Integrand size = 30, antiderivative size = 341 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}} \] Output:
-1/8*c/a^3/x^8+1/5*(-a*d+3*b*c)/a^4/x^5-1/2*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^ 2-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^4/(b*x^3+a)^2-1/18*(-5*a^3*f+11*a ^2*b*e-17*a*b^2*d+23*b^3*c)*x/a^5/(b*x^3+a)+1/27*(-5*a^3*f+20*a^2*b*e-44*a *b^2*d+77*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2) /a^(17/3)/b^(1/3)-1/27*(-5*a^3*f+20*a^2*b*e-44*a*b^2*d+77*b^3*c)*ln(a^(1/3 )+b^(1/3)*x)/a^(17/3)/b^(1/3)+1/54*(-5*a^3*f+20*a^2*b*e-44*a*b^2*d+77*b^3* c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)/b^(1/3)
Time = 0.18 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {135 a^{8/3} c}{x^8}-\frac {216 a^{5/3} (-3 b c+a d)}{x^5}-\frac {540 a^{2/3} \left (6 b^2 c-3 a b d+a^2 e\right )}{x^2}+\frac {180 a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{\left (a+b x^3\right )^2}+\frac {60 a^{2/3} \left (-23 b^3 c+17 a b^2 d-11 a^2 b e+5 a^3 f\right ) x}{a+b x^3}+\frac {40 \sqrt {3} \left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {40 \left (-77 b^3 c+44 a b^2 d-20 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {20 \left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{1080 a^{17/3}} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^3),x]
Output:
((-135*a^(8/3)*c)/x^8 - (216*a^(5/3)*(-3*b*c + a*d))/x^5 - (540*a^(2/3)*(6 *b^2*c - 3*a*b*d + a^2*e))/x^2 + (180*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b* e + a^3*f)*x)/(a + b*x^3)^2 + (60*a^(2/3)*(-23*b^3*c + 17*a*b^2*d - 11*a^2 *b*e + 5*a^3*f)*x)/(a + b*x^3) + (40*Sqrt[3]*(77*b^3*c - 44*a*b^2*d + 20*a ^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + ( 40*(-77*b^3*c + 44*a*b^2*d - 20*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x ])/b^(1/3) + (20*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3 ) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(1080*a^(17/3))
Time = 1.71 (sec) , antiderivative size = 372, 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, 25, 2368, 27, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {-\frac {5 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {6 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-6 b^3 \left (\frac {b c}{a}-d\right ) x^3+6 b^3 c}{x^9 \left (b x^3+a\right )^2}dx}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {5 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {6 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-6 b^3 \left (\frac {b c}{a}-d\right ) x^3+6 b^3 c}{x^9 \left (b x^3+a\right )^2}dx}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (-\frac {b^6 \left (-5 f a^3+11 b e a^2-17 b^2 d a+23 b^3 c\right ) x^9}{a^3}+\frac {9 b^6 \left (e a^2-2 b d a+3 b^2 c\right ) x^6}{a^2}-9 b^6 \left (\frac {2 b c}{a}-d\right ) x^3+9 b^6 c\right )}{x^9 \left (b x^3+a\right )}dx}{3 a b^3}-\frac {b^3 x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {-\frac {b^6 \left (-5 f a^3+11 b e a^2-17 b^2 d a+23 b^3 c\right ) x^9}{a^3}+\frac {9 b^6 \left (e a^2-2 b d a+3 b^2 c\right ) x^6}{a^2}-9 b^6 \left (\frac {2 b c}{a}-d\right ) x^3+9 b^6 c}{x^9 \left (b x^3+a\right )}dx}{3 a b^3}-\frac {b^3 x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\frac {2 \int \left (\frac {\left (5 f a^3-20 b e a^2+44 b^2 d a-77 b^3 c\right ) b^6}{a^3 \left (b x^3+a\right )}+\frac {9 \left (e a^2-3 b d a+6 b^2 c\right ) b^6}{a^3 x^3}+\frac {9 (a d-3 b c) b^6}{a^2 x^6}+\frac {9 c b^6}{a x^9}\right )dx}{3 a b^3}-\frac {b^3 x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 \left (\frac {9 b^6 (3 b c-a d)}{5 a^2 x^5}-\frac {9 b^6 \left (a^2 e-3 a b d+6 b^2 