Integrand size = 30, antiderivative size = 317 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=-\frac {c}{4 a^3 x^4}+\frac {3 b c-a d}{a^4 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac {\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{13/3} b^{5/3}}-\frac {\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3} b^{5/3}}+\frac {\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3} b^{5/3}} \]
-1/4*c/a^3/x^4+(-a*d+3*b*c)/a^4/x+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a ^3/b/(b*x^3+a)^2+1/9*(a^3*f+2*a^2*b*e-5*a*b^2*d+8*b^3*c)*x^2/a^4/b/(b*x^3+ a)-1/27*(a^3*f+2*a^2*b*e-14*a*b^2*d+35*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(13/ 3)/b^(5/3)+1/54*(a^3*f+2*a^2*b*e-14*a*b^2*d+35*b^3*c)*ln(a^(2/3)-a^(1/3)*b ^(1/3)*x+b^(2/3)*x^2)/a^(13/3)/b^(5/3)-1/27*(a^3*f+2*a^2*b*e-14*a*b^2*d+35 *b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(13/3)/b^(5/3) *3^(1/2)
Time = 0.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {27 a^{4/3} c}{x^4}-\frac {108 \sqrt [3]{a} (-3 b c+a d)}{x}-\frac {18 a^{4/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{b \left (a+b x^3\right )^2}+\frac {12 \sqrt [3]{a} \left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{b \left (a+b x^3\right )}-\frac {4 \sqrt {3} \left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}-\frac {4 \left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}+\frac {2 \left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}}{108 a^{13/3}} \]
((-27*a^(4/3)*c)/x^4 - (108*a^(1/3)*(-3*b*c + a*d))/x - (18*a^(4/3)*(-(b^3 *c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)^2) + (12*a^(1/3)*(8*b ^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)) - (4*Sqrt[3]*(3 5*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3 ))/Sqrt[3]])/b^(5/3) - (4*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[ a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^ 3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3))/(108*a^(13/3 ))
Time = 0.64 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2368, 27, 1808, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}-\frac {\int -\frac {2 \left (b^2 \left (\frac {2 c b^3}{a^2}-\frac {2 d b^2}{a}+2 e b+a f\right ) x^6-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c\right )}{x^5 \left (b x^3+a\right )^2}dx}{6 a b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^2 \left (\frac {2 c b^3}{a^2}-\frac {2 d b^2}{a}+2 e b+a f\right ) x^6-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c}{x^5 \left (b x^3+a\right )^2}dx}{3 a b^3}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1808 |
| \(\displaystyle \frac {\frac {\int \frac {b^4 \left (f a^3+2 b e a^2-5 b^2 d a+8 b^3 c\right ) x^6-9 a b^5 (2 b c-a d) x^3+9 a^2 b^5 c}{x^5 \left (b x^3+a\right )}dx}{3 a^3 b^2}+\frac {b^2 x^2 \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}}{3 a b^3}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\frac {\int \left (-\frac {9 (3 b c-a d) b^5}{x^2}+\frac {9 a c b^5}{x^5}+\frac {\left (f a^3+2 b e a^2-14 b^2 d a+35 b^3 c\right ) x b^4}{b x^3+a}\right )dx}{3 a^3 b^2}+\frac {b^2 x^2 \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}}{3 a b^3}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\frac {b^2 x^2 \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{3 a^3 \left (a+b x^3\right )}+\frac {-\frac {b^{10/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{3 \sqrt [3]{a}}+\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{6 \sqrt [3]{a}}+\frac {9 b^5 (3 b c-a d)}{x}-\frac {9 a b^5 c}{4 x^4}}{3 a^3 b^2}}{3 a b^3}\) |
