Integrand size = 27, antiderivative size = 264 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{10/3}}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}} \]
(-2*a*f+b*e)*x/b^3+1/4*f*x^4/b^2+1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a/b^ 3/(b*x^3+a)+1/9*(7*a^3*f-4*a^2*b*e+a*b^2*d+2*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/ a^(5/3)/b^(10/3)-1/18*(7*a^3*f-4*a^2*b*e+a*b^2*d+2*b^3*c)*ln(a^(2/3)-a^(1/ 3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(10/3)-1/9*(7*a^3*f-4*a^2*b*e+a*b^2*d+ 2*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/3)/b^(10/3 )*3^(1/2)
Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {36 \sqrt [3]{b} (b e-2 a f) x+9 b^{4/3} f x^4+\frac {12 \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{a \left (a+b x^3\right )}-\frac {4 \sqrt {3} \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {4 \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {2 \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{36 b^{10/3}} \]
(36*b^(1/3)*(b*e - 2*a*f)*x + 9*b^(4/3)*f*x^4 + (12*b^(1/3)*(b^3*c - a*b^2 *d + a^2*b*e - a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (4*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/ a^(5/3) - (2*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/ 3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(36*b^(10/3))
Time = 0.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.82, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2397, 25, 1741, 27, 913, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}-\frac {\int -\frac {3 a b^2 f x^6+3 a b (b e-a f) x^3+2 b^3 c+a b^2 d-a^2 b e+a^3 f}{b x^3+a}dx}{3 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a b^2 f x^6+3 a b (b e-a f) x^3+2 b^3 c+a b^2 d-a^2 b e+a^3 f}{b x^3+a}dx}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {\frac {\int \frac {4 b \left (f a^3-b e a^2+3 b (b e-2 a f) x^3 a+b^2 d a+2 b^3 c\right )}{b x^3+a}dx}{4 b}+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {f a^3-b e a^2+3 b (b e-2 a f) x^3 a+b^2 d a+2 b^3 c}{b x^3+a}dx+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \int \frac {1}{b x^3+a}dx+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac {\left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )+3 a x (b e-2 a f)+\frac {3}{4} a b f x^4}{3 a b^3}\) |
((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a*b^3*(a + b*x^3)) + (3*a*(b*e - 2*a*f)*x + (3*a*b*f*x^4)/4 + (2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*( Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2* b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/(3*a*b^3)
3.3.66.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[c*x^(n + 1)*((d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1))) , x] + Simp[1/(e*(n*(q + 2) + 1)) Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(\frac {f \,x^{4}}{4 b^{2}}-\frac {2 x a f}{b^{3}}+\frac {e x}{b^{2}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{3 a \,b^{3} \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (7 f \,a^{3}-4 a^{2} b e +a \,b^{2} d +2 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{4} a}\) | \(123\) |
| default | \(-\frac {-\frac {1}{4} b f \,x^{4}+2 a f x -b e x}{b^{3}}+\frac {-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {\left (7 f \,a^{3}-4 a^{2} b e +a \,b^{2} d +2 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}}{b^{3}}\) | \(191\) |
1/4*f*x^4/b^2-2/b^3*x*a*f+1/b^2*e*x-1/3*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a*x/ b^3/(b*x^3+a)+1/9/b^4/a*sum((7*a^3*f-4*a^2*b*e+a*b^2*d+2*b^3*c)/_R^2*ln(x- _R),_R=RootOf(_Z^3*b+a))
Time = 0.32 (sec) , antiderivative size = 861, normalized size of antiderivative = 3.26 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\left [\frac {9 \, a^{3} b^{3} f x^{7} + 9 \, {\left (4 \, a^{3} b^{3} e - 7 \, a^{4} b^{2} f\right )} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{2} b^{4} c + a^{3} b^{3} d - 4 \, a^{4} b^{2} e + 7 \, a^{5} b f + {\left (2 \, a b^{5} c + a^{2} b^{4} d - 4 \, a^{3} b^{3} e + 7 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 2 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 7 \, a^{5} b f\right )} x}{36 \, {\left (a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}, \frac {9 \, a^{3} b^{3} f x^{7} + 9 \, {\left (4 \, a^{3} b^{3} e - 7 \, a^{4} b^{2} f\right )} x^{4} + 12 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{2} b^{4} c + a^{3} b^{3} d - 4 \, a^{4} b^{2} e + 7 \, a^{5} b f + {\left (2 \, a b^{5} c + a^{2} b^{4} d - 4 \, a^{3} b^{3} e + 7 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 2 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 7 \, a^{5} b f\right )} x}{36 \, {\left (a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}\right ] \]
