Integrand size = 27, antiderivative size = 292 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\frac {f x}{b^3}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a b^3 \left (a+b x^3\right )^2}+\frac {\left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{18 a^2 b^3 \left (a+b x^3\right )}-\frac {\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{10/3}}+\frac {\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{10/3}}-\frac {\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{10/3}} \]
f*x/b^3+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a/b^3/(b*x^3+a)^2+1/18*(13*a^ 3*f-7*a^2*b*e+a*b^2*d+5*b^3*c)*x/a^2/b^3/(b*x^3+a)+1/27*(-14*a^3*f+2*a^2*b *e+a*b^2*d+5*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^(10/3)-1/54*(-14*a^3*f +2*a^2*b*e+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^(8 /3)/b^(10/3)-1/27*(-14*a^3*f+2*a^2*b*e+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^(8/3)/b^(10/3)*3^(1/2)
Time = 0.21 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\frac {54 \sqrt [3]{b} f x+\frac {9 \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{a \left (a+b x^3\right )^2}+\frac {3 \sqrt [3]{b} \left (5 b^3 c+a b^2 d-7 a^2 b e+13 a^3 f\right ) x}{a^2 \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{8/3}}+\frac {2 \left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}-\frac {\left (5 b^3 c+a b^2 d+2 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}}{54 b^{10/3}} \]
(54*b^(1/3)*f*x + (9*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a*(a + b*x^3)^2) + (3*b^(1/3)*(5*b^3*c + a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(a^ 2*(a + b*x^3)) - (2*Sqrt[3]*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Arc Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) + (2*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) - ((5*b^3*c + a* b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^ 2])/a^(8/3))/(54*b^(10/3))
Time = 0.54 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.87, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2397, 25, 1739, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}-\frac {\int -\frac {6 a b^2 f x^6+6 a b (b e-a f) x^3+5 b^3 c+a b^2 d-a^2 b e+a^3 f}{\left (b x^3+a\right )^2}dx}{6 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 a b^2 f x^6+6 a b (b e-a f) x^3+5 b^3 c+a b^2 d-a^2 b e+a^3 f}{\left (b x^3+a\right )^2}dx}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1739 |
| \(\displaystyle \frac {\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}-\frac {\int -\frac {2 b^2 \left (-5 f a^3+9 b f x^3 a^2+2 b e a^2+b^2 d a+5 b^3 c\right )}{b x^3+a}dx}{3 a b^2}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {-5 f a^3+9 b f x^3 a^2+2 b e a^2+b^2 d a+5 b^3 c}{b x^3+a}dx}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \int \frac {1}{b x^3+a}dx+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )+9 a^2 f x\right )}{3 a}+\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}}{6 a b^3}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac {\frac {x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{3 a \left (a+b x^3\right )}+\frac {2 \left (9 a^2 f x+\left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )\right )}{3 a}}{6 a b^3}\) |
((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a*b^3*(a + b*x^3)^2) + (((5*b^3 *c + a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(3*a*(a + b*x^3)) + (2*(9*a^2*f*x + (5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*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))/(6*a*b^3)
3.3.94.3.1 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*d^2 - b*d*e + a*e^2))*x*((d + e*x^n)^(q + 1)/(d* e^2*n*(q + 1))), x] + Simp[1/(n*(q + 1)*d*e^2) Int[(d + e*x^n)^(q + 1)*Si mp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && N eQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]
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 = 1.56 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {f x}{b^{3}}+\frac {\frac {b \left (13 f \,a^{3}-7 a^{2} b e +a \,b^{2} d +5 b^{3} c \right ) x^{4}}{18 a^{2}}+\frac {\left (5 f \,a^{3}-2 a^{2} b e -a \,b^{2} d +4 b^{3} c \right ) x}{9 a}}{b^{3} \left (b \,x^{3}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (14 f \,a^{3}-2 a^{2} b e -a \,b^{2} d -5 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{4} a^{2}}\) | \(146\) |
| default | \(\frac {f x}{b^{3}}-\frac {\frac {-\frac {b \left (13 f \,a^{3}-7 a^{2} b e +a \,b^{2} d +5 b^{3} c \right ) x^{4}}{18 a^{2}}-\frac {\left (5 f \,a^{3}-2 a^{2} b e -a \,b^{2} d +4 b^{3} c \right ) x}{9 a}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (14 f \,a^{3}-2 a^{2} b e -a \,b^{2} d -5 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9 a^{2}}}{b^{3}}\) | \(215\) |
f*x/b^3+(1/18*b*(13*a^3*f-7*a^2*b*e+a*b^2*d+5*b^3*c)/a^2*x^4+1/9*(5*a^3*f- 2*a^2*b*e-a*b^2*d+4*b^3*c)/a*x)/b^3/(b*x^3+a)^2-1/27/b^4/a^2*sum((14*a^3*f -2*a^2*b*e-a*b^2*d-5*b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (251) = 502\).