c\right )}{2 a^3 x^2}+\frac {b^{17/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt {3} a^{11/3}}+\frac {b^{17/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{6 a^{11/3}}-\frac {b^{17/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{3 a^{11/3}}-\frac {9 b^6 c}{8 a x^8}\right )}{3 a b^3}-\frac {b^3 x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{3 a^4 \left (a+b x^3\right )}}{6 a b^3}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(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^4*(a + b*x^3)^2) + (-1/3*( b^3*(23*b^3*c - 17*a*b^2*d + 11*a^2*b*e - 5*a^3*f)*x)/(a^4*(a + b*x^3)) + (2*((-9*b^6*c)/(8*a*x^8) + (9*b^6*(3*b*c - a*d))/(5*a^2*x^5) - (9*b^6*(6*b ^2*c - 3*a*b*d + a^2*e))/(2*a^3*x^2) + (b^(17/3)*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/( Sqrt[3]*a^(11/3)) - (b^(17/3)*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3* f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(11/3)) + (b^(17/3)*(77*b^3*c - 44*a*b^2 *d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]) /(6*a^(11/3))))/(3*a*b^3))/(6*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 = 0.16 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(\frac {\frac {\left (\frac {5}{18} a^{3} b f -\frac {11}{18} a^{2} b^{2} e +\frac {17}{18} a \,b^{3} d -\frac {23}{18} b^{4} c \right ) x^{4}+\frac {a \left (4 f \,a^{3}-7 e \,a^{2} b +10 d a \,b^{2}-13 b^{3} c \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 f \,a^{3}-20 e \,a^{2} b +44 d a \,b^{2}-77 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}}{a^{5}}-\frac {c}{8 a^{3} x^{8}}-\frac {a d -3 c b}{5 a^{4} x^{5}}-\frac {a^{2} e -3 d a b +6 b^{2} c}{2 a^{5} x^{2}}\) | \(252\) |
| risch | \(\frac {\frac {b \left (5 f \,a^{3}-20 e \,a^{2} b +44 d a \,b^{2}-77 b^{3} c \right ) x^{12}}{18 a^{5}}+\frac {4 \left (5 f \,a^{3}-20 e \,a^{2} b +44 d a \,b^{2}-77 b^{3} c \right ) x^{9}}{45 a^{4}}-\frac {\left (20 a^{2} e -44 d a b +77 b^{2} c \right ) x^{6}}{40 a^{3}}-\frac {\left (4 a d -7 c b \right ) x^{3}}{20 a^{2}}-\frac {c}{8 a}}{x^{8} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{17} b \,\textit {\_Z}^{3}-125 a^{9} f^{3}+1500 a^{8} b e \,f^{2}-3300 a^{7} b^{2} d \,f^{2}-6000 a^{7} b^{2} e^{2} f +5775 a^{6} b^{3} c \,f^{2}+26400 a^{6} b^{3} d e f +8000 a^{6} b^{3} e^{3}-46200 a^{5} b^{4} c e f -29040 a^{5} b^{4} d^{2} f -52800 a^{5} b^{4} d \,e^{2}+101640 a^{4} b^{5} c d f +92400 a^{4} b^{5} c \,e^{2}+116160 a^{4} b^{5} d^{2} e -88935 a^{3} b^{6} c^{2} f -406560 a^{3} b^{6} c d e -85184 a^{3} b^{6} d^{3}+355740 a^{2} b^{7} c^{2} e +447216 a^{2} b^{7} c \,d^{2}-782628 a \,b^{8} c^{2} d +456533 b^{9} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{17} b +375 a^{9} f^{3}-4500 a^{8} b e \,f^{2}+9900 a^{7} b^{2} d \,f^{2}+18000 a^{7} b^{2} e^{2} f -17325 a^{6} b^{3} c \,f^{2}-79200 a^{6} b^{3} d e f -24000 a^{6} b^{3} e^{3}+138600 a^{5} b^{4} c e f +87120 a^{5} b^{4} d^{2} f +158400 a^{5} b^{4} d \,e^{2}-304920 a^{4} b^{5} c d f -277200 a^{4} b^{5} c \,e^{2}-348480 a^{4} b^{5} d^{2} e +266805 a^{3} b^{6} c^{2} f +1219680 a^{3} b^{6} c d e +255552 a^{3} b^{6} d^{3}-1067220 a^{2} b^{7} c^{2} e -1341648 a^{2} b^{7} c \,d^{2}+2347884 a \,b^{8} c^{2} d -1369599 b^{9} c^{3}\right ) x +\left (-25 a^{12} f^{2}+200 a^{11} b e f -440 a^{10} b^{2} d f -400 a^{10} b^{2} e^{2}+770 a^{9} b^{3} c f +1760 a^{9} b^{3} d e -3080 a^{8} b^{4} c e -1936 a^{8} b^{4} d^{2}+6776 a^{7} b^{5} c d -5929 a^{6} b^{6} c^{2}\right ) \textit {\_R} \right )\right )}{27}\) | \(713\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^9/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/a^5*(((5/18*a^3*b*f-11/18*a^2*b^2*e+17/18*a*b^3*d-23/18*b^4*c)*x^4+1/9*a *(4*a^3*f-7*a^2*b*e+10*a*b^2*d-13*b^3*c)*x)/(b*x^3+a)^2+1/9*(5*a^3*f-20*a^ 2*b*e+44*a*b^2*d-77*b^3*c)*(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))))-1/8*c/a^3/x^8-1/5*(a*d-3*b*c)/a^4/x^5- 1/2*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^2
