Integrand size = 30, antiderivative size = 343 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=-\frac {c}{7 a^3 x^7}+\frac {3 b c-a d}{4 a^4 x^4}-\frac {6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac {\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{16/3} b^{2/3}}+\frac {\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{16/3} b^{2/3}}-\frac {\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{16/3} b^{2/3}} \] Output:
-1/7*c/a^3/x^7+1/4*(-a*d+3*b*c)/a^4/x^4-(a^2*e-3*a*b*d+6*b^2*c)/a^5/x-1/6* (-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a^4/(b*x^3+a)^2-1/9*(-2*a^3*f+5*a^2*b*e -8*a*b^2*d+11*b^3*c)*x^2/a^5/(b*x^3+a)+1/27*(-2*a^3*f+14*a^2*b*e-35*a*b^2* d+65*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(1 6/3)/b^(2/3)+1/27*(-2*a^3*f+14*a^2*b*e-35*a*b^2*d+65*b^3*c)*ln(a^(1/3)+b^( 1/3)*x)/a^(16/3)/b^(2/3)-1/54*(-2*a^3*f+14*a^2*b*e-35*a*b^2*d+65*b^3*c)*ln (a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(16/3)/b^(2/3)
Time = 0.17 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {108 a^{7/3} c}{x^7}-\frac {189 a^{4/3} (-3 b c+a d)}{x^4}-\frac {756 \sqrt [3]{a} \left (6 b^2 c-3 a b d+a^2 e\right )}{x}+\frac {126 a^{4/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{\left (a+b x^3\right )^2}+\frac {84 \sqrt [3]{a} \left (-11 b^3 c+8 a b^2 d-5 a^2 b e+2 a^3 f\right ) x^2}{a+b x^3}+\frac {28 \sqrt {3} \left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {28 \left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {14 \left (-65 b^3 c+35 a b^2 d-14 a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{756 a^{16/3}} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]
Output:
((-108*a^(7/3)*c)/x^7 - (189*a^(4/3)*(-3*b*c + a*d))/x^4 - (756*a^(1/3)*(6 *b^2*c - 3*a*b*d + a^2*e))/x + (126*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3)^2 + (84*a^(1/3)*(-11*b^3*c + 8*a*b^2*d - 5*a^2*b *e + 2*a^3*f)*x^2)/(a + b*x^3) + (28*Sqrt[3]*(65*b^3*c - 35*a*b^2*d + 14*a ^2*b*e - 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + ( 28*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x] )/b^(2/3) + (14*(-65*b^3*c + 35*a*b^2*d - 14*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^(2/3))/(756*a^(16/3))
Time = 1.65 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2368, 27, 2368, 25, 2373, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {2 \left (-\frac {2 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {3 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c\right )}{x^8 \left (b x^3+a\right )^2}dx}{6 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\frac {2 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {3 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c}{x^8 \left (b x^3+a\right )^2}dx}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\frac {b^6 \left (-2 f a^3+5 b e a^2-8 b^2 d a+11 b^3 c\right ) x^9}{a^3}+\frac {9 b^6 \left (e a^2-2 b d a+3 b^2 c\right ) x^6}{a^2}-9 b^6 \left (\frac {2 b c}{a}-d\right ) x^3+9 b^6 c}{x^8 \left (b x^3+a\right )}dx}{3 a b^3}-\frac {b^3 x^2 \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\frac {b^6 \left (-2 f a^3+5 b e a^2-8 b^2 d a+11 b^3 c\right ) x^9}{a^3}+\frac {9 b^6 \left (e a^2-2 b d a+3 b^2 c\right ) x^6}{a^2}-9 b^6 \left (\frac {2 b c}{a}-d\right ) x^3+9 b^6 c}{x^8 \left (b x^3+a\right )}dx}{3 a b^3}-\frac {b^3 x^2 \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\frac {\int \left (\frac {\left (2 f a^3-14 b e a^2+35 b^2 d a-65 b^3 c\right ) x b^6}{a^3 \left (b x^3+a\right )}+\frac {9 \left (e a^2-3 b d a+6 b^2 c\right ) b^6}{a^3 x^2}+\frac {9 (a d-3 b c) b^6}{a^2 x^5}+\frac {9 c b^6}{a x^8}\right )dx}{3 a b^3}-\frac {b^3 x^2 \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {9 b^6 (3 b c-a d)}{4 a^2 x^4}-\frac {9 b^6 \left (a^2 e-3 a b d+6 b^2 c\right )}{a^3 x}+\frac {b^{16/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{\sqrt {3} a^{10/3}}-\frac {b^{16/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{6 a^{10/3}}+\frac {b^{16/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{3 a^{10/3}}-\frac {9 b^6 c}{7 a x^7}}{3 a b^3}-\frac {b^3 x^2 \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]
Output:
-1/6*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a^4*(a + b*x^3)^2) + (-1/3 *(b^3*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*x^2)/(a^4*(a + b*x^3)) + ((-9*b^6*c)/(7*a*x^7) + (9*b^6*(3*b*c - a*d))/(4*a^2*x^4) - (9*b^6*(6*b^ 2*c - 3*a*b*d + a^2*e))/(a^3*x) + (b^(16/3)*(65*b^3*c - 35*a*b^2*d + 14*a^ 2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[ 3]*a^(10/3)) + (b^(16/3)*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Lo g[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)) - (b^(16/3)*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a ^(10/3)))/(3*a*b^3))/(3*a*b^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
Time = 0.17 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(\frac {\frac {\left (\frac {2}{9} a^{3} b f -\frac {5}{9} a^{2} b^{2} e +\frac {8}{9} a \,b^{3} d -\frac {11}{9} b^{4} c \right ) x^{5}+\frac {a \left (7 f \,a^{3}-13 e \,a^{2} b +19 d a \,b^{2}-25 b^{3} c \right ) x^{2}}{18}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {2}{9} f \,a^{3}-\frac {14}{9} e \,a^{2} b +\frac {35}{9} d a \,b^{2}-\frac {65}{9} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{5}}-\frac {c}{7 a^{3} x^{7}}-\frac {a d -3 c b}{4 a^{4} x^{4}}-\frac {a^{2} e -3 d a b +6 b^{2} c}{a^{5} x}\) | \(253\) |
| risch | \(\frac {\frac {b \left (2 f \,a^{3}-14 e \,a^{2} b +35 d a \,b^{2}-65 b^{3} c \right ) x^{12}}{9 a^{5}}+\frac {7 \left (2 f \,a^{3}-14 e \,a^{2} b +35 d a \,b^{2}-65 b^{3} c \right ) x^{9}}{36 a^{4}}-\frac {\left (14 a^{2} e -35 d a b +65 b^{2} c \right ) x^{6}}{14 a^{3}}-\frac {\left (7 a d -13 c b \right ) x^{3}}{28 a^{2}}-\frac {c}{7 a}}{x^{7} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{16} b^{2} \textit {\_Z}^{3}+8 a^{9} f^{3}-168 a^{8} b e \,f^{2}+420 a^{7} b^{2} d \,f^{2}+1176 a^{7} b^{2} e^{2} f -780 a^{6} b^{3} c \,f^{2}-5880 a^{6} b^{3} d e f -2744 a^{6} b^{3} e^{3}+10920 a^{5} b^{4} c e f +7350 a^{5} b^{4} d^{2} f +20580 a^{5} b^{4} d \,e^{2}-27300 a^{4} b^{5} c d f -38220 a^{4} b^{5} c \,e^{2}-51450 a^{4} b^{5} d^{2} e +25350 a^{3} b^{6} c^{2} f +191100 a^{3} b^{6} c d e +42875 a^{3} b^{6} d^{3}-177450 a^{2} b^{7} c^{2} e -238875 a^{2} b^{7} c \,d^{2}+443625 a \,b^{8} c^{2} d -274625 b^{9} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{16} b^{2}-24 a^{9} f^{3}+504 a^{8} b e \,f^{2}-1260 a^{7} b^{2} d \,f^{2}-3528 a^{7} b^{2} e^{2} f +2340 a^{6} b^{3} c \,f^{2}+17640 a^{6} b^{3} d e f +8232 a^{6} b^{3} e^{3}-32760 a^{5} b^{4} c e f -22050 a^{5} b^{4} d^{2} f -61740 a^{5} b^{4} d \,e^{2}+81900 a^{4} b^{5} c d f +114660 a^{4} b^{5} c \,e^{2}+154350 a^{4} b^{5} d^{2} e -76050 a^{3} b^{6} c^{2} f -573300 a^{3} b^{6} c d e -128625 a^{3} b^{6} d^{3}+532350 a^{2} b^{7} c^{2} e +716625 a^{2} b^{7} c \,d^{2}-1330875 a \,b^{8} c^{2} d +823875 b^{9} c^{3}\right ) x +\left (2 a^{14} b f -14 a^{13} b^{2} e +35 a^{12} b^{3} d -65 a^{11} b^{4} c \right ) \textit {\_R}^{2}\right )\right )}{27}\) | \(654\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/a^5*(((2/9*a^3*b*f-5/9*a^2*b^2*e+8/9*a*b^3*d-11/9*b^4*c)*x^5+1/18*a*(7*a ^3*f-13*a^2*b*e+19*a*b^2*d-25*b^3*c)*x^2)/(b*x^3+a)^2+(2/9*f*a^3-14/9*e*a^ 2*b+35/9*d*a*b^2-65/9*b^3*c)*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/( a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arc tan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))-1/7*c/a^3/x^7-1/4*(a*d-3*b*c)/a^4/x ^4-(a^2*e-3*a*b*d+6*b^2*c)/a^5/x
Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (298) = 596\).
Time = 0.14 (sec) , antiderivative size = 1340, normalized size of antiderivative = 3.91 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[-1/756*(84*(65*a*b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3*f)*x^12 + 147*(65*a^2*b^5*c - 35*a^3*b^4*d + 14*a^4*b^3*e - 2*a^5*b^2*f)*x^9 + 108 *a^5*b^2*c + 54*(65*a^3*b^4*c - 35*a^4*b^3*d + 14*a^5*b^2*e)*x^6 - 27*(13* a^4*b^3*c - 7*a^5*b^2*d)*x^3 + 42*sqrt(1/3)*((65*a*b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3*f)*x^13 + 2*(65*a^2*b^5*c - 35*a^3*b^4*d + 14*a^4 *b^3*e - 2*a^5*b^2*f)*x^10 + (65*a^3*b^4*c - 35*a^4*b^3*d + 14*a^5*b^2*e - 2*a^6*b*f)*x^7)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3) *(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 14*((65*b^5*c - 35*a*b^4*d + 14*a^2*b ^3*e - 2*a^3*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2 *a^4*b*f)*x^10 + (65*a^2*b^3*c - 35*a^3*b^2*d + 14*a^4*b*e - 2*a^5*f)*x^7) *(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 28*(( 65*b^5*c - 35*a*b^4*d + 14*a^2*b^3*e - 2*a^3*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^10 + (65*a^2*b^3*c - 35*a^3*b^ 2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3))) /(a^6*b^4*x^13 + 2*a^7*b^3*x^10 + a^8*b^2*x^7), -1/756*(84*(65*a*b^6*c - 3 5*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3*f)*x^12 + 147*(65*a^2*b^5*c - 35*a^ 3*b^4*d + 14*a^4*b^3*e - 2*a^5*b^2*f)*x^9 + 108*a^5*b^2*c + 54*(65*a^3*b^4 *c - 35*a^4*b^3*d + 14*a^5*b^2*e)*x^6 - 27*(13*a^4*b^3*c - 7*a^5*b^2*d)*x^ 3 + 84*sqrt(1/3)*((65*a*b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3...