((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^3*b*(a + b*x^3)^2) + ((b^2* (8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*x^2)/(3*a^3*(a + b*x^3)) + ((-9* a*b^5*c)/(4*x^4) + (9*b^5*(3*b*c - a*d))/x - (b^(10/3)*(35*b^3*c - 14*a*b^ 2*d + 2*a^2*b*e + a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] )/(Sqrt[3]*a^(1/3)) - (b^(10/3)*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f )*Log[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)) + (b^(10/3)*(35*b^3*c - 14*a*b^2*d + 2*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 ^(1/3)))/(3*a^3*b^2))/(3*a*b^3)
3.3.97.3.1 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 = 1.66 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {c}{4 a^{3} x^{4}}-\frac {a d -3 b c}{a^{4} x}+\frac {\frac {\left (\frac {1}{9} f \,a^{3}+\frac {2}{9} a^{2} b e -\frac {5}{9} a \,b^{2} d +\frac {8}{9} b^{3} c \right ) x^{5}-\frac {a \left (f \,a^{3}-7 a^{2} b e +13 a \,b^{2} d -19 b^{3} c \right ) x^{2}}{18 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (f \,a^{3}+2 a^{2} b e -14 a \,b^{2} d +35 b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 b}}{a^{4}}\) | \(230\) |
| risch | \(\frac {\frac {\left (f \,a^{3}+2 a^{2} b e -14 a \,b^{2} d +35 b^{3} c \right ) x^{9}}{9 a^{4}}-\frac {\left (2 f \,a^{3}-14 a^{2} b e +98 a \,b^{2} d -245 b^{3} c \right ) x^{6}}{36 a^{3} b}-\frac {\left (2 a d -5 b c \right ) x^{3}}{2 a^{2}}-\frac {c}{4 a}}{x^{4} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} b^{5} \textit {\_Z}^{3}+a^{9} f^{3}+6 a^{8} b e \,f^{2}-42 a^{7} b^{2} d \,f^{2}+12 a^{7} b^{2} e^{2} f +105 a^{6} b^{3} c \,f^{2}-168 a^{6} b^{3} d e f +8 a^{6} b^{3} e^{3}+420 a^{5} b^{4} c e f +588 a^{5} b^{4} d^{2} f -168 a^{5} b^{4} d \,e^{2}-2940 a^{4} b^{5} c d f +420 a^{4} b^{5} c \,e^{2}+1176 a^{4} b^{5} d^{2} e +3675 a^{3} b^{6} c^{2} f -5880 a^{3} b^{6} c d e -2744 a^{3} b^{6} d^{3}+7350 a^{2} b^{7} c^{2} e +20580 a^{2} b^{7} c \,d^{2}-51450 a \,b^{8} c^{2} d +42875 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{13} b^{5}-3 a^{9} f^{3}-18 a^{8} b e \,f^{2}+126 a^{7} b^{2} d \,f^{2}-36 a^{7} b^{2} e^{2} f -315 a^{6} b^{3} c \,f^{2}+504 a^{6} b^{3} d e f -24 a^{6} b^{3} e^{3}-1260 a^{5} b^{4} c e f -1764 a^{5} b^{4} d^{2} f +504 a^{5} b^{4} d \,e^{2}+8820 a^{4} b^{5} c d f -1260 a^{4} b^{5} c \,e^{2}-3528 a^{4} b^{5} d^{2} e -11025 a^{3} b^{6} c^{2} f +17640 a^{3} b^{6} c d e +8232 a^{3} b^{6} d^{3}-22050 a^{2} b^{7} c^{2} e -61740 a^{2} b^{7} c \,d^{2}+154350 a \,b^{8} c^{2} d -128625 c^{3} b^{9}\right ) x +\left (a^{12} b^{3} f +2 b^{4} e \,a^{11}-14 b^{5} d \,a^{10}+35 b^{6} c \,a^{9}\right ) \textit {\_R}^{2}\right )\right )}{27}\) | \(629\) |
-1/4*c/a^3/x^4-(a*d-3*b*c)/a^4/x+1/a^4*(((1/9*f*a^3+2/9*a^2*b*e-5/9*a*b^2* d+8/9*b^3*c)*x^5-1/18*a*(a^3*f-7*a^2*b*e+13*a*b^2*d-19*b^3*c)/b*x^2)/(b*x^ 3+a)^2+1/9*(a^3*f+2*a^2*b*e-14*a*b^2*d+35*b^3*c)/b*(-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))))
Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (274) = 548\).
