Integrand size = 30, antiderivative size = 345 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{2 b^5}+\frac {(b e-3 a f) x^5}{5 b^4}+\frac {f x^8}{8 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{9 b^5 \left (a+b x^3\right )}-\frac {\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{17/3}}-\frac {\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{17/3}}+\frac {\left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{17/3}} \]
1/2*(6*a^2*f-3*a*b*e+b^2*d)*x^2/b^5+1/5*(-3*a*f+b*e)*x^5/b^4+1/8*f*x^8/b^3 +1/6*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/b^5/(b*x^3+a)^2-1/9*(-13*a^3*f+1 0*a^2*b*e-7*a*b^2*d+4*b^3*c)*x^2/b^5/(b*x^3+a)-1/27*(-77*a^3*f+44*a^2*b*e- 20*a*b^2*d+5*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/b^(17/3)+1/54*(-77*a^3*f +44*a^2*b*e-20*a*b^2*d+5*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/ a^(1/3)/b^(17/3)-1/27*(-77*a^3*f+44*a^2*b*e-20*a*b^2*d+5*b^3*c)*arctan(1/3 *(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(1/3)/b^(17/3)*3^(1/2)
Time = 0.25 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.95 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {540 b^{2/3} \left (b^2 d-3 a b e+6 a^2 f\right ) x^2+216 b^{5/3} (b e-3 a f) x^5+135 b^{8/3} f x^8+\frac {180 a b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{\left (a+b x^3\right )^2}-\frac {120 b^{2/3} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) x^2}{a+b x^3}+\frac {40 \sqrt {3} \left (-5 b^3 c+20 a b^2 d-44 a^2 b e+77 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {40 \left (-5 b^3 c+20 a b^2 d-44 a^2 b e+77 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {20 \left (5 b^3 c-20 a b^2 d+44 a^2 b e-77 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}}{1080 b^{17/3}} \]
(540*b^(2/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^2 + 216*b^(5/3)*(b*e - 3*a*f)*x ^5 + 135*b^(8/3)*f*x^8 + (180*a*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f )*x^2)/(a + b*x^3)^2 - (120*b^(2/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13 *a^3*f)*x^2)/(a + b*x^3) + (40*Sqrt[3]*(-5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(- 5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^ (1/3) + (20*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(1080*b^(17/3))
Time = 1.24 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2367, 27, 2390, 2367, 2029, 2375, 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.
| \(\displaystyle \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\int \frac {2 \left (-3 a b^5 f x^{13}-3 a b^4 (b e-a f) x^{10}-3 a b^3 \left (f a^2-b e a+b^2 d\right ) x^7-3 a b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^4+a^2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x\right )}{\left (b x^3+a\right )^2}dx}{6 a b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\int \frac {-3 a b^5 f x^{13}-3 a b^4 (b e-a f) x^{10}-3 a b^3 \left (f a^2-b e a+b^2 d\right ) x^7-3 a b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^4+a^2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x}{\left (b x^3+a\right )^2}dx}{3 a b^6}\) |
| \(\Big \downarrow \) 2390 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\int \frac {x \left (-3 a b^5 f x^{12}-3 a b^4 (b e-a f) x^9-3 a b^3 \left (f a^2-b e a+b^2 d\right ) x^6-3 a b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )\right )}{\left (b x^3+a\right )^2}dx}{3 a b^6}\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a b x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {9 a^2 b^9 f x^{10}+9 a^2 b^8 (b e-2 a f) x^7+9 a^2 b^7 \left (3 f a^2-2 b e a+b^2 d\right ) x^4+a^2 b^6 \left (-23 f a^3+17 b e a^2-11 b^2 d a+5 b^3 c\right ) x}{b x^3+a}dx}{3 a b^5}}{3 a b^6}\) |
