Integrand size = 30, antiderivative size = 316 \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {(b e-3 a f) x^2}{2 b^4}+\frac {f x^5}{5 b^3}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac {\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{9 a b^4 \left (a+b x^3\right )}-\frac {\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{14/3}}-\frac {\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{14/3}}+\frac {\left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{14/3}} \]
1/2*(-3*a*f+b*e)*x^2/b^4+1/5*f*x^5/b^3-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)* x^2/b^4/(b*x^3+a)^2+1/9*(-10*a^3*f+7*a^2*b*e-4*a*b^2*d+b^3*c)*x^2/a/b^4/(b *x^3+a)-1/27*(44*a^3*f-20*a^2*b*e+5*a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a ^(4/3)/b^(14/3)+1/54*(44*a^3*f-20*a^2*b*e+5*a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1 /3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)/b^(14/3)-1/27*(44*a^3*f-20*a^2*b*e+5*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^(4/3)/b^( 14/3)*3^(1/2)
Time = 0.22 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {135 b^{2/3} (b e-3 a f) x^2+54 b^{5/3} f x^5-\frac {45 b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{\left (a+b x^3\right )^2}+\frac {30 b^{2/3} \left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) x^2}{a \left (a+b x^3\right )}-\frac {10 \sqrt {3} \left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {10 \left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {5 \left (b^3 c+5 a b^2 d-20 a^2 b e+44 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}}{270 b^{14/3}} \]
(135*b^(2/3)*(b*e - 3*a*f)*x^2 + 54*b^(5/3)*f*x^5 - (45*b^(2/3)*(b^3*c - a *b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3)^2 + (30*b^(2/3)*(b^3*c - 4*a*b^ 2*d + 7*a^2*b*e - 10*a^3*f)*x^2)/(a*(a + b*x^3)) - (10*Sqrt[3]*(b^3*c + 5* a*b^2*d - 20*a^2*b*e + 44*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3 ]])/a^(4/3) - (10*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log [a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(270*b^(14/3))
Time = 0.88 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2367, 27, 2029, 2367, 25, 2028, 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.
| \(\displaystyle \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle -\frac {\int -\frac {2 \left (3 a b^4 f x^{10}+3 a b^3 (b e-a f) x^7+3 a b^2 \left (f a^2-b e a+b^2 d\right ) x^4+a b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x\right )}{\left (b x^3+a\right )^2}dx}{6 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a b^4 f x^{10}+3 a b^3 (b e-a f) x^7+3 a b^2 \left (f a^2-b e a+b^2 d\right ) x^4+a b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x}{\left (b x^3+a\right )^2}dx}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2029 |
| \(\displaystyle \frac {\int \frac {x \left (3 a b^4 f x^9+3 a b^3 (b e-a f) x^6+3 a b^2 \left (f a^2-b e a+b^2 d\right ) x^3+a b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )\right )}{\left (b x^3+a\right )^2}dx}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {\frac {b x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int -\frac {9 a^2 b^7 f x^7+9 a^2 b^6 (b e-2 a f) x^4+a b^5 \left (17 f a^3-11 b e a^2+5 b^2 d a+b^3 c\right ) x}{b x^3+a}dx}{3 a b^4}}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {9 a^2 b^7 f x^7+9 a^2 b^6 (b e-2 a f) x^4+a b^5 \left (17 f a^3-11 b e a^2+5 b^2 d a+b^3 c\right ) x}{b x^3+a}dx}{3 a b^4}+\frac {b x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \frac {\frac {\int \frac {x \left (9 a^2 f x^6 b^7+9 a^2 (b e-2 a f) x^3 b^6+a \left (17 f a^3-11 b e a^2+5 b^2 d a+b^3 c\right ) b^5\right )}{b x^3+a}dx}{3 a b^4}+\frac {b x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\frac {\int \left (9 a^2 f x^4 b^6+9 a^2 (b e-3 a f) x b^5+\frac {\left (a c b^8+5 a^2 d b^7-20 a^3 e b^6+44 a^4 f b^5\right ) x}{b x^3+a}\right )dx}{3 a b^4}+\frac {b x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \left (a+b x^3\right )}+\frac {\frac {9}{5} a^2 b^6 f x^5+\frac {9}{2} a^2 b^5 x^2 (b e-3 a f)-\frac {a^{2/3} b^{13/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{\sqrt {3}}+\frac {1}{6} a^{2/3} b^{13/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )-\frac {1}{3} a^{2/3} b^{13/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{3 a b^4}}{3 a b^5}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}\) |
