Integrand size = 30, antiderivative size = 375 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{19/3}}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}} \]
-1/13*c/a^2/x^13+1/10*(-a*d+2*b*c)/a^3/x^10+1/7*(-a^2*e+2*a*b*d-3*b^2*c)/a ^4/x^7+1/4*(-a^3*f+2*a^2*b*e-3*a*b^2*d+4*b^3*c)/a^5/x^4-b*(-2*a^3*f+3*a^2* b*e-4*a*b^2*d+5*b^3*c)/a^6/x-1/3*b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a^ 6/(b*x^3+a)+1/9*b^(4/3)*(-7*a^3*f+10*a^2*b*e-13*a*b^2*d+16*b^3*c)*ln(a^(1/ 3)+b^(1/3)*x)/a^(19/3)-1/18*b^(4/3)*(-7*a^3*f+10*a^2*b*e-13*a*b^2*d+16*b^3 *c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(19/3)+1/9*b^(4/3)*(-7*a^3 *f+10*a^2*b*e-13*a*b^2*d+16*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 = 0.39 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}+\frac {b \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^6 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{19/3}}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}+\frac {b^{4/3} \left (-16 b^3 c+13 a b^2 d-10 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}} \]
-1/13*c/(a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^ 2*e)/(7*a^4*x^7) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) + (b*(-5*b^3*c + 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3* c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/ Sqrt[3]])/(3*Sqrt[3]*a^(19/3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2* b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(19/3)) + (b^(4/3)*(-16*b^3* c + 13*a*b^2*d - 10*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b ^(2/3)*x^2])/(18*a^(19/3))
Time = 1.06 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2368, 25, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {-\frac {b^5 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{15}}{a^5}+\frac {3 b^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{12}}{a^4}-\frac {3 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {3 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c}{x^{14} \left (b x^3+a\right )}dx}{3 a b^3}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {b^5 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{15}}{a^5}+\frac {3 b^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{12}}{a^4}-\frac {3 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {3 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c}{x^{14} \left (b x^3+a\right )}dx}{3 a b^3}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\int \left (\frac {\left (7 f a^3-10 b e a^2+13 b^2 d a-16 b^3 c\right ) x b^5}{a^5 \left (b x^3+a\right )}-\frac {3 \left (2 f a^3-3 b e a^2+4 b^2 d a-5 b^3 c\right ) b^4}{a^5 x^2}+\frac {3 \left (f a^3-2 b e a^2+3 b^2 d a-4 b^3 c\right ) b^3}{a^4 x^5}+\frac {3 \left (e a^2-2 b d a+3 b^2 c\right ) b^3}{a^3 x^8}+\frac {3 (a d-2 b c) b^3}{a^2 x^{11}}+\frac {3 c b^3}{a x^{14}}\right )dx}{3 a b^3}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 b^3 (2 b c-a d)}{10 a^2 x^{10}}-\frac {3 b^3 \left (a^2 e-2 a b d+3 b^2 c\right )}{7 a^3 x^7}+\frac {b^{13/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{\sqrt {3} a^{16/3}}-\frac {b^{13/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{6 a^{16/3}}+\frac {b^{13/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 a^{16/3}}-\frac {3 b^4 \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^5 x}+\frac {3 b^3 \left (a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right )}{4 a^4 x^4}-\frac {3 b^3 c}{13 a x^{13}}}{3 a b^3}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}\) |
-1/3*(b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a^6*(a + b*x^3)) + ((- 3*b^3*c)/(13*a*x^13) + (3*b^3*(2*b*c - a*d))/(10*a^2*x^10) - (3*b^3*(3*b^2 *c - 2*a*b*d + a^2*e))/(7*a^3*x^7) + (3*b^3*(4*b^3*c - 3*a*b^2*d + 2*a^2*b *e - a^3*f))/(4*a^4*x^4) - (3*b^4*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3 *f))/(a^5*x) + (b^(13/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Ar cTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(16/3)) + (b^( 13/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3) *x])/(3*a^(16/3)) - (b^(13/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3* f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(16/3)))/(3*a*b^3)
3.3.75.3.1 Defintions of rubi rules used
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
