Integrand size = 30, antiderivative size = 301 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{7/3}}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{7/3}} \] Output:
-1/2*c/a^3/x^2-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^2/b^2/(b*x^3+a)^2-1/ 18*(7*a^3*f-a^2*b*e-5*a*b^2*d+11*b^3*c)*x/a^3/b^2/(b*x^3+a)+1/27*(-2*a^3*f -a^2*b*e-5*a*b^2*d+20*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1 /3))*3^(1/2)/a^(11/3)/b^(7/3)-1/27*(-2*a^3*f-a^2*b*e-5*a*b^2*d+20*b^3*c)*l n(a^(1/3)+b^(1/3)*x)/a^(11/3)/b^(7/3)+1/54*(-2*a^3*f-a^2*b*e-5*a*b^2*d+20* b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/3)/b^(7/3)
Time = 0.15 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {27 a^{2/3} c}{x^2}+\frac {9 a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{b^2 \left (a+b x^3\right )^2}-\frac {3 a^{2/3} \left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{b^2 \left (a+b x^3\right )}+\frac {2 \sqrt {3} \left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{7/3}}+\frac {2 \left (-20 b^3 c+5 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{7/3}}-\frac {\left (-20 b^3 c+5 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{7/3}}}{54 a^{11/3}} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^3),x]
Output:
((-27*a^(2/3)*c)/x^2 + (9*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x )/(b^2*(a + b*x^3)^2) - (3*a^(2/3)*(11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3 *f)*x)/(b^2*(a + b*x^3)) + (2*Sqrt[3]*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2* a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(7/3) + (2*(-20*b^3* c + 5*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(7/3) - ((- 20*b^3*c + 5*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(7/3))/(54*a^(11/3))
Time = 1.03 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.87, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2368, 25, 1808, 27, 955, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {6 a b^2 f x^6-b \left (\frac {5 c b^3}{a}-5 d b^2-a e b+a^2 f\right ) x^3+6 b^3 c}{x^3 \left (b x^3+a\right )^2}dx}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 a b^2 f x^6-b \left (\frac {5 c b^3}{a}-5 d b^2-a e b+a^2 f\right ) x^3+6 b^3 c}{x^3 \left (b x^3+a\right )^2}dx}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1808 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 b^3 \left (9 a b^2 c-\left (-2 f a^3-b e a^2-5 b^2 d a+11 b^3 c\right ) x^3\right )}{x^3 \left (b x^3+a\right )}dx}{3 a^2 b^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \int \frac {9 a b^2 c-\left (-2 f a^3-b e a^2-5 b^2 d a+11 b^3 c\right ) x^3}{x^3 \left (b x^3+a\right )}dx}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \int \frac {1}{b x^3+a}dx-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 b \left (-\left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )-\frac {9 b^2 c}{2 x^2}\right )}{3 a^2}-\frac {b x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^3),x]
Output:
-1/6*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a^2*b^2*(a + b*x^3)^2) + (-1 /3*(b*(11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3*f)*x)/(a^2*(a + b*x^3)) + (2 *b*((-9*b^2*c)/(2*x^2) - (20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*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^2))/(6*a*b^3)
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
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[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)* (q + 1)) Int[x^m*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^n)) *(n*(-d)^(-(m - Mod[m, n])/n + 1)*e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d *(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\frac {\frac {-\frac {\left (7 f \,a^{3}-e \,a^{2} b -5 d a \,b^{2}+11 b^{3} c \right ) x^{4}}{18 b}-\frac {a \left (2 f \,a^{3}+e \,a^{2} b -4 d a \,b^{2}+7 b^{3} c \right ) x}{9 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 f \,a^{3}+e \,a^{2} b +5 d a \,b^{2}-20 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9 b^{2}}}{a^{3}}-\frac {c}{2 a^{3} x^{2}}\) | \(216\) |