-1/6*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(b^4*(a + b*x^3)^2) + ((b*( b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^2)/(3*(a + b*x^3)) + ((9*a^2*b ^5*(b*e - 3*a*f)*x^2)/2 + (9*a^2*b^6*f*x^5)/5 - (a^(2/3)*b^(13/3)*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3 ]*a^(1/3))])/Sqrt[3] - (a^(2/3)*b^(13/3)*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/3 + (a^(2/3)*b^(13/3)*(b^3*c + 5*a*b^ 2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2 ])/6)/(3*a*b^4))/(3*a*b^5)
3.3.91.3.1 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[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(Fx_.)*((d_.)*(x_)^(q_.) + (a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)* (x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r) + d*x^(q - r))^p*Fx, x] /; FreeQ[{a, b, c, d, r, s, t, q}, x] && IntegerQ[p ] && PosQ[s - r] && PosQ[t - r] && PosQ[q - r] && !(EqQ[p, 1] && EqQ[u, 1] )
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) *x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) I nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 ] && LtQ[p, -1] && IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {f \,x^{5}}{5 b^{3}}-\frac {3 x^{2} a f}{2 b^{4}}+\frac {e \,x^{2}}{2 b^{3}}+\frac {-\frac {b \left (10 f \,a^{3}-7 a^{2} b e +4 a \,b^{2} d -b^{3} c \right ) x^{5}}{9 a}+\left (-\frac {17}{18} f \,a^{3}+\frac {11}{18} a^{2} b e -\frac {5}{18} a \,b^{2} d -\frac {1}{18} b^{3} c \right ) x^{2}}{b^{4} \left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (44 f \,a^{3}-20 a^{2} b e +5 a \,b^{2} d +b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{27 b^{5} a}\) | \(166\) |
| default | \(-\frac {-\frac {b f \,x^{5}}{5}+\frac {\left (3 a f -b e \right ) x^{2}}{2}}{b^{4}}+\frac {\frac {-\frac {b \left (10 f \,a^{3}-7 a^{2} b e +4 a \,b^{2} d -b^{3} c \right ) x^{5}}{9 a}+\left (-\frac {17}{18} f \,a^{3}+\frac {11}{18} a^{2} b e -\frac {5}{18} a \,b^{2} d -\frac {1}{18} b^{3} c \right ) x^{2}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (44 f \,a^{3}-20 a^{2} b e +5 a \,b^{2} d +b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a}}{b^{4}}\) | \(233\) |
1/5*f*x^5/b^3-3/2/b^4*x^2*a*f+1/2/b^3*e*x^2+(-1/9*b*(10*a^3*f-7*a^2*b*e+4* a*b^2*d-b^3*c)/a*x^5+(-17/18*f*a^3+11/18*a^2*b*e-5/18*a*b^2*d-1/18*b^3*c)* x^2)/b^4/(b*x^3+a)^2+1/27/b^5/a*sum((44*a^3*f-20*a^2*b*e+5*a*b^2*d+b^3*c)/ _R*ln(x-_R),_R=RootOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (271) = 542\).
Time = 0.29 (sec) , antiderivative size = 1224, normalized size of antiderivative = 3.87 \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/270*(54*a^2*b^5*f*x^11 + 27*(5*a^2*b^5*e - 11*a^3*b^4*f)*x^8 + 6*(5*a*b ^6*c - 20*a^2*b^5*d + 80*a^3*b^4*e - 176*a^4*b^3*f)*x^5 - 15*(a^2*b^5*c + 5*a^3*b^4*d - 20*a^4*b^3*e + 44*a^5*b^2*f)*x^2 + 15*sqrt(1/3)*(a^3*b^4*c + 5*a^4*b^3*d - 20*a^5*b^2*e + 44*a^6*b*f + (a*b^6*c + 5*a^2*b^5*d - 20*a^3 *b^4*e + 44*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 5*a^3*b^4*d - 20*a^4*b^3*e + 4 4*a^5*b^2*f)*x^3)*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)) + 5*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f + 2 *(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2*e + 44*a^4*b*f)*x^3)*(-a*b^2)^(2/3)*l og(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 10*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f + 2*(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2*e + 44*a^4*b*f)*x^3)*( -a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^8*x^6 + 2*a^3*b^7*x^3 + a^ 4*b^6), 1/270*(54*a^2*b^5*f*x^11 + 27*(5*a^2*b^5*e - 11*a^3*b^4*f)*x^8 + 6 *(5*a*b^6*c - 20*a^2*b^5*d + 80*a^3*b^4*e - 176*a^4*b^3*f)*x^5 - 15*(a^2*b ^5*c + 5*a^3*b^4*d - 20*a^4*b^3*e + 44*a^5*b^2*f)*x^2 + 30*sqrt(1/3)*(a^3* b^4*c + 5*a^4*b^3*d - 20*a^5*b^2*e + 44*a^6*b*f + (a*b^6*c + 5*a^2*b^5*d - 20*a^3*b^4*e + 44*a^4*b^3*f)*x^6 + 2*(a^2*b^5*c + 5*a^3*b^4*d - 20*a^4*b^ 3*e + 44*a^5*b^2*f)*x^3)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*...