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 = 1.56 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {c}{13 a^{2} x^{13}}-\frac {a d -2 b c}{10 a^{3} x^{10}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{7 a^{4} x^{7}}-\frac {f \,a^{3}-2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c}{4 a^{5} x^{4}}+\frac {b \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right )}{a^{6} x}+\frac {b^{2} \left (\frac {\left (\frac {1}{3} f \,a^{3}-\frac {1}{3} a^{2} b e +\frac {1}{3} a \,b^{2} d -\frac {1}{3} b^{3} c \right ) x^{2}}{b \,x^{3}+a}+\left (\frac {7}{3} f \,a^{3}-\frac {10}{3} a^{2} b e +\frac {13}{3} a \,b^{2} d -\frac {16}{3} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{a^{6}}\) | \(287\) |
| risch | \(\frac {-\frac {c}{13 a}-\frac {\left (13 a d -16 b c \right ) x^{3}}{130 a^{2}}-\frac {\left (10 a^{2} e -13 a b d +16 b^{2} c \right ) x^{6}}{70 a^{3}}-\frac {\left (7 f \,a^{3}-10 a^{2} b e +13 a \,b^{2} d -16 b^{3} c \right ) x^{9}}{28 a^{4}}+\frac {b \left (7 f \,a^{3}-10 a^{2} b e +13 a \,b^{2} d -16 b^{3} c \right ) x^{12}}{4 a^{5}}+\frac {b^{2} \left (7 f \,a^{3}-10 a^{2} b e +13 a \,b^{2} d -16 b^{3} c \right ) x^{15}}{3 a^{6}}}{x^{13} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{19} \textit {\_Z}^{3}+343 a^{9} b^{4} f^{3}-1470 a^{8} b^{5} e \,f^{2}+1911 a^{7} b^{6} d \,f^{2}+2100 a^{7} b^{6} e^{2} f -2352 a^{6} b^{7} c \,f^{2}-5460 a^{6} b^{7} d e f -1000 a^{6} b^{7} e^{3}+6720 a^{5} b^{8} c e f +3549 a^{5} b^{8} d^{2} f +3900 a^{5} b^{8} d \,e^{2}-8736 a^{4} b^{9} c d f -4800 a^{4} b^{9} c \,e^{2}-5070 a^{4} b^{9} d^{2} e +5376 a^{3} b^{10} c^{2} f +12480 a^{3} b^{10} c d e +2197 a^{3} b^{10} d^{3}-7680 a^{2} b^{11} c^{2} e -8112 a^{2} b^{11} c \,d^{2}+9984 a \,b^{12} c^{2} d -4096 b^{13} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{19}-1029 a^{9} b^{4} f^{3}+4410 a^{8} b^{5} e \,f^{2}-5733 a^{7} b^{6} d \,f^{2}-6300 a^{7} b^{6} e^{2} f +7056 a^{6} b^{7} c \,f^{2}+16380 a^{6} b^{7} d e f +3000 a^{6} b^{7} e^{3}-20160 a^{5} b^{8} c e f -10647 a^{5} b^{8} d^{2} f -11700 a^{5} b^{8} d \,e^{2}+26208 a^{4} b^{9} c d f +14400 a^{4} b^{9} c \,e^{2}+15210 a^{4} b^{9} d^{2} e -16128 a^{3} b^{10} c^{2} f -37440 a^{3} b^{10} c d e -6591 a^{3} b^{10} d^{3}+23040 a^{2} b^{11} c^{2} e +24336 a^{2} b^{11} c \,d^{2}-29952 a \,b^{12} c^{2} d +12288 b^{13} c^{3}\right ) x +\left (7 a^{16} b f -10 a^{15} b^{2} e +13 a^{14} b^{3} d -16 a^{13} b^{4} c \right ) \textit {\_R}^{2}\right )\right )}{9}\) | \(696\) |
-1/13*c/a^2/x^13-1/10*(a*d-2*b*c)/a^3/x^10-1/7*(a^2*e-2*a*b*d+3*b^2*c)/a^4 /x^7-1/4*(a^3*f-2*a^2*b*e+3*a*b^2*d-4*b^3*c)/a^5/x^4+b*(2*a^3*f-3*a^2*b*e+ 4*a*b^2*d-5*b^3*c)/a^6/x+b^2/a^6*((1/3*f*a^3-1/3*a^2*b*e+1/3*a*b^2*d-1/3*b ^3*c)*x^2/(b*x^3+a)+(7/3*f*a^3-10/3*a^2*b*e+13/3*a*b^2*d-16/3*b^3*c)*(-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 ))))
Time = 0.28 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=-\frac {5460 \, {\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 4095 \, {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 585 \, {\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 234 \, {\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 1260 \, a^{5} c - 126 \, {\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3} + 1820 \, \sqrt {3} {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 910 \, {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 1820 \, {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{16380 \, {\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} \]
-1/16380*(5460*(16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^15 + 4095*(16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^12 - 585*(1 6*a^2*b^3*c - 13*a^3*b^2*d + 10*a^4*b*e - 7*a^5*f)*x^9 + 234*(16*a^3*b^2*c - 13*a^4*b*d + 10*a^5*e)*x^6 + 1260*a^5*c - 126*(16*a^4*b*c - 13*a^5*d)*x ^3 + 1820*sqrt(3)*((16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^ 16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/a)^( 1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 910*((16*b^5*c - 1 3*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3 ) - a*(-b/a)^(1/3)) + 1820*((16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3* b^2*f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13) *(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)))/(a^6*b*x^16 + a^7*x^13)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=-\frac {1820 \, {\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 1365 \, {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 195 \, {\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 78 \, {\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 420 \, a^{5} c - 42 \, {\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3}}{5460 \, {\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} - \frac {\sqrt {3} {\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
