Integrand size = 28, antiderivative size = 301 \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {f x^2}{2 b^3}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}-\frac {\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{11/3}}-\frac {\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{11/3}}+\frac {\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{11/3}} \]
1/2*f*x^2/b^3+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a/b^3/(b*x^3+a)^2+1/9 *(7*a^3*f-4*a^2*b*e+a*b^2*d+2*b^3*c)*x^2/a^2/b^3/(b*x^3+a)-1/27*(-20*a^3*f +5*a^2*b*e+a*b^2*d+2*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(7/3)/b^(11/3)+1/54*(- 20*a^3*f+5*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^(7/3)/b^(11/3)-1/27*(-20*a^3*f+5*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^(7/3)/b^(11/3)*3^(1/2)
Time = 0.22 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {27 b^{2/3} f x^2+\frac {9 b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{a \left (a+b x^3\right )^2}+\frac {6 b^{2/3} \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{a^2 \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{7/3}}-\frac {2 \left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3}}+\frac {\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3}}}{54 b^{11/3}} \]
(27*b^(2/3)*f*x^2 + (9*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a *(a + b*x^3)^2) + (6*b^(2/3)*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2 )/(a^2*(a + b*x^3)) - (2*Sqrt[3]*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f )*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(7/3) - (2*(2*b^3*c + a*b ^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(7/3) + ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/ 3)*x^2])/a^(7/3))/(54*b^(11/3))
Time = 0.64 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.87, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {2367, 27, 2028, 1806, 25, 27, 959, 821, 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.
| \(\displaystyle \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}-\frac {\int -\frac {2 \left (3 a b^3 f x^7+3 a b^2 (b e-a f) x^4+b \left (f a^3-b e a^2+b^2 d a+2 b^3 c\right ) x\right )}{\left (b x^3+a\right )^2}dx}{6 a b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a b^3 f x^7+3 a b^2 (b e-a f) x^4+b \left (f a^3-b e a^2+b^2 d a+2 b^3 c\right ) x}{\left (b x^3+a\right )^2}dx}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \frac {\int \frac {x \left (3 a b^3 f x^6+3 a b^2 (b e-a f) x^3+b \left (f a^3-b e a^2+b^2 d a+2 b^3 c\right )\right )}{\left (b x^3+a\right )^2}dx}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1806 |
| \(\displaystyle \frac {\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}-\frac {\int -\frac {b^3 x \left (\frac {2 c b^3}{a}+d b^2+9 a f x^3 b+5 a e b-11 a^2 f\right )}{b x^3+a}dx}{3 b^2}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b^3 x \left (\frac {2 c b^3}{a}+d b^2+9 a f x^3 b+5 a e b-11 a^2 f\right )}{b x^3+a}dx}{3 b^2}+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} b \int \frac {x \left (\frac {2 c b^3}{a}+d b^2+9 a f x^3 b+5 a e b-11 a^2 f\right )}{b x^3+a}dx+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \int \frac {x}{b x^3+a}dx+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{3} b \left (\left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac {\frac {1}{3} b \left (\left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right ) \left (-20 a^2 f+\frac {2 b^3 c}{a}+5 a b e+b^2 d\right )+\frac {9}{2} a f x^2\right )+\frac {b x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 a \left (a+b x^3\right )}}{3 a b^4}\) |