| \(\Big \downarrow \) 2029 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a b x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {x \left (9 a^2 b^9 f x^9+9 a^2 b^8 (b e-2 a f) x^6+9 a^2 b^7 \left (3 f a^2-2 b e a+b^2 d\right ) x^3+a^2 b^6 \left (-23 f a^3+17 b e a^2-11 b^2 d a+5 b^3 c\right )\right )}{b x^3+a}dx}{3 a b^5}}{3 a b^6}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a b x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\int \frac {8 x \left (9 a^2 (b e-3 a f) x^6 b^9+9 a^2 \left (3 f a^2-2 b e a+b^2 d\right ) x^3 b^8+a^2 \left (-23 f a^3+17 b e a^2-11 b^2 d a+5 b^3 c\right ) b^7\right )}{b x^3+a}dx}{8 b}+\frac {9}{8} a^2 b^8 f x^8}{3 a b^5}}{3 a b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a b x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\int \frac {x \left (9 a^2 (b e-3 a f) x^6 b^9+9 a^2 \left (3 f a^2-2 b e a+b^2 d\right ) x^3 b^8+a^2 \left (-23 f a^3+17 b e a^2-11 b^2 d a+5 b^3 c\right ) b^7\right )}{b x^3+a}dx}{b}+\frac {9}{8} a^2 b^8 f x^8}{3 a b^5}}{3 a b^6}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a b x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {\int \left (9 a^2 (b e-3 a f) x^4 b^8+9 a^2 \left (6 f a^2-3 b e a+b^2 d\right ) x b^7+\frac {\left (5 a^2 c b^{10}-20 a^3 d b^9+44 a^4 e b^8-77 a^5 f b^7\right ) x}{b x^3+a}\right )dx}{b}+\frac {9}{8} a^2 b^8 f x^8}{3 a b^5}}{3 a b^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a b x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\frac {9}{8} a^2 b^8 f x^8+\frac {\frac {9}{5} a^2 b^8 x^5 (b e-3 a f)+\frac {9}{2} a^2 b^7 x^2 \left (6 a^2 f-3 a b e+b^2 d\right )-\frac {a^{5/3} b^{19/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{\sqrt {3}}+\frac {1}{6} a^{5/3} b^{19/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )-\frac {1}{3} a^{5/3} b^{19/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{b}}{3 a b^5}}{3 a b^6}\) |
(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^5*(a + b*x^3)^2) - ((a*b* (4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(3*(a + b*x^3)) - ((9*a ^2*b^8*f*x^8)/8 + ((9*a^2*b^7*(b^2*d - 3*a*b*e + 6*a^2*f)*x^2)/2 + (9*a^2* b^8*(b*e - 3*a*f)*x^5)/5 - (a^(5/3)*b^(19/3)*(5*b^3*c - 20*a*b^2*d + 44*a^ 2*b*e - 77*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/Sqrt[ 3] - (a^(5/3)*b^(19/3)*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[ a^(1/3) + b^(1/3)*x])/3 + (a^(5/3)*b^(19/3)*(5*b^3*c - 20*a*b^2*d + 44*a^2 *b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/6)/b)/(3* a*b^5))/(3*a*b^6)
3.3.89.3.1 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_.)*((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]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Int[x*PolynomialQuot ient[Pq, x, x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(\frac {f \,x^{8}}{8 b^{3}}-\frac {3 x^{5} a f}{5 b^{4}}+\frac {x^{5} e}{5 b^{3}}+\frac {3 a^{2} f \,x^{2}}{b^{5}}-\frac {3 a e \,x^{2}}{2 b^{4}}+\frac {x^{2} d}{2 b^{3}}+\frac {\left (\frac {13}{9} a^{3} b f -\frac {10}{9} a^{2} e \,b^{2}+\frac {7}{9} a \,b^{3} d -\frac {4}{9} b^{4} c \right ) x^{5}+\frac {a \left (23 f \,a^{3}-17 a^{2} b e +11 a \,b^{2} d -5 b^{3} c \right ) x^{2}}{18}}{b^{5} \left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-77 f \,a^{3}+44 a^{2} b e -20 a \,b^{2} d +5 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{27 b^{6}}\) | \(195\) |
| default | \(\frac {\frac {b^{2} f \,x^{8}}{8}+\frac {\left (-3 a f b +b^{2} e \right ) x^{5}}{5}+\frac {\left (6 a^{2} f -3 a e b +b^{2} d \right ) x^{2}}{2}}{b^{5}}-\frac {\frac {\left (-\frac {13}{9} a^{3} b f +\frac {10}{9} a^{2} e \,b^{2}-\frac {7}{9} a \,b^{3} d +\frac {4}{9} b^{4} c \right ) x^{5}-\frac {a \left (23 f \,a^{3}-17 a^{2} b e +11 a \,b^{2} d -5 b^{3} c \right ) x^{2}}{18}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {77}{9} f \,a^{3}-\frac {44}{9} a^{2} b e +\frac {20}{9} a \,b^{2} d -\frac {5}{9} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{5}}\) | \(256\) |
1/8*f*x^8/b^3-3/5/b^4*x^5*a*f+1/5/b^3*x^5*e+3/b^5*a^2*f*x^2-3/2/b^4*a*e*x^ 2+1/2/b^3*x^2*d+((13/9*a^3*b*f-10/9*a^2*e*b^2+7/9*a*b^3*d-4/9*b^4*c)*x^5+1 /18*a*(23*a^3*f-17*a^2*b*e+11*a*b^2*d-5*b^3*c)*x^2)/b^5/(b*x^3+a)^2+1/27/b ^6*sum((-77*a^3*f+44*a^2*b*e-20*a*b^2*d+5*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. 616 vs. \(2 (298) = 596\).