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**8/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=-\frac {28 \, {\left (65 \, b^{4} c - 35 \, a b^{3} d + 14 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{12} + 49 \, {\left (65 \, a b^{3} c - 35 \, a^{2} b^{2} d + 14 \, a^{3} b e - 2 \, a^{4} f\right )} x^{9} + 18 \, {\left (65 \, a^{2} b^{2} c - 35 \, a^{3} b d + 14 \, a^{4} e\right )} x^{6} + 36 \, a^{4} c - 9 \, {\left (13 \, a^{3} b c - 7 \, a^{4} d\right )} x^{3}}{252 \, {\left (a^{5} b^{2} x^{13} + 2 \, a^{6} b x^{10} + a^{7} x^{7}\right )}} - \frac {\sqrt {3} {\left (65 \, b^{3} c - 35 \, a b^{2} d + 14 \, a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (65 \, b^{3} c - 35 \, a b^{2} d + 14 \, a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (65 \, b^{3} c - 35 \, a b^{2} d + 14 \, a^{2} b e - 2 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/252*(28*(65*b^4*c - 35*a*b^3*d + 14*a^2*b^2*e - 2*a^3*b*f)*x^12 + 49*(6 5*a*b^3*c - 35*a^2*b^2*d + 14*a^3*b*e - 2*a^4*f)*x^9 + 18*(65*a^2*b^2*c - 35*a^3*b*d + 14*a^4*e)*x^6 + 36*a^4*c - 9*(13*a^3*b*c - 7*a^4*d)*x^3)/(a^5 *b^2*x^13 + 2*a^6*b*x^10 + a^7*x^7) - 1/27*sqrt(3)*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3) )/(a^5*b*(a/b)^(1/3)) - 1/54*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f )*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*b*(a/b)^(1/3)) + 1/27*(65*b^ 3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*log(x + (a/b)^(1/3))/(a^5*b*(a/b) ^(1/3))
Time = 0.13 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (65 \, b^{3} c - 35 \, a b^{2} d + 14 \, a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{5}} + \frac {{\left (65 \, b^{3} c - 35 \, a b^{2} d + 14 \, a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{5}} + \frac {{\left (65 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 35 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 14 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{6}} - \frac {22 \, b^{4} c x^{5} - 16 \, a b^{3} d x^{5} + 10 \, a^{2} b^{2} e x^{5} - 4 \, a^{3} b f x^{5} + 25 \, a b^{3} c x^{2} - 19 \, a^{2} b^{2} d x^{2} + 13 \, a^{3} b e x^{2} - 7 \, a^{4} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{5}} - \frac {168 \, b^{2} c x^{6} - 84 \, a b d x^{6} + 28 \, a^{2} e x^{6} - 21 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{5} x^{7}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x, algorithm="giac")
Output:
-1/27*sqrt(3)*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*arctan(1/3*sq rt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(1/3)*a^5) + 1/54*(65*b ^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b )^(2/3))/((-a*b^2)^(1/3)*a^5) + 1/27*(65*b^3*c*(-a/b)^(1/3) - 35*a*b^2*d*( -a/b)^(1/3) + 14*a^2*b*e*(-a/b)^(1/3) - 2*a^3*f*(-a/b)^(1/3))*(-a/b)^(1/3) *log(abs(x - (-a/b)^(1/3)))/a^6 - 1/18*(22*b^4*c*x^5 - 16*a*b^3*d*x^5 + 10 *a^2*b^2*e*x^5 - 4*a^3*b*f*x^5 + 25*a*b^3*c*x^2 - 19*a^2*b^2*d*x^2 + 13*a^ 3*b*e*x^2 - 7*a^4*f*x^2)/((b*x^3 + a)^2*a^5) - 1/28*(168*b^2*c*x^6 - 84*a* b*d*x^6 + 28*a^2*e*x^6 - 21*a*b*c*x^3 + 7*a^2*d*x^3 + 4*a^2*c)/(a^5*x^7)
