Integrand size = 30, antiderivative size = 316 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{2 a^4 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}-\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{14/3} b^{4/3}}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3} b^{4/3}}-\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{14/3} b^{4/3}} \]
-1/5*c/a^3/x^5+1/2*(-a*d+3*b*c)/a^4/x^2+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c) *x/a^3/b/(b*x^3+a)^2+1/18*(a^3*f+5*a^2*b*e-11*a*b^2*d+17*b^3*c)*x/a^4/b/(b *x^3+a)+1/27*(a^3*f+5*a^2*b*e-20*a*b^2*d+44*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a ^(14/3)/b^(4/3)-1/54*(a^3*f+5*a^2*b*e-20*a*b^2*d+44*b^3*c)*ln(a^(2/3)-a^(1 /3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3)/b^(4/3)-1/27*(a^3*f+5*a^2*b*e-20*a*b^2 *d+44*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(14/3)/b^ (4/3)*3^(1/2)
Time = 0.24 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {54 a^{5/3} c}{x^5}-\frac {135 a^{2/3} (-3 b c+a d)}{x^2}-\frac {45 a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{b \left (a+b x^3\right )^2}+\frac {15 a^{2/3} \left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{b \left (a+b x^3\right )}-\frac {10 \sqrt {3} \left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{4/3}}+\frac {10 \left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}-\frac {5 \left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{4/3}}}{270 a^{14/3}} \]
((-54*a^(5/3)*c)/x^5 - (135*a^(2/3)*(-3*b*c + a*d))/x^2 - (45*a^(5/3)*(-(b ^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)^2) + (15*a^(2/3)*(17* b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)) - (10*Sqrt[3]*( 44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/ 3))/Sqrt[3]])/b^(4/3) + (10*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Lo g[a^(1/3) + b^(1/3)*x])/b^(4/3) - (5*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3))/(270*a^(14 /3))
Time = 0.63 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2368, 25, 1808, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}-\frac {\int -\frac {b^2 \left (\frac {5 c b^3}{a^2}-\frac {5 d b^2}{a}+5 e b+a f\right ) x^6-6 b^3 \left (\frac {b c}{a}-d\right ) x^3+6 b^3 c}{x^6 \left (b x^3+a\right )^2}dx}{6 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b^2 \left (\frac {5 c b^3}{a^2}-\frac {5 d b^2}{a}+5 e b+a f\right ) x^6-6 b^3 \left (\frac {b c}{a}-d\right ) x^3+6 b^3 c}{x^6 \left (b x^3+a\right )^2}dx}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1808 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (b^4 \left (f a^3+5 b e a^2-11 b^2 d a+17 b^3 c\right ) x^6-9 a b^5 (2 b c-a d) x^3+9 a^2 b^5 c\right )}{x^6 \left (b x^3+a\right )}dx}{3 a^3 b^2}+\frac {b^2 x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {b^4 \left (f a^3+5 b e a^2-11 b^2 d a+17 b^3 c\right ) x^6-9 a b^5 (2 b c-a d) x^3+9 a^2 b^5 c}{x^6 \left (b x^3+a\right )}dx}{3 a^3 b^2}+\frac {b^2 x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\frac {2 \int \left (-\frac {9 (3 b c-a d) b^5}{x^3}+\frac {9 a c b^5}{x^6}+\frac {\left (f a^3+5 b e a^2-20 b^2 d a+44 b^3 c\right ) b^4}{b x^3+a}\right )dx}{3 a^3 b^2}+\frac {b^2 x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\frac {b^2 x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}+\frac {2 \left (-\frac {b^{11/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{\sqrt {3} a^{2/3}}-\frac {b^{11/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{6 a^{2/3}}+\frac {b^{11/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{3 a^{2/3}}+\frac {9 b^5 (3 b c-a d)}{2 x^2}-\frac {9 a b^5 c}{5 x^5}\right )}{3 a^3 b^2}}{6 a b^3}\) |