Timed out. \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.98 \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{5} - {\left (a b^{3} c + 5 \, a^{2} b^{2} d - 11 \, a^{3} b e + 17 \, a^{4} f\right )} x^{2}}{18 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}} + \frac {2 \, b f x^{5} + 5 \, {\left (b e - 3 \, a f\right )} x^{2}}{10 \, b^{4}} + \frac {\sqrt {3} {\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/18*(2*(b^4*c - 4*a*b^3*d + 7*a^2*b^2*e - 10*a^3*b*f)*x^5 - (a*b^3*c + 5* a^2*b^2*d - 11*a^3*b*e + 17*a^4*f)*x^2)/(a*b^6*x^6 + 2*a^2*b^5*x^3 + a^3*b ^4) + 1/10*(2*b*f*x^5 + 5*(b*e - 3*a*f)*x^2)/b^4 + 1/27*sqrt(3)*(b^3*c + 5 *a*b^2*d - 20*a^2*b*e + 44*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/( a/b)^(1/3))/(a*b^5*(a/b)^(1/3)) + 1/54*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 4 4*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^5*(a/b)^(1/3)) - 1/27 *(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*log(x + (a/b)^(1/3))/(a*b^5*( a/b)^(1/3))
Time = 0.27 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.14 \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{4}} - \frac {{\left (b^{3} c + 5 \, a b^{2} d - 20 \, a^{2} b e + 44 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{4}} - \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 44 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{4}} + \frac {2 \, b^{4} c x^{5} - 8 \, a b^{3} d x^{5} + 14 \, a^{2} b^{2} e x^{5} - 20 \, a^{3} b f x^{5} - a b^{3} c x^{2} - 5 \, a^{2} b^{2} d x^{2} + 11 \, a^{3} b e x^{2} - 17 \, a^{4} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac {2 \, b^{12} f x^{5} + 5 \, b^{12} e x^{2} - 15 \, a b^{11} f x^{2}}{10 \, b^{15}} \]
1/27*sqrt(3)*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*arctan(1/3*sqrt(3 )*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(1/3)*a*b^4) - 1/54*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/ 3))/((-a*b^2)^(1/3)*a*b^4) - 1/27*(b^3*c*(-a/b)^(1/3) + 5*a*b^2*d*(-a/b)^( 1/3) - 20*a^2*b*e*(-a/b)^(1/3) + 44*a^3*f*(-a/b)^(1/3))*(-a/b)^(1/3)*log(a bs(x - (-a/b)^(1/3)))/(a^2*b^4) + 1/18*(2*b^4*c*x^5 - 8*a*b^3*d*x^5 + 14*a ^2*b^2*e*x^5 - 20*a^3*b*f*x^5 - a*b^3*c*x^2 - 5*a^2*b^2*d*x^2 + 11*a^3*b*e *x^2 - 17*a^4*f*x^2)/((b*x^3 + a)^2*a*b^4) + 1/10*(2*b^12*f*x^5 + 5*b^12*e *x^2 - 15*a*b^11*f*x^2)/b^15
Time = 9.26 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^2\,\left (\frac {e}{2\,b^3}-\frac {3\,a\,f}{2\,b^4}\right )-\frac {x^2\,\left (\frac {17\,f\,a^3}{18}-\frac {11\,e\,a^2\,b}{18}+\frac {5\,d\,a\,b^2}{18}+\frac {c\,b^3}{18}\right )-\frac {x^5\,\left (-10\,f\,a^3\,b+7\,e\,a^2\,b^2-4\,d\,a\,b^3+c\,b^4\right )}{9\,a}}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+\frac {f\,x^5}{5\,b^3}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (44\,f\,a^3-20\,e\,a^2\,b+5\,d\,a\,b^2+c\,b^3\right )}{27\,a^{4/3}\,b^{14/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (44\,f\,a^3-20\,e\,a^2\,b+5\,d\,a\,b^2+c\,b^3\right )}{27\,a^{4/3}\,b^{14/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (44\,f\,a^3-20\,e\,a^2\,b+5\,d\,a\,b^2+c\,b^3\right )}{27\,a^{4/3}\,b^{14/3}} \]
x^2*(e/(2*b^3) - (3*a*f)/(2*b^4)) - (x^2*((b^3*c)/18 + (17*a^3*f)/18 + (5* a*b^2*d)/18 - (11*a^2*b*e)/18) - (x^5*(b^4*c + 7*a^2*b^2*e - 4*a*b^3*d - 1 0*a^3*b*f))/(9*a))/(a^2*b^4 + b^6*x^6 + 2*a*b^5*x^3) + (f*x^5)/(5*b^3) - ( log(b^(1/3)*x + a^(1/3))*(b^3*c + 44*a^3*f + 5*a*b^2*d - 20*a^2*b*e))/(27* a^(4/3)*b^(14/3)) + (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 + 44*a^3*f + 5*a*b^2*d - 20*a^2*b*e))/(27*a^(4/3) *b^(14/3)) - (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 + 44*a^3*f + 5*a*b^2*d - 20*a^2*b*e))/(27*a^(4/3)*b^(14/ 3))