[1/36*(9*a^3*b^3*f*x^7 + 9*(4*a^3*b^3*e - 7*a^4*b^2*f)*x^4 + 6*sqrt(1/3)*( 2*a^2*b^4*c + a^3*b^3*d - 4*a^4*b^2*e + 7*a^5*b*f + (2*a*b^5*c + a^2*b^4*d - 4*a^3*b^3*e + 7*a^4*b^2*f)*x^3)*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)) - 2*(2*a*b^3*c + a^2*b ^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*c + a*b^3*d - 4*a^2*b^2*e + 7*a^3*b*f) *x^3)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*( 2*a*b^3*c + a^2*b^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*c + a*b^3*d - 4*a^2*b ^2*e + 7*a^3*b*f)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 12*(a^2* b^4*c - a^3*b^3*d + 4*a^4*b^2*e - 7*a^5*b*f)*x)/(a^3*b^5*x^3 + a^4*b^4), 1 /36*(9*a^3*b^3*f*x^7 + 9*(4*a^3*b^3*e - 7*a^4*b^2*f)*x^4 + 12*sqrt(1/3)*(2 *a^2*b^4*c + a^3*b^3*d - 4*a^4*b^2*e + 7*a^5*b*f + (2*a*b^5*c + a^2*b^4*d - 4*a^3*b^3*e + 7*a^4*b^2*f)*x^3)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*( 2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 2*(2*a*b ^3*c + a^2*b^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*c + a*b^3*d - 4*a^2*b^2*e + 7*a^3*b*f)*x^3)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1 /3)*a) + 4*(2*a*b^3*c + a^2*b^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*c + a*b^3 *d - 4*a^2*b^2*e + 7*a^3*b*f)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3) ) + 12*(a^2*b^4*c - a^3*b^3*d + 4*a^4*b^2*e - 7*a^5*b*f)*x)/(a^3*b^5*x^3 + a^4*b^4)]
Time = 3.10 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=x \left (- \frac {2 a f}{b^{3}} + \frac {e}{b^{2}}\right ) + \frac {x \left (- a^{3} f + a^{2} b e - a b^{2} d + b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{10} - 343 a^{9} f^{3} + 588 a^{8} b e f^{2} - 147 a^{7} b^{2} d f^{2} - 336 a^{7} b^{2} e^{2} f - 294 a^{6} b^{3} c f^{2} + 168 a^{6} b^{3} d e f + 64 a^{6} b^{3} e^{3} + 336 a^{5} b^{4} c e f - 21 a^{5} b^{4} d^{2} f - 48 a^{5} b^{4} d e^{2} - 84 a^{4} b^{5} c d f - 96 a^{4} b^{5} c e^{2} + 12 a^{4} b^{5} d^{2} e - 84 a^{3} b^{6} c^{2} f + 48 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 48 a^{2} b^{7} c^{2} e - 6 a^{2} b^{7} c d^{2} - 12 a b^{8} c^{2} d - 8 b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t a^{2} b^{3}}{7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c} + x \right )} \right )\right )} + \frac {f x^{4}}{4 b^{2}} \]
x*(-2*a*f/b**3 + e/b**2) + x*(-a**3*f + a**2*b*e - a*b**2*d + b**3*c)/(3*a **2*b**3 + 3*a*b**4*x**3) + RootSum(729*_t**3*a**5*b**10 - 343*a**9*f**3 + 588*a**8*b*e*f**2 - 147*a**7*b**2*d*f**2 - 336*a**7*b**2*e**2*f - 294*a** 6*b**3*c*f**2 + 168*a**6*b**3*d*e*f + 64*a**6*b**3*e**3 + 336*a**5*b**4*c* e*f - 21*a**5*b**4*d**2*f - 48*a**5*b**4*d*e**2 - 84*a**4*b**5*c*d*f - 96* a**4*b**5*c*e**2 + 12*a**4*b**5*d**2*e - 84*a**3*b**6*c**2*f + 48*a**3*b** 6*c*d*e - a**3*b**6*d**3 + 48*a**2*b**7*c**2*e - 6*a**2*b**7*c*d**2 - 12*a *b**8*c**2*d - 8*b**9*c**3, Lambda(_t, _t*log(9*_t*a**2*b**3/(7*a**3*f - 4 *a**2*b*e + a*b**2*d + 2*b**3*c) + x))) + f*x**4/(4*b**2)
Time = 0.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{3 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {b f x^{4} + 4 \, {\left (b e - 2 \, a f\right )} x}{4 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x/(a*b^4*x^3 + a^2*b^3) + 1/4*(b*f *x^4 + 4*(b*e - 2*a*f)*x)/b^3 + 1/9*sqrt(3)*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^4*(a/ b)^(2/3)) - 1/18*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*log(x^2 - x*(a/ b)^(1/3) + (a/b)^(2/3))/(a*b^4*(a/b)^(2/3)) + 1/9*(2*b^3*c + a*b^2*d - 4*a ^2*b*e + 7*a^3*f)*log(x + (a/b)^(1/3))/(a*b^4*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac {b^{3} c x - a b^{2} d x + a^{2} b e x - a^{3} f x}{3 \, {\left (b x^{3} + a\right )} a b^{3}} + \frac {b^{6} f x^{4} + 4 \, b^{6} e x - 8 \, a b^{5} f x}{4 \, b^{8}} \]
-1/9*sqrt(3)*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*arctan(1/3*sqrt(3)* (2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b^2) - 1/18*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/ ((-a*b^2)^(2/3)*a*b^2) - 1/9*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*(-a /b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/3*(b^3*c*x - a*b^2*d*x + a^2*b*e*x - a^3*f*x)/((b*x^3 + a)*a*b^3) + 1/4*(b^6*f*x^4 + 4*b^6*e*x - 8*a*b^5*f*x)/b^8
Time = 9.77 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=x\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )+\frac {f\,x^4}{4\,b^2}+\frac {x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a\,\left (b^4\,x^3+a\,b^3\right )}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (7\,f\,a^3-4\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{5/3}\,b^{10/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,f\,a^3-4\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{5/3}\,b^{10/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,f\,a^3-4\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{5/3}\,b^{10/3}} \]
x*(e/b^2 - (2*a*f)/b^3) + (f*x^4)/(4*b^2) + (x*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a*(a*b^3 + b^4*x^3)) + (log(b^(1/3)*x + a^(1/3))*(2*b^3*c + 7 *a^3*f + a*b^2*d - 4*a^2*b*e))/(9*a^(5/3)*b^(10/3)) + (log(3^(1/2)*a^(1/3) *1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(2*b^3*c + 7*a^3*f + a *b^2*d - 4*a^2*b*e))/(9*a^(5/3)*b^(10/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^ (1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(2*b^3*c + 7*a^3*f + a*b^2*d - 4 *a^2*b*e))/(9*a^(5/3)*b^(10/3))