Time = 0.29 (sec) , antiderivative size = 1254, normalized size of antiderivative = 3.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/108*(12*(35*a*b^6*c - 14*a^2*b^5*d + 2*a^3*b^4*e + a^4*b^3*f)*x^9 - 27* a^4*b^3*c + 3*(245*a^2*b^5*c - 98*a^3*b^4*d + 14*a^4*b^3*e - 2*a^5*b^2*f)* x^6 + 54*(5*a^3*b^4*c - 2*a^4*b^3*d)*x^3 + 6*sqrt(1/3)*((35*a*b^6*c - 14*a ^2*b^5*d + 2*a^3*b^4*e + a^4*b^3*f)*x^10 + 2*(35*a^2*b^5*c - 14*a^3*b^4*d + 2*a^4*b^3*e + a^5*b^2*f)*x^7 + (35*a^3*b^4*c - 14*a^4*b^3*d + 2*a^5*b^2* e + a^6*b*f)*x^4)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3 )*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 2*((35*b^5*c - 14*a*b^4*d + 2*a^2*b^ 3*e + a^3*b^2*f)*x^10 + 2*(35*a*b^4*c - 14*a^2*b^3*d + 2*a^3*b^2*e + a^4*b *f)*x^7 + (35*a^2*b^3*c - 14*a^3*b^2*d + 2*a^4*b*e + a^5*f)*x^4)*(-a*b^2)^ (2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 4*((35*b^5*c - 14*a*b^4*d + 2*a^2*b^3*e + a^3*b^2*f)*x^10 + 2*(35*a*b^4*c - 14*a^2*b^3*d + 2*a^3*b^2*e + a^4*b*f)*x^7 + (35*a^2*b^3*c - 14*a^3*b^2*d + 2*a^4*b*e + a^5*f)*x^4)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^5*b^5*x^10 + 2*a^ 6*b^4*x^7 + a^7*b^3*x^4), 1/108*(12*(35*a*b^6*c - 14*a^2*b^5*d + 2*a^3*b^4 *e + a^4*b^3*f)*x^9 - 27*a^4*b^3*c + 3*(245*a^2*b^5*c - 98*a^3*b^4*d + 14* a^4*b^3*e - 2*a^5*b^2*f)*x^6 + 54*(5*a^3*b^4*c - 2*a^4*b^3*d)*x^3 + 12*sqr t(1/3)*((35*a*b^6*c - 14*a^2*b^5*d + 2*a^3*b^4*e + a^4*b^3*f)*x^10 + 2*(35 *a^2*b^5*c - 14*a^3*b^4*d + 2*a^4*b^3*e + a^5*b^2*f)*x^7 + (35*a^3*b^4*c - 14*a^4*b^3*d + 2*a^5*b^2*e + a^6*b*f)*x^4)*sqrt(-(-a*b^2)^(1/3)/a)*arc...