Time = 0.28 (sec) , antiderivative size = 1184, normalized size of antiderivative = 4.05 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/54*(54*a^4*b^3*f*x^7 + 3*(5*a^2*b^5*c + a^3*b^4*d - 7*a^4*b^3*e + 49*a^ 5*b^2*f)*x^4 - 3*sqrt(1/3)*(5*a^3*b^4*c + a^4*b^3*d + 2*a^5*b^2*e - 14*a^6 *b*f + (5*a*b^6*c + a^2*b^5*d + 2*a^3*b^4*e - 14*a^4*b^3*f)*x^6 + 2*(5*a^2 *b^5*c + a^3*b^4*d + 2*a^4*b^3*e - 14*a^5*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)) - ((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5*a^2*b^3*c + a^ 3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a^3*b^2*e - 14*a^4*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) + 2*((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5*a^2 *b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a ^3*b^2*e - 14*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 6 *(4*a^3*b^4*c - a^4*b^3*d - 2*a^5*b^2*e + 14*a^6*b*f)*x)/(a^4*b^6*x^6 + 2* a^5*b^5*x^3 + a^6*b^4), 1/54*(54*a^4*b^3*f*x^7 + 3*(5*a^2*b^5*c + a^3*b^4* d - 7*a^4*b^3*e + 49*a^5*b^2*f)*x^4 + 6*sqrt(1/3)*(5*a^3*b^4*c + a^4*b^3*d + 2*a^5*b^2*e - 14*a^6*b*f + (5*a*b^6*c + a^2*b^5*d + 2*a^3*b^4*e - 14*a^ 4*b^3*f)*x^6 + 2*(5*a^2*b^5*c + a^3*b^4*d + 2*a^4*b^3*e - 14*a^5*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) - ((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f +...
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (5 \, b^{4} c + a b^{3} d - 7 \, a^{2} b^{2} e + 13 \, a^{3} b f\right )} x^{4} + 2 \, {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x}{18 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}} + \frac {f x}{b^{3}} + \frac {\sqrt {3} {\left (5 \, b^{3} c + a b^{2} d + 2 \, a^{2} b e - 14 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (5 \, b^{3} c + a b^{2} d + 2 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (5 \, b^{3} c + a b^{2} d + 2 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/18*((5*b^4*c + a*b^3*d - 7*a^2*b^2*e + 13*a^3*b*f)*x^4 + 2*(4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)/(a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3 ) + f*x/b^3 + 1/27*sqrt(3)*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*arct an(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^4*(a/b)^(2/3)) - 1/ 54*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a /b)^(2/3))/(a^2*b^4*(a/b)^(2/3)) + 1/27*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 1 4*a^3*f)*log(x + (a/b)^(1/3))/(a^2*b^4*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\frac {f x}{b^{3}} - \frac {\sqrt {3} {\left (5 \, b^{3} c + a b^{2} d + 2 \, a^{2} b e - 14 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b^{2}} - \frac {{\left (5 \, b^{3} c + a b^{2} d + 2 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b^{2}} - \frac {{\left (5 \, b^{3} c + a b^{2} d + 2 \, a^{2} b e - 14 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} + \frac {5 \, b^{4} c x^{4} + a b^{3} d x^{4} - 7 \, a^{2} b^{2} e x^{4} + 13 \, a^{3} b f x^{4} + 8 \, a b^{3} c x - 2 \, a^{2} b^{2} d x - 4 \, a^{3} b e x + 10 \, a^{4} f x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} \]
f*x/b^3 - 1/27*sqrt(3)*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*arctan(1 /3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a^2*b^2) - 1 /54*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a^2*b^2) - 1/27*(5*b^3*c + a*b^2*d + 2*a^2*b *e - 14*a^3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^3) + 1/18*(5 *b^4*c*x^4 + a*b^3*d*x^4 - 7*a^2*b^2*e*x^4 + 13*a^3*b*f*x^4 + 8*a*b^3*c*x - 2*a^2*b^2*d*x - 4*a^3*b*e*x + 10*a^4*f*x)/((b*x^3 + a)^2*a^2*b^3)
Time = 9.15 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {x\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a}+\frac {x^4\,\left (13\,f\,a^3\,b-7\,e\,a^2\,b^2+d\,a\,b^3+5\,c\,b^4\right )}{18\,a^2}}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {f\,x}{b^3}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-14\,f\,a^3+2\,e\,a^2\,b+d\,a\,b^2+5\,c\,b^3\right )}{27\,a^{8/3}\,b^{10/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+2\,e\,a^2\,b+d\,a\,b^2+5\,c\,b^3\right )}{27\,a^{8/3}\,b^{10/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+2\,e\,a^2\,b+d\,a\,b^2+5\,c\,b^3\right )}{27\,a^{8/3}\,b^{10/3}} \]
((x*(4*b^3*c + 5*a^3*f - a*b^2*d - 2*a^2*b*e))/(9*a) + (x^4*(5*b^4*c - 7*a ^2*b^2*e + a*b^3*d + 13*a^3*b*f))/(18*a^2))/(a^2*b^3 + b^5*x^6 + 2*a*b^4*x ^3) + (f*x)/b^3 + (log(b^(1/3)*x + a^(1/3))*(5*b^3*c - 14*a^3*f + a*b^2*d + 2*a^2*b*e))/(27*a^(8/3)*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)*(5*b^3*c - 14*a^3*f + a*b^2*d + 2*a^2* b*e))/(27*a^(8/3)*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)*(5*b^3*c - 14*a^3*f + a*b^2*d + 2*a^2*b*e))/(2 7*a^(8/3)*b^(10/3))