((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^3*b*(a + b*x^3)^2) + ((b^2*(1 7*b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(3*a^3*(a + b*x^3)) + (2*((-9 *a*b^5*c)/(5*x^5) + (9*b^5*(3*b*c - a*d))/(2*x^2) - (b^(11/3)*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^ (1/3))])/(Sqrt[3]*a^(2/3)) + (b^(11/3)*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)) - (b^(11/3)*(44*b^3*c - 20* a*b^2*d + 5*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^(2/3))))/(3*a^3*b^2))/(6*a*b^3)
3.3.98.3.1 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[(x_)^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e _.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*((d + e*x^n)^(q + 1)/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1))), x] + Simp[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)* (q + 1)) Int[x^m*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^n)) *(n*(-d)^(-(m - Mod[m, n])/n + 1)*e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d *(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]
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_)*(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]
Time = 1.56 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {c}{5 a^{3} x^{5}}-\frac {a d -3 b c}{2 a^{4} x^{2}}+\frac {\frac {\left (\frac {1}{18} f \,a^{3}+\frac {5}{18} a^{2} b e -\frac {11}{18} a \,b^{2} d +\frac {17}{18} b^{3} c \right ) x^{4}-\frac {a \left (f \,a^{3}-4 a^{2} b e +7 a \,b^{2} d -10 b^{3} c \right ) x}{9 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (f \,a^{3}+5 a^{2} b e -20 a \,b^{2} d +44 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9 b}}{a^{4}}\) | \(228\) |
| risch | \(\frac {\frac {\left (f \,a^{3}+5 a^{2} b e -20 a \,b^{2} d +44 b^{3} c \right ) x^{9}}{18 a^{4}}-\frac {\left (5 f \,a^{3}-20 a^{2} b e +80 a \,b^{2} d -176 b^{3} c \right ) x^{6}}{45 a^{3} b}-\frac {\left (5 a d -11 b c \right ) x^{3}}{10 a^{2}}-\frac {c}{5 a}}{x^{5} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{14} b^{4} \textit {\_Z}^{3}-a^{9} f^{3}-15 a^{8} b e \,f^{2}+60 a^{7} b^{2} d \,f^{2}-75 a^{7} b^{2} e^{2} f -132 a^{6} b^{3} c \,f^{2}+600 a^{6} b^{3} d e f -125 a^{6} b^{3} e^{3}-1320 a^{5} b^{4} c e f -1200 a^{5} b^{4} d^{2} f +1500 a^{5} b^{4} d \,e^{2}+5280 a^{4} b^{5} c d f -3300 a^{4} b^{5} c \,e^{2}-6000 a^{4} b^{5} d^{2} e -5808 a^{3} b^{6} c^{2} f +26400 a^{3} b^{6} c d e +8000 a^{3} b^{6} d^{3}-29040 a^{2} b^{7} c^{2} e -52800 a^{2} b^{7} c \,d^{2}+116160 a \,b^{8} c^{2} d -85184 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{14} b^{4}+3 a^{9} f^{3}+45 a^{8} b e \,f^{2}-180 a^{7} b^{2} d \,f^{2}+225 a^{7} b^{2} e^{2} f +396 a^{6} b^{3} c \,f^{2}-1800 a^{6} b^{3} d e f +375 a^{6} b^{3} e^{3}+3960 a^{5} b^{4} c e f +3600 a^{5} b^{4} d^{2} f -4500 a^{5} b^{4} d \,e^{2}-15840 a^{4} b^{5} c d f +9900 a^{4} b^{5} c \,e^{2}+18000 a^{4} b^{5} d^{2} e +17424 a^{3} b^{6} c^{2} f -79200 a^{3} b^{6} c d e -24000 a^{3} b^{6} d^{3}+87120 a^{2} b^{7} c^{2} e +158400 a^{2} b^{7} c \,d^{2}-348480 a \,b^{8} c^{2} d +255552 c^{3} b^{9}\right ) x +\left (-f^{2} a^{11} b -10 b^{2} e f \,a^{10}+40 b^{3} d f \,a^{9}-25 b^{3} e^{2} a^{9}-88 b^{4} c f \,a^{8}+200 b^{4} d e \,a^{8}-440 b^{5} c e \,a^{7}-400 b^{5} d^{2} a^{7}+1760 b^{6} c d \,a^{6}-1936 b^{7} c^{2} a^{5}\right ) \textit {\_R} \right )\right )}{27}\) | \(695\) |
-1/5*c/a^3/x^5-1/2*(a*d-3*b*c)/a^4/x^2+1/a^4*(((1/18*f*a^3+5/18*a^2*b*e-11 /18*a*b^2*d+17/18*b^3*c)*x^4-1/9*a*(a^3*f-4*a^2*b*e+7*a*b^2*d-10*b^3*c)/b* x)/(b*x^3+a)^2+1/9*(a^3*f+5*a^2*b*e-20*a*b^2*d+44*b^3*c)/b*(1/3/b/(a/b)^(2 /3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+ 1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))
Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (271) = 542\).