| risch | \(\frac {-\frac {\left (7 f \,a^{3}-e \,a^{2} b -5 d a \,b^{2}+20 b^{3} c \right ) x^{6}}{18 a^{3} b}-\frac {\left (2 f \,a^{3}+e \,a^{2} b -4 d a \,b^{2}+16 b^{3} c \right ) x^{3}}{9 a^{2} b^{2}}-\frac {c}{2 a}}{x^{2} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} b^{7} \textit {\_Z}^{3}-8 a^{9} f^{3}-12 a^{8} b e \,f^{2}-60 a^{7} b^{2} d \,f^{2}-6 a^{7} b^{2} e^{2} f +240 a^{6} b^{3} c \,f^{2}-60 a^{6} b^{3} d e f -a^{6} b^{3} e^{3}+240 a^{5} b^{4} c e f -150 a^{5} b^{4} d^{2} f -15 a^{5} b^{4} d \,e^{2}+1200 a^{4} b^{5} c d f +60 a^{4} b^{5} c \,e^{2}-75 a^{4} b^{5} d^{2} e -2400 a^{3} b^{6} c^{2} f +600 a^{3} b^{6} c d e -125 a^{3} b^{6} d^{3}-1200 a^{2} b^{7} c^{2} e +1500 a^{2} b^{7} c \,d^{2}-6000 a \,b^{8} c^{2} d +8000 b^{9} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b^{7}+24 a^{9} f^{3}+36 a^{8} b e \,f^{2}+180 a^{7} b^{2} d \,f^{2}+18 a^{7} b^{2} e^{2} f -720 a^{6} b^{3} c \,f^{2}+180 a^{6} b^{3} d e f +3 a^{6} b^{3} e^{3}-720 a^{5} b^{4} c e f +450 a^{5} b^{4} d^{2} f +45 a^{5} b^{4} d \,e^{2}-3600 a^{4} b^{5} c d f -180 a^{4} b^{5} c \,e^{2}+225 a^{4} b^{5} d^{2} e +7200 a^{3} b^{6} c^{2} f -1800 a^{3} b^{6} c d e +375 a^{3} b^{6} d^{3}+3600 a^{2} b^{7} c^{2} e -4500 a^{2} b^{7} c \,d^{2}+18000 a \,b^{8} c^{2} d -24000 b^{9} c^{3}\right ) x +\left (-4 a^{10} f^{2} b^{2}-4 e f \,a^{9} b^{3}-20 d f \,a^{8} b^{4}-e^{2} a^{8} b^{4}+80 b^{5} c f \,a^{7}-10 b^{5} d e \,a^{7}+40 b^{6} c e \,a^{6}-25 b^{6} d^{2} a^{6}+200 b^{7} c d \,a^{5}-400 b^{8} c^{2} a^{4}\right ) \textit {\_R} \right )\right )}{27}\) | \(683\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^3/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/a^3*((-1/18*(7*a^3*f-a^2*b*e-5*a*b^2*d+11*b^3*c)/b*x^4-1/9*a*(2*a^3*f+a^ 2*b*e-4*a*b^2*d+7*b^3*c)/b^2*x)/(b*x^3+a)^2+1/9*(2*a^3*f+a^2*b*e+5*a*b^2*d -20*b^3*c)/b^2*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x ^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2) *(2/(a/b)^(1/3)*x-1))))-1/2*c/a^3/x^2
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (258) = 516\).
Time = 0.11 (sec) , antiderivative size = 1217, normalized size of antiderivative = 4.04 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^3/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[-1/54*(27*a^4*b^3*c + 3*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e + 7*a^5*b ^2*f)*x^6 + 6*(16*a^3*b^4*c - 4*a^4*b^3*d + a^5*b^2*e + 2*a^6*b*f)*x^3 + 3 *sqrt(1/3)*((20*a*b^6*c - 5*a^2*b^5*d - a^3*b^4*e - 2*a^4*b^3*f)*x^8 + 2*( 20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e - 2*a^5*b^2*f)*x^5 + (20*a^3*b^4*c - 5*a^4*b^3*d - a^5*b^2*e - 2*a^6*b*f)*x^2)*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)) - ((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^4*b*e - 2*a^ 5*f)*x^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 2*((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^4*b*e - 2*a^5*f)*x^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^5*b^5 *x^8 + 2*a^6*b^4*x^5 + a^7*b^3*x^2), -1/54*(27*a^4*b^3*c + 3*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e + 7*a^5*b^2*f)*x^6 + 6*(16*a^3*b^4*c - 4*a^4*b^ 3*d + a^5*b^2*e + 2*a^6*b*f)*x^3 + 6*sqrt(1/3)*((20*a*b^6*c - 5*a^2*b^5*d - a^3*b^4*e - 2*a^4*b^3*f)*x^8 + 2*(20*a^2*b^5*c - 5*a^3*b^4*d - a^4*b^3*e - 2*a^5*b^2*f)*x^5 + (20*a^3*b^4*c - 5*a^4*b^3*d - a^5*b^2*e - 2*a^6*b*f) *x^2)*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) - ((20*b^5*c - 5*a*b^4*d - a^2*b^3*...
