c \,d^{2}+9 a \,b^{15} c^{2} d -3 b^{16} c^{3}\right ) x +\left (-a^{16} b^{2} f +a^{15} b^{3} e -a^{14} b^{4} d +a^{13} b^{5} c \right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(676\) |
-1/16*c/a/x^16-1/13*(a*d-b*c)/a^2/x^13-1/10*(a^2*e-a*b*d+b^2*c)/a^3/x^10-1 /7*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^4/x^7-(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^6 *b^2/x+1/4*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^5*b/x^4-(-1/3/b/(a/b)^(1/3)*ln( x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^( 1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))*b^3*(a^3*f-a^2 *b*e+a*b^2*d-b^3*c)/a^6
Time = 0.30 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=\frac {7280 \, \sqrt {3} {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{16} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 3640 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{16} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 7280 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{16} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 21840 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} - 5460 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 3120 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{9} - 2184 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{6} - 1365 \, a^{5} c + 1680 \, {\left (a^{4} b c - a^{5} d\right )} x^{3}}{21840 \, a^{6} x^{16}} \]
1/21840*(7280*sqrt(3)*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^16*(b/a) ^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3*sqrt(3)) + 3640*(b^5*c - a*b ^4*d + a^2*b^3*e - a^3*b^2*f)*x^16*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 7280*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^16*(b /a)^(1/3)*log(b*x + a*(b/a)^(2/3)) + 21840*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^15 - 5460*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^12 + 3120*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^9 - 2184*(a^3*b^2*c - a^4 *b*d + a^5*e)*x^6 - 1365*a^5*c + 1680*(a^4*b*c - a^5*d)*x^3)/(a^6*x^16)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=\frac {\sqrt {3} {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {7280 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} - 1820 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 1040 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{9} - 728 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{6} - 455 \, a^{5} c + 560 \, {\left (a^{4} b c - a^{5} d\right )} x^{3}}{7280 \, a^{6} x^{16}} \]
1/3*sqrt(3)*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*arctan(1/3*sqrt(3)*( 2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^6*(a/b)^(1/3)) + 1/6*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^6*(a/b)^ (1/3)) - 1/3*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*log(x + (a/b)^(1/3) )/(a^6*(a/b)^(1/3)) + 1/7280*(7280*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2* f)*x^15 - 1820*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^12 + 1040*(a^ 2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^9 - 728*(a^3*b^2*c - a^4*b*d + a^ 5*e)*x^6 - 455*a^5*c + 560*(a^4*b*c - a^5*d)*x^3)/(a^6*x^16)
Time = 0.28 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.33 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3} d + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b^{2} e - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{7}} - \frac {{\left (b^{6} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{5} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b^{4} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{7}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{4} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3} d + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b^{2} e - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{7}} + \frac {7280 \, b^{5} c x^{15} - 7280 \, a b^{4} d x^{15} + 7280 \, a^{2} b^{3} e x^{15} - 7280 \, a^{3} b^{2} f x^{15} - 1820 \, a b^{4} c x^{12} + 1820 \, a^{2} b^{3} d x^{12} - 1820 \, a^{3} b^{2} e x^{12} + 1820 \, a^{4} b f x^{12} + 1040 \, a^{2} b^{3} c x^{9} - 1040 \, a^{3} b^{2} d x^{9} + 1040 \, a^{4} b e x^{9} - 1040 \, a^{5} f x^{9} - 728 \, a^{3} b^{2} c x^{6} + 728 \, a^{4} b d x^{6} - 728 \, a^{5} e x^{6} + 560 \, a^{4} b c x^{3} - 560 \, a^{5} d x^{3} - 455 \, a^{5} c}{7280 \, a^{6} x^{16}} \]
-1/3*sqrt(3)*((-a*b^2)^(2/3)*b^4*c - (-a*b^2)^(2/3)*a*b^3*d + (-a*b^2)^(2/ 3)*a^2*b^2*e - (-a*b^2)^(2/3)*a^3*b*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1 /3))/(-a/b)^(1/3))/a^7 - 1/3*(b^6*c*(-a/b)^(1/3) - a*b^5*d*(-a/b)^(1/3) + a^2*b^4*e*(-a/b)^(1/3) - a^3*b^3*f*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^7 + 1/6*((-a*b^2)^(2/3)*b^4*c - (-a*b^2)^(2/3)*a*b^3*d + (-a*b^2)^(2/3)*a^2*b^2*e - (-a*b^2)^(2/3)*a^3*b*f)*log(x^2 + x*(-a/b)^(1/3 ) + (-a/b)^(2/3))/a^7 + 1/7280*(7280*b^5*c*x^15 - 7280*a*b^4*d*x^15 + 7280 *a^2*b^3*e*x^15 - 7280*a^3*b^2*f*x^15 - 1820*a*b^4*c*x^12 + 1820*a^2*b^3*d *x^12 - 1820*a^3*b^2*e*x^12 + 1820*a^4*b*f*x^12 + 1040*a^2*b^3*c*x^9 - 104 0*a^3*b^2*d*x^9 + 1040*a^4*b*e*x^9 - 1040*a^5*f*x^9 - 728*a^3*b^2*c*x^6 + 728*a^4*b*d*x^6 - 728*a^5*e*x^6 + 560*a^4*b*c*x^3 - 560*a^5*d*x^3 - 455*a^ 5*c)/(a^6*x^16)
Time = 10.00 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{17} \left (a+b x^3\right )} \, dx=-\frac {\frac {c}{16\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{7\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{13\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{10\,a^3}+\frac {b\,x^{12}\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{4\,a^5}-\frac {b^2\,x^{15}\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^6}}{x^{16}}-\frac {b^{7/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{19/3}}+\frac {b^{7/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{19/3}}-\frac {b^{7/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{19/3}} \]
(b^(7/3)*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*a^(19/3)) - (b^(7/3)*log(b^( 1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(19/3)) - (c/( 16*a) - (x^9*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(7*a^4) + (x^3*(a*d - b* c))/(13*a^2) + (x^6*(b^2*c + a^2*e - a*b*d))/(10*a^3) + (b*x^12*(b^3*c - a ^3*f - a*b^2*d + a^2*b*e))/(4*a^5) - (b^2*x^15*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/a^6)/x^16 - (b^(7/3)*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*a^(19/ 3))