-1/5460*(1820*(16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^15 + 1365*(16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^12 - 195*(16 *a^2*b^3*c - 13*a^3*b^2*d + 10*a^4*b*e - 7*a^5*f)*x^9 + 78*(16*a^3*b^2*c - 13*a^4*b*d + 10*a^5*e)*x^6 + 420*a^5*c - 42*(16*a^4*b*c - 13*a^5*d)*x^3)/ (a^6*b*x^16 + a^7*x^13) - 1/9*sqrt(3)*(16*b^4*c - 13*a*b^3*d + 10*a^2*b^2* e - 7*a^3*b*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^6*(a /b)^(1/3)) - 1/18*(16*b^4*c - 13*a*b^3*d + 10*a^2*b^2*e - 7*a^3*b*f)*log(x ^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^6*(a/b)^(1/3)) + 1/9*(16*b^4*c - 13*a *b^3*d + 10*a^2*b^2*e - 7*a^3*b*f)*log(x + (a/b)^(1/3))/(a^6*(a/b)^(1/3))
Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.27 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} {\left (16 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{7}} + \frac {{\left (16 \, b^{5} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 13 \, a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 10 \, a^{2} b^{3} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, a^{3} b^{2} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{7}} - \frac {{\left (16 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{7}} - \frac {b^{5} c x^{2} - a b^{4} d x^{2} + a^{2} b^{3} e x^{2} - a^{3} b^{2} f x^{2}}{3 \, {\left (b x^{3} + a\right )} a^{6}} - \frac {9100 \, b^{4} c x^{12} - 7280 \, a b^{3} d x^{12} + 5460 \, a^{2} b^{2} e x^{12} - 3640 \, a^{3} b f x^{12} - 1820 \, a b^{3} c x^{9} + 1365 \, a^{2} b^{2} d x^{9} - 910 \, a^{3} b e x^{9} + 455 \, a^{4} f x^{9} + 780 \, a^{2} b^{2} c x^{6} - 520 \, a^{3} b d x^{6} + 260 \, a^{4} e x^{6} - 364 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{6} x^{13}} \]
1/9*sqrt(3)*(16*(-a*b^2)^(2/3)*b^3*c - 13*(-a*b^2)^(2/3)*a*b^2*d + 10*(-a* b^2)^(2/3)*a^2*b*e - 7*(-a*b^2)^(2/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a /b)^(1/3))/(-a/b)^(1/3))/a^7 + 1/9*(16*b^5*c*(-a/b)^(1/3) - 13*a*b^4*d*(-a /b)^(1/3) + 10*a^2*b^3*e*(-a/b)^(1/3) - 7*a^3*b^2*f*(-a/b)^(1/3))*(-a/b)^( 1/3)*log(abs(x - (-a/b)^(1/3)))/a^7 - 1/18*(16*(-a*b^2)^(2/3)*b^3*c - 13*( -a*b^2)^(2/3)*a*b^2*d + 10*(-a*b^2)^(2/3)*a^2*b*e - 7*(-a*b^2)^(2/3)*a^3*f )*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^7 - 1/3*(b^5*c*x^2 - a*b^4*d* x^2 + a^2*b^3*e*x^2 - a^3*b^2*f*x^2)/((b*x^3 + a)*a^6) - 1/1820*(9100*b^4* c*x^12 - 7280*a*b^3*d*x^12 + 5460*a^2*b^2*e*x^12 - 3640*a^3*b*f*x^12 - 182 0*a*b^3*c*x^9 + 1365*a^2*b^2*d*x^9 - 910*a^3*b*e*x^9 + 455*a^4*f*x^9 + 780 *a^2*b^2*c*x^6 - 520*a^3*b*d*x^6 + 260*a^4*e*x^6 - 364*a^3*b*c*x^3 + 182*a ^4*d*x^3 + 140*a^4*c)/(a^6*x^13)
Time = 9.72 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx=\frac {b^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}}-\frac {\frac {c}{13\,a}-\frac {x^9\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{28\,a^4}+\frac {x^3\,\left (13\,a\,d-16\,b\,c\right )}{130\,a^2}+\frac {x^6\,\left (10\,e\,a^2-13\,d\,a\,b+16\,c\,b^2\right )}{70\,a^3}+\frac {b\,x^{12}\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{4\,a^5}+\frac {b^2\,x^{15}\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{3\,a^6}}{b\,x^{16}+a\,x^{13}}-\frac {b^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}}+\frac {b^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}} \]
(b^(4/3)*log(b^(1/3)*x + a^(1/3))*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^ 2*b*e))/(9*a^(19/3)) - (c/(13*a) - (x^9*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(28*a^4) + (x^3*(13*a*d - 16*b*c))/(130*a^2) + (x^6*(16*b^2* c + 10*a^2*e - 13*a*b*d))/(70*a^3) + (b*x^12*(16*b^3*c - 7*a^3*f - 13*a*b^ 2*d + 10*a^2*b*e))/(4*a^5) + (b^2*x^15*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(3*a^6))/(a*x^13 + b*x^16) - (b^(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)*(16*b^3*c - 7*a^3*f - 13*a *b^2*d + 10*a^2*b*e))/(9*a^(19/3)) + (b^(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)*(16*b^3*c - 7*a^3*f - 13*a*b^2* d + 10*a^2*b*e))/(9*a^(19/3))