((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a*b^3*(a + b*x^3)^2) + ((b*(2 *b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2)/(3*a*(a + b*x^3)) + (b*((9*a* f*x^2)/2 + ((2*b^3*c)/a + b^2*d + 5*a*b*e - 20*a^2*f)*(-1/3*Log[a^(1/3) + b^(1/3)*x]/(a^(1/3)*b^(2/3)) + (-((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^(1/3)*b^(1/3)))))/3)/(3*a*b^4)
3.3.93.3.1 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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + 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[(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[1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1 )) Int[x^Mod[m, n]*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^n ))*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)*x^(m - Mod[m, n])*(a + b*x^n + c* x^(2*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d*(Mod [m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IG tQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 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[(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 = 1.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {f \,x^{2}}{2 b^{3}}+\frac {\frac {b \left (7 f \,a^{3}-4 a^{2} b e +a \,b^{2} d +2 b^{3} c \right ) x^{5}}{9 a^{2}}+\frac {\left (11 f \,a^{3}-5 a^{2} b e -a \,b^{2} d +7 b^{3} c \right ) x^{2}}{18 a}}{b^{3} \left (b \,x^{3}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (20 f \,a^{3}-5 a^{2} b e -a \,b^{2} d -2 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{27 b^{4} a^{2}}\) | \(151\) |
| default | \(\frac {f \,x^{2}}{2 b^{3}}-\frac {\frac {-\frac {b \left (7 f \,a^{3}-4 a^{2} b e +a \,b^{2} d +2 b^{3} c \right ) x^{5}}{9 a^{2}}-\frac {\left (11 f \,a^{3}-5 a^{2} b e -a \,b^{2} d +7 b^{3} c \right ) x^{2}}{18 a}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (20 f \,a^{3}-5 a^{2} b e -a \,b^{2} d -2 b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a^{2}}}{b^{3}}\) | \(220\) |
1/2*f*x^2/b^3+(1/9*b*(7*a^3*f-4*a^2*b*e+a*b^2*d+2*b^3*c)/a^2*x^5+1/18*(11* a^3*f-5*a^2*b*e-a*b^2*d+7*b^3*c)/a*x^2)/b^3/(b*x^3+a)^2-1/27/b^4/a^2*sum(( 20*a^3*f-5*a^2*b*e-a*b^2*d-2*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. 556 vs. \(2 (258) = 516\).
Time = 0.29 (sec) , antiderivative size = 1158, normalized size of antiderivative = 3.85 \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
[1/54*(27*a^3*b^4*f*x^8 + 6*(2*a*b^6*c + a^2*b^5*d - 4*a^3*b^4*e + 16*a^4* b^3*f)*x^5 + 3*(7*a^2*b^5*c - a^3*b^4*d - 5*a^4*b^3*e + 20*a^5*b^2*f)*x^2 - 3*sqrt(1/3)*(2*a^3*b^4*c + a^4*b^3*d + 5*a^5*b^2*e - 20*a^6*b*f + (2*a*b ^6*c + a^2*b^5*d + 5*a^3*b^4*e - 20*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c + a^3* b^4*d + 5*a^4*b^3*e - 20*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)*s qrt(-(a*b^2)^(1/3)/a) - 3*(a*b^2)^(2/3)*x)/(b*x^3 + a)) + ((2*b^5*c + a*b^ 4*d + 5*a^2*b^3*e - 20*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b* e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3*b^2*e - 20*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)) - 2*((2*b^5* c + a*b^4*d + 5*a^2*b^3*e - 20*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3*b^2*e - 20*a^4*b*f )*x^3)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^3*b^7*x^6 + 2*a^4*b^6*x^ 3 + a^5*b^5), 1/54*(27*a^3*b^4*f*x^8 + 6*(2*a*b^6*c + a^2*b^5*d - 4*a^3*b^ 4*e + 16*a^4*b^3*f)*x^5 + 3*(7*a^2*b^5*c - a^3*b^4*d - 5*a^4*b^3*e + 20*a^ 5*b^2*f)*x^2 - 6*sqrt(1/3)*(2*a^3*b^4*c + a^4*b^3*d + 5*a^5*b^2*e - 20*a^6 *b*f + (2*a*b^6*c + a^2*b^5*d + 5*a^3*b^4*e - 20*a^4*b^3*f)*x^6 + 2*(2*a^2 *b^5*c + a^3*b^4*d + 5*a^4*b^3*e - 20*a^5*b^2*f)*x^3)*sqrt((a*b^2)^(1/3)/a )*arctan(-sqrt(1/3)*(2*b*x - (a*b^2)^(1/3))*sqrt((a*b^2)^(1/3)/a)/b) + ((2 *b^5*c + a*b^4*d + 5*a^2*b^3*e - 20*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*...