Time = 0.29 (sec) , antiderivative size = 1278, normalized size of antiderivative = 3.70 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/1080*(135*a*b^6*f*x^14 + 54*(4*a*b^6*e - 7*a^2*b^5*f)*x^11 + 27*(20*a*b ^6*d - 44*a^2*b^5*e + 77*a^3*b^4*f)*x^8 - 96*(5*a*b^6*c - 20*a^2*b^5*d + 4 4*a^3*b^4*e - 77*a^4*b^3*f)*x^5 - 60*(5*a^2*b^5*c - 20*a^3*b^4*d + 44*a^4* b^3*e - 77*a^5*b^2*f)*x^2 - 60*sqrt(1/3)*(5*a^3*b^4*c - 20*a^4*b^3*d + 44* a^5*b^2*e - 77*a^6*b*f + (5*a*b^6*c - 20*a^2*b^5*d + 44*a^3*b^4*e - 77*a^4 *b^3*f)*x^6 + 2*(5*a^2*b^5*c - 20*a^3*b^4*d + 44*a^4*b^3*e - 77*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)) + 20*((5*b^5*c - 20*a*b^4*d + 44*a^2*b^3*e - 77*a^3*b ^2*f)*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^ 4*c - 20*a^2*b^3*d + 44*a^3*b^2*e - 77*a^4*b*f)*x^3)*(a*b^2)^(2/3)*log(b^2 *x^2 - (a*b^2)^(1/3)*b*x + (a*b^2)^(2/3)) - 40*((5*b^5*c - 20*a*b^4*d + 44 *a^2*b^3*e - 77*a^3*b^2*f)*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 20*a^2*b^3*d + 44*a^3*b^2*e - 77*a^4*b*f)*x^3)* (a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a*b^9*x^6 + 2*a^2*b^8*x^3 + a^3*b ^7), 1/1080*(135*a*b^6*f*x^14 + 54*(4*a*b^6*e - 7*a^2*b^5*f)*x^11 + 27*(20 *a*b^6*d - 44*a^2*b^5*e + 77*a^3*b^4*f)*x^8 - 96*(5*a*b^6*c - 20*a^2*b^5*d + 44*a^3*b^4*e - 77*a^4*b^3*f)*x^5 - 60*(5*a^2*b^5*c - 20*a^3*b^4*d + 44* a^4*b^3*e - 77*a^5*b^2*f)*x^2 - 120*sqrt(1/3)*(5*a^3*b^4*c - 20*a^4*b^3*d + 44*a^5*b^2*e - 77*a^6*b*f + (5*a*b^6*c - 20*a^2*b^5*d + 44*a^3*b^4*e ...
Timed out. \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.96 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {2 \, {\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{5} + {\left (5 \, a b^{3} c - 11 \, a^{2} b^{2} d + 17 \, a^{3} b e - 23 \, a^{4} f\right )} x^{2}}{18 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {\sqrt {3} {\left (5 \, b^{3} c - 20 \, a b^{2} d + 44 \, a^{2} b e - 77 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, b^{2} f x^{8} + 8 \, {\left (b^{2} e - 3 \, a b f\right )} x^{5} + 20 \, {\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x^{2}}{40 \, b^{5}} + \frac {{\left (5 \, b^{3} c - 20 \, a b^{2} d + 44 \, a^{2} b e - 77 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (5 \, b^{3} c - 20 \, a b^{2} d + 44 \, a^{2} b e - 77 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
-1/18*(2*(4*b^4*c - 7*a*b^3*d + 10*a^2*b^2*e - 13*a^3*b*f)*x^5 + (5*a*b^3* c - 11*a^2*b^2*d + 17*a^3*b*e - 23*a^4*f)*x^2)/(b^7*x^6 + 2*a*b^6*x^3 + a^ 2*b^5) + 1/27*sqrt(3)*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*arcta n(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(1/3)) + 1/40*(5 *b^2*f*x^8 + 8*(b^2*e - 3*a*b*f)*x^5 + 20*(b^2*d - 3*a*b*e + 6*a^2*f)*x^2) /b^5 + 1/54*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*log(x^2 - x*(a/ b)^(1/3) + (a/b)^(2/3))/(b^6*(a/b)^(1/3)) - 1/27*(5*b^3*c - 20*a*b^2*d + 4 4*a^2*b*e - 77*a^3*f)*log(x + (a/b)^(1/3))/(b^6*(a/b)^(1/3))