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {4 \, {\left (35 \, b^{4} c - 14 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{9} + {\left (245 \, a b^{3} c - 98 \, a^{2} b^{2} d + 14 \, a^{3} b e - 2 \, a^{4} f\right )} x^{6} - 9 \, a^{3} b c + 18 \, {\left (5 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} x^{3}}{36 \, {\left (a^{4} b^{3} x^{10} + 2 \, a^{5} b^{2} x^{7} + a^{6} b x^{4}\right )}} + \frac {\sqrt {3} {\left (35 \, b^{3} c - 14 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (35 \, b^{3} c - 14 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (35 \, b^{3} c - 14 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/36*(4*(35*b^4*c - 14*a*b^3*d + 2*a^2*b^2*e + a^3*b*f)*x^9 + (245*a*b^3*c - 98*a^2*b^2*d + 14*a^3*b*e - 2*a^4*f)*x^6 - 9*a^3*b*c + 18*(5*a^2*b^2*c - 2*a^3*b*d)*x^3)/(a^4*b^3*x^10 + 2*a^5*b^2*x^7 + a^6*b*x^4) + 1/27*sqrt(3 )*(35*b^3*c - 14*a*b^2*d + 2*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)^(1/3)) + 1/54*(35*b^3*c - 14*a*b^2* d + 2*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)^(1/3)) - 1/27*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*log(x + (a/b) ^(1/3))/(a^4*b^2*(a/b)^(1/3))
Time = 0.28 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (35 \, b^{3} c - 14 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} b} - \frac {{\left (35 \, b^{3} c - 14 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} b} - \frac {{\left (35 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{5} b} + \frac {16 \, b^{4} c x^{5} - 10 \, a b^{3} d x^{5} + 4 \, a^{2} b^{2} e x^{5} + 2 \, a^{3} b f x^{5} + 19 \, a b^{3} c x^{2} - 13 \, a^{2} b^{2} d x^{2} + 7 \, a^{3} b e x^{2} - a^{4} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4} b} + \frac {12 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{4} x^{4}} \]
1/27*sqrt(3)*(35*b^3*c - 14*a*b^2*d + 2*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)^(1/3)*a^4*b) - 1/54*(35*b^3 *c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/ 3))/((-a*b^2)^(1/3)*a^4*b) - 1/27*(35*b^3*c*(-a/b)^(1/3) - 14*a*b^2*d*(-a/ b)^(1/3) + 2*a^2*b*e*(-a/b)^(1/3) + a^3*f*(-a/b)^(1/3))*(-a/b)^(1/3)*log(a bs(x - (-a/b)^(1/3)))/(a^5*b) + 1/18*(16*b^4*c*x^5 - 10*a*b^3*d*x^5 + 4*a^ 2*b^2*e*x^5 + 2*a^3*b*f*x^5 + 19*a*b^3*c*x^2 - 13*a^2*b^2*d*x^2 + 7*a^3*b* e*x^2 - a^4*f*x^2)/((b*x^3 + a)^2*a^4*b) + 1/4*(12*b*c*x^3 - 4*a*d*x^3 - a *c)/(a^4*x^4)
Time = 9.33 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx=-\frac {\frac {c}{4\,a}-\frac {x^9\,\left (f\,a^3+2\,e\,a^2\,b-14\,d\,a\,b^2+35\,c\,b^3\right )}{9\,a^4}+\frac {x^3\,\left (2\,a\,d-5\,b\,c\right )}{2\,a^2}-\frac {x^6\,\left (-2\,f\,a^3+14\,e\,a^2\,b-98\,d\,a\,b^2+245\,c\,b^3\right )}{36\,a^3\,b}}{a^2\,x^4+2\,a\,b\,x^7+b^2\,x^{10}}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (f\,a^3+2\,e\,a^2\,b-14\,d\,a\,b^2+35\,c\,b^3\right )}{27\,a^{13/3}\,b^{5/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+2\,e\,a^2\,b-14\,d\,a\,b^2+35\,c\,b^3\right )}{27\,a^{13/3}\,b^{5/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+2\,e\,a^2\,b-14\,d\,a\,b^2+35\,c\,b^3\right )}{27\,a^{13/3}\,b^{5/3}} \]
(log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(3 5*b^3*c + a^3*f - 14*a*b^2*d + 2*a^2*b*e))/(27*a^(13/3)*b^(5/3)) - (log(b^ (1/3)*x + a^(1/3))*(35*b^3*c + a^3*f - 14*a*b^2*d + 2*a^2*b*e))/(27*a^(13/ 3)*b^(5/3)) - (c/(4*a) - (x^9*(35*b^3*c + a^3*f - 14*a*b^2*d + 2*a^2*b*e)) /(9*a^4) + (x^3*(2*a*d - 5*b*c))/(2*a^2) - (x^6*(245*b^3*c - 2*a^3*f - 98* a*b^2*d + 14*a^2*b*e))/(36*a^3*b))/(a^2*x^4 + b^2*x^10 + 2*a*b*x^7) - (log (3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(35*b^ 3*c + a^3*f - 14*a*b^2*d + 2*a^2*b*e))/(27*a^(13/3)*b^(5/3))