Timed out. \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.98 \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{5} + {\left (7 \, a b^{3} c - a^{2} b^{2} d - 5 \, a^{3} b e + 11 \, a^{4} f\right )} x^{2}}{18 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}} + \frac {f x^{2}}{2 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/18*(2*(2*b^4*c + a*b^3*d - 4*a^2*b^2*e + 7*a^3*b*f)*x^5 + (7*a*b^3*c - a ^2*b^2*d - 5*a^3*b*e + 11*a^4*f)*x^2)/(a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b ^3) + 1/2*f*x^2/b^3 + 1/27*sqrt(3)*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3 *f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^4*(a/b)^(1/ 3)) + 1/54*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*log(x^2 - x*(a/b)^(1 /3) + (a/b)^(2/3))/(a^2*b^4*(a/b)^(1/3)) - 1/27*(2*b^3*c + a*b^2*d + 5*a^2 *b*e - 20*a^3*f)*log(x + (a/b)^(1/3))/(a^2*b^4*(a/b)^(1/3))
Time = 0.27 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {f x^{2}}{2 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{3}} - \frac {{\left (2 \, b^{3} c + a b^{2} d + 5 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{3}} - \frac {{\left (2 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} + \frac {4 \, b^{4} c x^{5} + 2 \, a b^{3} d x^{5} - 8 \, a^{2} b^{2} e x^{5} + 14 \, a^{3} b f x^{5} + 7 \, a b^{3} c x^{2} - a^{2} b^{2} d x^{2} - 5 \, a^{3} b e x^{2} + 11 \, a^{4} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} \]
1/2*f*x^2/b^3 + 1/27*sqrt(3)*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*ar ctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(1/3)*a^2*b^ 3) - 1/54*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*log(x^2 + x*(-a/b)^(1 /3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^2*b^3) - 1/27*(2*b^3*c*(-a/b)^(1/3) + a*b^2*d*(-a/b)^(1/3) + 5*a^2*b*e*(-a/b)^(1/3) - 20*a^3*f*(-a/b)^(1/3))*( -a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^3) + 1/18*(4*b^4*c*x^5 + 2*a *b^3*d*x^5 - 8*a^2*b^2*e*x^5 + 14*a^3*b*f*x^5 + 7*a*b^3*c*x^2 - a^2*b^2*d* x^2 - 5*a^3*b*e*x^2 + 11*a^4*f*x^2)/((b*x^3 + a)^2*a^2*b^3)
Time = 9.21 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.93 \[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {x^2\,\left (11\,f\,a^3-5\,e\,a^2\,b-d\,a\,b^2+7\,c\,b^3\right )}{18\,a}+\frac {x^5\,\left (7\,f\,a^3\,b-4\,e\,a^2\,b^2+d\,a\,b^3+2\,c\,b^4\right )}{9\,a^2}}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {f\,x^2}{2\,b^3}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-20\,f\,a^3+5\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{7/3}\,b^{11/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+5\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{7/3}\,b^{11/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+5\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{7/3}\,b^{11/3}} \]
((x^2*(7*b^3*c + 11*a^3*f - a*b^2*d - 5*a^2*b*e))/(18*a) + (x^5*(2*b^4*c - 4*a^2*b^2*e + a*b^3*d + 7*a^3*b*f))/(9*a^2))/(a^2*b^3 + b^5*x^6 + 2*a*b^4 *x^3) + (f*x^2)/(2*b^3) - (log(b^(1/3)*x + a^(1/3))*(2*b^3*c - 20*a^3*f + a*b^2*d + 5*a^2*b*e))/(27*a^(7/3)*b^(11/3)) + (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 - 20*a^3*f + a*b^2*d + 5*a^2*b*e))/(27*a^(7/3)*b^(11/3)) - (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 - 20*a^3*f + a*b^2*d + 5*a^2* b*e))/(27*a^(7/3)*b^(11/3))