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**3/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {{\left (20 \, b^{4} c - 5 \, a b^{3} d - a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{6} + 9 \, a^{2} b^{2} c + 2 \, {\left (16 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{3}}{18 \, {\left (a^{3} b^{4} x^{8} + 2 \, a^{4} b^{3} x^{5} + a^{5} b^{2} x^{2}\right )}} - \frac {\sqrt {3} {\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^3/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/18*((20*b^4*c - 5*a*b^3*d - a^2*b^2*e + 7*a^3*b*f)*x^6 + 9*a^2*b^2*c + 2*(16*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*f)*x^3)/(a^3*b^4*x^8 + 2*a^4 *b^3*x^5 + a^5*b^2*x^2) - 1/27*sqrt(3)*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2 *a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b^3*(a/b) ^(2/3)) + 1/54*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*log(x^2 - x*(a/b )^(1/3) + (a/b)^(2/3))/(a^3*b^3*(a/b)^(2/3)) - 1/27*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*log(x + (a/b)^(1/3))/(a^3*b^3*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b} + \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b} + \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} b^{2}} - \frac {20 \, b^{4} c x^{6} - 5 \, a b^{3} d x^{6} - a^{2} b^{2} e x^{6} + 7 \, a^{3} b f x^{6} + 32 \, a b^{3} c x^{3} - 8 \, a^{2} b^{2} d x^{3} + 2 \, a^{3} b e x^{3} + 4 \, a^{4} f x^{3} + 9 \, a^{2} b^{2} c}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3} b^{2}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^3/(b*x^3+a)^3,x, algorithm="giac")
Output:
1/27*sqrt(3)*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*arctan(1/3*sqrt(3) *(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a^3*b) + 1/54*(20*b^3* c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3) )/((-a*b^2)^(2/3)*a^3*b) + 1/27*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f) *(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b^2) - 1/18*(20*b^4*c*x^6 - 5*a*b^3*d*x^6 - a^2*b^2*e*x^6 + 7*a^3*b*f*x^6 + 32*a*b^3*c*x^3 - 8*a^2*b^2 *d*x^3 + 2*a^3*b*e*x^3 + 4*a^4*f*x^3 + 9*a^2*b^2*c)/((b*x^4 + a*x)^2*a^3*b ^2)
Time = 6.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (2\,f\,a^3+e\,a^2\,b+5\,d\,a\,b^2-20\,c\,b^3\right )}{27\,a^{11/3}\,b^{7/3}}-\frac {\frac {c}{2\,a}+\frac {x^3\,\left (2\,f\,a^3+e\,a^2\,b-4\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^2\,b^2}+\frac {x^6\,\left (7\,f\,a^3-e\,a^2\,b-5\,d\,a\,b^2+20\,c\,b^3\right )}{18\,a^3\,b}}{a^2\,x^2+2\,a\,b\,x^5+b^2\,x^8}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,f\,a^3+e\,a^2\,b+5\,d\,a\,b^2-20\,c\,b^3\right )}{27\,a^{11/3}\,b^{7/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,f\,a^3+e\,a^2\,b+5\,d\,a\,b^2-20\,c\,b^3\right )}{27\,a^{11/3}\,b^{7/3}} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^3),x)
Output:
(log(b^(1/3)*x + a^(1/3))*(2*a^3*f - 20*b^3*c + 5*a*b^2*d + a^2*b*e))/(27* a^(11/3)*b^(7/3)) - (c/(2*a) + (x^3*(16*b^3*c + 2*a^3*f - 4*a*b^2*d + a^2* b*e))/(9*a^2*b^2) + (x^6*(20*b^3*c + 7*a^3*f - 5*a*b^2*d - a^2*b*e))/(18*a ^3*b))/(a^2*x^2 + b^2*x^8 + 2*a*b*x^5) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/ 3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(2*a^3*f - 20*b^3*c + 5*a*b^2*d + a ^2*b*e))/(27*a^(11/3)*b^(7/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a ^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(2*a^3*f - 20*b^3*c + 5*a*b^2*d + a^2*b*e)) /(27*a^(11/3)*b^(7/3))
Time = 0.15 (sec) , antiderivative size = 1170, normalized size of antiderivative = 3.89 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^3/(b*x^3+a)^3,x)
Output:
( - 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* a**5*f*x**2 - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)* sqrt(3)))*a**4*b*e*x**2 - 8*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x )/(a**(1/3)*sqrt(3)))*a**4*b*f*x**5 - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2*d*x**2 - 4*a**(1/3)*sqrt(3)*a tan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2*e*x**5 - 4*a** (1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2 *f*x**8 + 40*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqr t(3)))*a**2*b**3*c*x**2 - 20*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)* x)/(a**(1/3)*sqrt(3)))*a**2*b**3*d*x**5 - 2*a**(1/3)*sqrt(3)*atan((a**(1/3 ) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**3*e*x**8 + 80*a**(1/3)*sqrt( 3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*c*x**5 - 10*a **(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4* d*x**8 + 40*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt (3)))*b**5*c*x**8 - 2*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/ 3)*x**2)*a**5*f*x**2 - a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2 /3)*x**2)*a**4*b*e*x**2 - 4*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*b*f*x**5 - 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3) *x + b**(2/3)*x**2)*a**3*b**2*d*x**2 - 2*a**(1/3)*log(a**(2/3) - b**(1/3)* a**(1/3)*x + b**(2/3)*x**2)*a**3*b**2*e*x**5 - 2*a**(1/3)*log(a**(2/3) ...