Time = 0.28 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.11 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (5 \, b^{3} c - 20 \, a b^{2} d + 44 \, a^{2} b e - 77 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{5}} - \frac {{\left (5 \, b^{3} c - 20 \, a b^{2} d + 44 \, a^{2} b e - 77 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{5}} - \frac {{\left (5 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 44 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 77 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{5}} - \frac {8 \, b^{4} c x^{5} - 14 \, a b^{3} d x^{5} + 20 \, a^{2} b^{2} e x^{5} - 26 \, a^{3} b f x^{5} + 5 \, a b^{3} c x^{2} - 11 \, a^{2} b^{2} d x^{2} + 17 \, a^{3} b e x^{2} - 23 \, a^{4} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{5}} + \frac {5 \, b^{21} f x^{8} + 8 \, b^{21} e x^{5} - 24 \, a b^{20} f x^{5} + 20 \, b^{21} d x^{2} - 60 \, a b^{20} e x^{2} + 120 \, a^{2} b^{19} f x^{2}}{40 \, b^{24}} \]
1/27*sqrt(3)*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*arctan(1/3*sqr t(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(1/3)*b^5) - 1/54*(5*b^3 *c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b) ^(2/3))/((-a*b^2)^(1/3)*b^5) - 1/27*(5*b^3*c*(-a/b)^(1/3) - 20*a*b^2*d*(-a /b)^(1/3) + 44*a^2*b*e*(-a/b)^(1/3) - 77*a^3*f*(-a/b)^(1/3))*(-a/b)^(1/3)* log(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/18*(8*b^4*c*x^5 - 14*a*b^3*d*x^5 + 20*a^2*b^2*e*x^5 - 26*a^3*b*f*x^5 + 5*a*b^3*c*x^2 - 11*a^2*b^2*d*x^2 + 17* a^3*b*e*x^2 - 23*a^4*f*x^2)/((b*x^3 + a)^2*b^5) + 1/40*(5*b^21*f*x^8 + 8*b ^21*e*x^5 - 24*a*b^20*f*x^5 + 20*b^21*d*x^2 - 60*a*b^20*e*x^2 + 120*a^2*b^ 19*f*x^2)/b^24
Time = 9.66 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.98 \[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^5\,\left (\frac {e}{5\,b^3}-\frac {3\,a\,f}{5\,b^4}\right )+\frac {x^2\,\left (\frac {23\,f\,a^4}{18}-\frac {17\,e\,a^3\,b}{18}+\frac {11\,d\,a^2\,b^2}{18}-\frac {5\,c\,a\,b^3}{18}\right )-x^5\,\left (-\frac {13\,f\,a^3\,b}{9}+\frac {10\,e\,a^2\,b^2}{9}-\frac {7\,d\,a\,b^3}{9}+\frac {4\,c\,b^4}{9}\right )}{a^2\,b^5+2\,a\,b^6\,x^3+b^7\,x^6}-x^2\,\left (\frac {3\,a^2\,f}{2\,b^5}-\frac {d}{2\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{2\,b}\right )+\frac {f\,x^8}{8\,b^3}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-77\,f\,a^3+44\,e\,a^2\,b-20\,d\,a\,b^2+5\,c\,b^3\right )}{27\,a^{1/3}\,b^{17/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-77\,f\,a^3+44\,e\,a^2\,b-20\,d\,a\,b^2+5\,c\,b^3\right )}{27\,a^{1/3}\,b^{17/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-77\,f\,a^3+44\,e\,a^2\,b-20\,d\,a\,b^2+5\,c\,b^3\right )}{27\,a^{1/3}\,b^{17/3}} \]
x^5*(e/(5*b^3) - (3*a*f)/(5*b^4)) + (x^2*((23*a^4*f)/18 + (11*a^2*b^2*d)/1 8 - (5*a*b^3*c)/18 - (17*a^3*b*e)/18) - x^5*((4*b^4*c)/9 + (10*a^2*b^2*e)/ 9 - (7*a*b^3*d)/9 - (13*a^3*b*f)/9))/(a^2*b^5 + b^7*x^6 + 2*a*b^6*x^3) - x ^2*((3*a^2*f)/(2*b^5) - d/(2*b^3) + (3*a*(e/b^3 - (3*a*f)/b^4))/(2*b)) + ( f*x^8)/(8*b^3) - (log(b^(1/3)*x + a^(1/3))*(5*b^3*c - 77*a^3*f - 20*a*b^2* d + 44*a^2*b*e))/(27*a^(1/3)*b^(17/3)) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/ 3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(5*b^3*c - 77*a^3*f - 20*a*b^2*d + 44*a^2*b*e))/(27*a^(1/3)*b^(17/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(5*b^3*c - 77*a^3*f - 20*a*b^2*d + 44*a ^2*b*e))/(27*a^(1/3)*b^(17/3))