Time = 6.64 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-2\,f\,a^3+14\,e\,a^2\,b-35\,d\,a\,b^2+65\,c\,b^3\right )}{27\,a^{16/3}\,b^{2/3}}-\frac {\frac {c}{7\,a}+\frac {7\,x^9\,\left (-2\,f\,a^3+14\,e\,a^2\,b-35\,d\,a\,b^2+65\,c\,b^3\right )}{36\,a^4}+\frac {x^3\,\left (7\,a\,d-13\,b\,c\right )}{28\,a^2}+\frac {x^6\,\left (14\,e\,a^2-35\,d\,a\,b+65\,c\,b^2\right )}{14\,a^3}+\frac {b\,x^{12}\,\left (-2\,f\,a^3+14\,e\,a^2\,b-35\,d\,a\,b^2+65\,c\,b^3\right )}{9\,a^5}}{a^2\,x^7+2\,a\,b\,x^{10}+b^2\,x^{13}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-2\,f\,a^3+14\,e\,a^2\,b-35\,d\,a\,b^2+65\,c\,b^3\right )}{27\,a^{16/3}\,b^{2/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-2\,f\,a^3+14\,e\,a^2\,b-35\,d\,a\,b^2+65\,c\,b^3\right )}{27\,a^{16/3}\,b^{2/3}} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x)
Output:
(log(b^(1/3)*x + a^(1/3))*(65*b^3*c - 2*a^3*f - 35*a*b^2*d + 14*a^2*b*e))/ (27*a^(16/3)*b^(2/3)) - (c/(7*a) + (7*x^9*(65*b^3*c - 2*a^3*f - 35*a*b^2*d + 14*a^2*b*e))/(36*a^4) + (x^3*(7*a*d - 13*b*c))/(28*a^2) + (x^6*(65*b^2* c + 14*a^2*e - 35*a*b*d))/(14*a^3) + (b*x^12*(65*b^3*c - 2*a^3*f - 35*a*b^ 2*d + 14*a^2*b*e))/(9*a^5))/(a^2*x^7 + b^2*x^13 + 2*a*b*x^10) - (log(3^(1/ 2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(65*b^3*c - 2*a^3*f - 35*a*b^2*d + 14*a^2*b*e))/(27*a^(16/3)*b^(2/3)) + (log(3^(1/2)*a ^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(65*b^3*c - 2*a^ 3*f - 35*a*b^2*d + 14*a^2*b*e))/(27*a^(16/3)*b^(2/3))
Time = 0.15 (sec) , antiderivative size = 1211, normalized size of antiderivative = 3.53 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x)
Output:
( - 56*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**5*f*x **7 + 392*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4* b*e*x**7 - 112*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* a**4*b*f*x**10 - 980*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt (3)))*a**3*b**2*d*x**7 + 784*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*a**3*b**2*e*x**10 - 56*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x )/(a**(1/3)*sqrt(3)))*a**3*b**2*f*x**13 + 1820*sqrt(3)*atan((a**(1/3) - 2* b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**3*c*x**7 - 1960*sqrt(3)*atan((a**( 1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**3*d*x**10 + 392*sqrt(3)*a tan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**3*e*x**13 + 3640 *sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*c*x**10 - 980*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*d *x**13 + 1820*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b **5*c*x**13 - 108*b**(2/3)*a**(1/3)*a**4*c - 189*b**(2/3)*a**(1/3)*a**4*d* x**3 - 756*b**(2/3)*a**(1/3)*a**4*e*x**6 + 294*b**(2/3)*a**(1/3)*a**4*f*x* *9 + 351*b**(2/3)*a**(1/3)*a**3*b*c*x**3 + 1890*b**(2/3)*a**(1/3)*a**3*b*d *x**6 - 2058*b**(2/3)*a**(1/3)*a**3*b*e*x**9 + 168*b**(2/3)*a**(1/3)*a**3* b*f*x**12 - 3510*b**(2/3)*a**(1/3)*a**2*b**2*c*x**6 + 5145*b**(2/3)*a**(1/ 3)*a**2*b**2*d*x**9 - 1176*b**(2/3)*a**(1/3)*a**2*b**2*e*x**12 - 9555*b**( 2/3)*a**(1/3)*a*b**3*c*x**9 + 2940*b**(2/3)*a**(1/3)*a*b**3*d*x**12 - 5...