Time = 0.30 (sec) , antiderivative size = 1247, normalized size of antiderivative = 3.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/270*(15*(44*a^2*b^5*c - 20*a^3*b^4*d + 5*a^4*b^3*e + a^5*b^2*f)*x^9 - 5 4*a^5*b^2*c + 6*(176*a^3*b^4*c - 80*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)* x^6 + 27*(11*a^4*b^3*c - 5*a^5*b^2*d)*x^3 + 15*sqrt(1/3)*((44*a*b^6*c - 20 *a^2*b^5*d + 5*a^3*b^4*e + a^4*b^3*f)*x^11 + 2*(44*a^2*b^5*c - 20*a^3*b^4* d + 5*a^4*b^3*e + a^5*b^2*f)*x^8 + (44*a^3*b^4*c - 20*a^4*b^3*d + 5*a^5*b^ 2*e + a^6*b*f)*x^5)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3 )*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)* sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 5*((44*b^5*c - 20*a*b^4*d + 5*a^2*b ^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c - 20*a^2*b^3*d + 5*a^3*b^2*e + a^4* b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*(a^2*b)^ (2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 10*((44*b^5*c - 2 0*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c - 20*a^2*b^3*d + 5*a^3*b^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a ^5*f)*x^5)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^4*x^11 + 2*a^7 *b^3*x^8 + a^8*b^2*x^5), 1/270*(15*(44*a^2*b^5*c - 20*a^3*b^4*d + 5*a^4*b^ 3*e + a^5*b^2*f)*x^9 - 54*a^5*b^2*c + 6*(176*a^3*b^4*c - 80*a^4*b^3*d + 20 *a^5*b^2*e - 5*a^6*b*f)*x^6 + 27*(11*a^4*b^3*c - 5*a^5*b^2*d)*x^3 + 30*sqr t(1/3)*((44*a*b^6*c - 20*a^2*b^5*d + 5*a^3*b^4*e + a^4*b^3*f)*x^11 + 2*(44 *a^2*b^5*c - 20*a^3*b^4*d + 5*a^4*b^3*e + a^5*b^2*f)*x^8 + (44*a^3*b^4*c - 20*a^4*b^3*d + 5*a^5*b^2*e + a^6*b*f)*x^5)*sqrt((a^2*b)^(1/3)/b)*arcta...
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=\frac {5 \, {\left (44 \, b^{4} c - 20 \, a b^{3} d + 5 \, a^{2} b^{2} e + a^{3} b f\right )} x^{9} + 2 \, {\left (176 \, a b^{3} c - 80 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{6} - 18 \, a^{3} b c + 9 \, {\left (11 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} x^{3}}{90 \, {\left (a^{4} b^{3} x^{11} + 2 \, a^{5} b^{2} x^{8} + a^{6} b x^{5}\right )}} + \frac {\sqrt {3} {\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/90*(5*(44*b^4*c - 20*a*b^3*d + 5*a^2*b^2*e + a^3*b*f)*x^9 + 2*(176*a*b^3 *c - 80*a^2*b^2*d + 20*a^3*b*e - 5*a^4*f)*x^6 - 18*a^3*b*c + 9*(11*a^2*b^2 *c - 5*a^3*b*d)*x^3)/(a^4*b^3*x^11 + 2*a^5*b^2*x^8 + a^6*b*x^5) + 1/27*sqr t(3)*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b^2*(a/b)^(2/3)) - 1/54*(44*b^3*c - 20*a*b ^2*d + 5*a^2*b*e + a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*b^2* (a/b)^(2/3)) + 1/27*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*log(x + (a /b)^(1/3))/(a^4*b^2*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{4}} - \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{4}} - \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{5} b} + \frac {17 \, b^{4} c x^{4} - 11 \, a b^{3} d x^{4} + 5 \, a^{2} b^{2} e x^{4} + a^{3} b f x^{4} + 20 \, a b^{3} c x - 14 \, a^{2} b^{2} d x + 8 \, a^{3} b e x - 2 \, a^{4} f x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4} b} + \frac {15 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{4} x^{5}} \]
-1/27*sqrt(3)*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*arctan(1/3*sqrt( 3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a^4) - 1/54*(44*b^3* c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3 ))/((-a*b^2)^(2/3)*a^4) - 1/27*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f) *(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*b) + 1/18*(17*b^4*c*x^4 - 11 *a*b^3*d*x^4 + 5*a^2*b^2*e*x^4 + a^3*b*f*x^4 + 20*a*b^3*c*x - 14*a^2*b^2*d *x + 8*a^3*b*e*x - 2*a^4*f*x)/((b*x^3 + a)^2*a^4*b) + 1/10*(15*b*c*x^3 - 5 *a*d*x^3 - 2*a*c)/(a^4*x^5)
Time = 0.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{27\,a^{14/3}\,b^{4/3}}-\frac {\frac {c}{5\,a}-\frac {x^9\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{18\,a^4}+\frac {x^3\,\left (5\,a\,d-11\,b\,c\right )}{10\,a^2}-\frac {x^6\,\left (-5\,f\,a^3+20\,e\,a^2\,b-80\,d\,a\,b^2+176\,c\,b^3\right )}{45\,a^3\,b}}{a^2\,x^5+2\,a\,b\,x^8+b^2\,x^{11}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{27\,a^{14/3}\,b^{4/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{27\,a^{14/3}\,b^{4/3}} \]
(log(b^(1/3)*x + a^(1/3))*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*a^2*b*e))/(27 *a^(14/3)*b^(4/3)) - (c/(5*a) - (x^9*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*a^ 2*b*e))/(18*a^4) + (x^3*(5*a*d - 11*b*c))/(10*a^2) - (x^6*(176*b^3*c - 5*a ^3*f - 80*a*b^2*d + 20*a^2*b*e))/(45*a^3*b))/(a^2*x^5 + b^2*x^11 + 2*a*b*x ^8) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1 /2)*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*a^2*b*e))/(27*a^(14/3)*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)*(44 *b^3*c + a^3*f - 20*a*b^2*d + 5*a^2*b*e))/(27*a^(14/3)*b^(4/3))