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (294) = 588\).
Time = 0.12 (sec) , antiderivative size = 1317, normalized size of antiderivative = 3.86 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^9/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[-1/1080*(60*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^ 12 + 96*(77*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^9 + 135 *a^6*b*c + 27*(77*a^4*b^3*c - 44*a^5*b^2*d + 20*a^6*b*e)*x^6 - 54*(7*a^5*b ^2*c - 4*a^6*b*d)*x^3 + 60*sqrt(1/3)*((77*a*b^6*c - 44*a^2*b^5*d + 20*a^3* b^4*e - 5*a^4*b^3*f)*x^14 + 2*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^11 + (77*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6* b*f)*x^8)*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*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5 *a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f )*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b) ^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3 *d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d + 20*a^ 4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^7*b^3*x ^14 + 2*a^8*b^2*x^11 + a^9*b*x^8), -1/1080*(60*(77*a^2*b^5*c - 44*a^3*b^4* d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^12 + 96*(77*a^3*b^4*c - 44*a^4*b^3*d + 2 0*a^5*b^2*e - 5*a^6*b*f)*x^9 + 135*a^6*b*c + 27*(77*a^4*b^3*c - 44*a^5*b^2 *d + 20*a^6*b*e)*x^6 - 54*(7*a^5*b^2*c - 4*a^6*b*d)*x^3 + 120*sqrt(1/3)*(( 77*a*b^6*c - 44*a^2*b^5*d + 20*a^3*b^4*e - 5*a^4*b^3*f)*x^14 + 2*(77*a^...
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**9/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=-\frac {20 \, {\left (77 \, b^{4} c - 44 \, a b^{3} d + 20 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 32 \, {\left (77 \, a b^{3} c - 44 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} + 9 \, {\left (77 \, a^{2} b^{2} c - 44 \, a^{3} b d + 20 \, a^{4} e\right )} x^{6} + 45 \, a^{4} c - 18 \, {\left (7 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}}{360 \, {\left (a^{5} b^{2} x^{14} + 2 \, a^{6} b x^{11} + a^{7} x^{8}\right )}} - \frac {\sqrt {3} {\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^9/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/360*(20*(77*b^4*c - 44*a*b^3*d + 20*a^2*b^2*e - 5*a^3*b*f)*x^12 + 32*(7 7*a*b^3*c - 44*a^2*b^2*d + 20*a^3*b*e - 5*a^4*f)*x^9 + 9*(77*a^2*b^2*c - 4 4*a^3*b*d + 20*a^4*e)*x^6 + 45*a^4*c - 18*(7*a^3*b*c - 4*a^4*d)*x^3)/(a^5* b^2*x^14 + 2*a^6*b*x^11 + a^7*x^8) - 1/27*sqrt(3)*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3)) /(a^5*b*(a/b)^(2/3)) + 1/54*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f) *log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*b*(a/b)^(2/3)) - 1/27*(77*b^3 *c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*log(x + (a/b)^(1/3))/(a^5*b*(a/b)^ (2/3))
Time = 0.14 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{6}} - \frac {\sqrt {3} {\left (77 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} b} - \frac {{\left (77 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} b} - \frac {23 \, b^{4} c x^{4} - 17 \, a b^{3} d x^{4} + 11 \, a^{2} b^{2} e x^{4} - 5 \, a^{3} b f x^{4} + 26 \, a b^{3} c x - 20 \, a^{2} b^{2} d x + 14 \, a^{3} b e x - 8 \, a^{4} f x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{5}} - \frac {120 \, b^{2} c x^{6} - 60 \, a b d x^{6} + 20 \, a^{2} e x^{6} - 24 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{5} x^{8}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^9/(b*x^3+a)^3,x, algorithm="giac")
Output:
1/27*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 - 1/27*sqrt(3)*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*b^2 )^(1/3)*a*b^2*d + 20*(-a*b^2)^(1/3)*a^2*b*e - 5*(-a*b^2)^(1/3)*a^3*f)*arct an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b) - 1/54*(77*(-a*b ^2)^(1/3)*b^3*c - 44*(-a*b^2)^(1/3)*a*b^2*d + 20*(-a*b^2)^(1/3)*a^2*b*e - 5*(-a*b^2)^(1/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^6*b) - 1/18*(23*b^4*c*x^4 - 17*a*b^3*d*x^4 + 11*a^2*b^2*e*x^4 - 5*a^3*b*f*x^4 + 26*a*b^3*c*x - 20*a^2*b^2*d*x + 14*a^3*b*e*x - 8*a^4*f*x)/((b*x^3 + a)^2*a ^5) - 1/40*(120*b^2*c*x^6 - 60*a*b*d*x^6 + 20*a^2*e*x^6 - 24*a*b*c*x^3 + 8 *a^2*d*x^3 + 5*a^2*c)/(a^5*x^8)
Time = 6.36 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=-\frac {\frac {c}{8\,a}+\frac {4\,x^9\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{45\,a^4}+\frac {x^3\,\left (4\,a\,d-7\,b\,c\right )}{20\,a^2}+\frac {x^6\,\left (20\,e\,a^2-44\,d\,a\,b+77\,c\,b^2\right )}{40\,a^3}+\frac {b\,x^{12}\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{18\,a^5}}{a^2\,x^8+2\,a\,b\,x^{11}+b^2\,x^{14}}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^3),x)
Output:
(log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(7 7*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(27*a^(17/3)*b^(1/3)) - (log (b^(1/3)*x + a^(1/3))*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(27* a^(17/3)*b^(1/3)) - (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^( 1/2)*1i)/2 - 1/2)*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(27*a^(1 7/3)*b^(1/3)) - (c/(8*a) + (4*x^9*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^ 2*b*e))/(45*a^4) + (x^3*(4*a*d - 7*b*c))/(20*a^2) + (x^6*(77*b^2*c + 20*a^ 2*e - 44*a*b*d))/(40*a^3) + (b*x^12*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20* a^2*b*e))/(18*a^5))/(a^2*x^8 + b^2*x^14 + 2*a*b*x^11)
Time = 0.15 (sec) , antiderivative size = 1230, normalized size of antiderivative = 3.61 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^9/(b*x^3+a)^3,x)
Output:
( - 200*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)) )*a**5*f*x**8 + 800*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*a**4*b*e*x**8 - 400*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*x)/(a**(1/3)*sqrt(3)))*a**4*b*f*x**11 - 1760*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**8 + 1600*a**(1/ 3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2*e* x**11 - 200*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**14 + 3080*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3 )*x)/(a**(1/3)*sqrt(3)))*a**2*b**3*c*x**8 - 3520*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**11 + 800*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 **14 + 6160*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt (3)))*a*b**4*c*x**11 - 1760*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x )/(a**(1/3)*sqrt(3)))*a*b**4*d*x**14 + 3080*a**(1/3)*sqrt(3)*atan((a**(1/3 ) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**5*c*x**14 - 100*a**(1/3)*log(a**( 2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**5*f*x**8 + 400*a**(1/3)*log (a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*b*e*x**8 - 200*a**(1 /3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*b*f*x**11 - 8 80*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**8 + 800*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**...