Integrand size = 30, antiderivative size = 380 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}-\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{20/3}}+\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{20/3}} \] Output:
-1/11*c/a^3/x^11+1/8*(-a*d+3*b*c)/a^4/x^8-1/5*(a^2*e-3*a*b*d+6*b^2*c)/a^5/ x^5+1/2*(-a^3*f+3*a^2*b*e-6*a*b^2*d+10*b^3*c)/a^6/x^2+1/6*b*(-a^3*f+a^2*b* e-a*b^2*d+b^3*c)*x/a^5/(b*x^3+a)^2+1/18*b*(-11*a^3*f+17*a^2*b*e-23*a*b^2*d +29*b^3*c)*x/a^6/(b*x^3+a)-1/27*b^(2/3)*(-20*a^3*f+44*a^2*b*e-77*a*b^2*d+1 19*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(20/ 3)+1/27*b^(2/3)*(-20*a^3*f+44*a^2*b*e-77*a*b^2*d+119*b^3*c)*ln(a^(1/3)+b^( 1/3)*x)/a^(20/3)-1/54*b^(2/3)*(-20*a^3*f+44*a^2*b*e-77*a*b^2*d+119*b^3*c)* ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(20/3)
Time = 0.33 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {b^{2/3} \left (-119 b^3 c+77 a b^2 d-44 a^2 b e+20 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{20/3}}+\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}+\frac {b^{2/3} \left (-119 b^3 c+77 a b^2 d-44 a^2 b e+20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{20/3}} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]
Output:
-1/11*c/(a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2* e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b ^3*c - 23*a*b^2*d + 17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) + (b^(2 /3)*(-119*b^3*c + 77*a*b^2*d - 44*a^2*b*e + 20*a^3*f)*ArcTan[(1 - (2*b^(1/ 3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c - 77*a *b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2*d - 44*a^2*b*e + 20*a^3*f)*Log[a^(2/3) - a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))
Time = 2.33 (sec) , antiderivative size = 414, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}-\frac {\int -\frac {\frac {5 b^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{12}}{a^4}-\frac {6 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {6 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-6 b^3 \left (\frac {b c}{a}-d\right ) x^3+6 b^3 c}{x^{12} \left (b x^3+a\right )^2}dx}{6 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {5 b^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{12}}{a^4}-\frac {6 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {6 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-6 b^3 \left (\frac {b c}{a}-d\right ) x^3+6 b^3 c}{x^{12} \left (b x^3+a\right )^2}dx}{6 a b^3}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {\frac {b^4 x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{3 a^5 \left (a+b x^3\right )}-\frac {\int -\frac {2 \left (\frac {b^8 \left (-11 f a^3+17 b e a^2-23 b^2 d a+29 b^3 c\right ) x^{12}}{a^4}-\frac {9 b^7 \left (-f a^3+2 b e a^2-3 b^2 d a+4 b^3 c\right ) x^9}{a^3}+\frac {9 b^7 \left (e a^2-2 b d a+3 b^2 c\right ) x^6}{a^2}-9 b^7 \left (\frac {2 b c}{a}-d\right ) x^3+9 b^7 c\right )}{x^{12} \left (b x^3+a\right )}dx}{3 a b^4}}{6 a b^3}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {\frac {b^8 \left (-11 f a^3+17 b e a^2-23 b^2 d a+29 b^3 c\right ) x^{12}}{a^4}-\frac {9 b^7 \left (-f a^3+2 b e a^2-3 b^2 d a+4 b^3 c\right ) x^9}{a^3}+\frac {9 b^7 \left (e a^2-2 b d a+3 b^2 c\right ) x^6}{a^2}-9 b^7 \left (\frac {2 b c}{a}-d\right ) x^3+9 b^7 c}{x^{12} \left (b x^3+a\right )}dx}{3 a b^4}+\frac {b^4 x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{3 a^5 \left (a+b x^3\right )}}{6 a b^3}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\frac {2 \int \left (-\frac {\left (20 f a^3-44 b e a^2+77 b^2 d a-119 b^3 c\right ) b^8}{a^4 \left (b x^3+a\right )}+\frac {9 \left (f a^3-3 b e a^2+6 b^2 d a-10 b^3 c\right ) b^7}{a^4 x^3}+\frac {9 \left (e a^2-3 b d a+6 b^2 c\right ) b^7}{a^3 x^6}+\frac {9 (a d-3 b c) b^7}{a^2 x^9}+\frac {9 c b^7}{a x^{12}}\right )dx}{3 a b^4}+\frac {b^4 x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{3 a^5 \left (a+b x^3\right )}}{6 a b^3}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac {\frac {b^4 x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {2 \left (\frac {9 b^7 (3 b c-a d)}{8 a^2 x^8}-\frac {9 b^7 \left (a^2 e-3 a b d+6 b^2 c\right )}{5 a^3 x^5}-\frac {b^{23/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{\sqrt {3} a^{14/3}}-\frac {b^{23/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{6 a^{14/3}}+\frac {b^{23/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{3 a^{14/3}}+\frac {9 b^7 \left (a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right )}{2 a^4 x^2}-\frac {9 b^7 c}{11 a x^{11}}\right )}{3 a b^4}}{6 a b^3}\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]
Output:
(b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + ((b^4*(2 9*b^3*c - 23*a*b^2*d + 17*a^2*b*e - 11*a^3*f)*x)/(3*a^5*(a + b*x^3)) + (2* ((-9*b^7*c)/(11*a*x^11) + (9*b^7*(3*b*c - a*d))/(8*a^2*x^8) - (9*b^7*(6*b^ 2*c - 3*a*b*d + a^2*e))/(5*a^3*x^5) + (9*b^7*(10*b^3*c - 6*a*b^2*d + 3*a^2 *b*e - a^3*f))/(2*a^4*x^2) - (b^(23/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b* e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]* a^(14/3)) + (b^(23/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log [a^(1/3) + b^(1/3)*x])/(3*a^(14/3)) - (b^(23/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6* a^(14/3))))/(3*a*b^4))/(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 = 0.14 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(-\frac {b \left (\frac {\left (\frac {11}{18} a^{3} b f -\frac {17}{18} a^{2} b^{2} e +\frac {23}{18} a \,b^{3} d -\frac {29}{18} b^{4} c \right ) x^{4}+\frac {a \left (7 f \,a^{3}-10 e \,a^{2} b +13 d a \,b^{2}-16 b^{3} c \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (20 f \,a^{3}-44 e \,a^{2} b +77 d a \,b^{2}-119 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}\right )}{a^{6}}-\frac {c}{11 a^{3} x^{11}}-\frac {a d -3 c b}{8 a^{4} x^{8}}-\frac {a^{2} e -3 d a b +6 b^{2} c}{5 a^{5} x^{5}}-\frac {f \,a^{3}-3 e \,a^{2} b +6 d a \,b^{2}-10 b^{3} c}{2 a^{6} x^{2}}\) | \(288\) |
| risch | \(\frac {-\frac {c}{11 a}-\frac {\left (11 a d -17 c b \right ) x^{3}}{88 a^{2}}-\frac {\left (44 a^{2} e -77 d a b +119 b^{2} c \right ) x^{6}}{220 a^{3}}-\frac {\left (20 f \,a^{3}-44 e \,a^{2} b +77 d a \,b^{2}-119 b^{3} c \right ) x^{9}}{40 a^{4}}-\frac {4 b \left (20 f \,a^{3}-44 e \,a^{2} b +77 d a \,b^{2}-119 b^{3} c \right ) x^{12}}{45 a^{5}}-\frac {b^{2} \left (20 f \,a^{3}-44 e \,a^{2} b +77 d a \,b^{2}-119 b^{3} c \right ) x^{15}}{18 a^{6}}}{x^{11} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{20} \textit {\_Z}^{3}+8000 a^{9} b^{2} f^{3}-52800 a^{8} b^{3} e \,f^{2}+92400 a^{7} b^{4} d \,f^{2}+116160 a^{7} b^{4} e^{2} f -142800 a^{6} b^{5} c \,f^{2}-406560 a^{6} b^{5} d e f -85184 a^{6} b^{5} e^{3}+628320 a^{5} b^{6} c e f +355740 a^{5} b^{6} d^{2} f +447216 a^{5} b^{6} d \,e^{2}-1099560 a^{4} b^{7} c d f -691152 a^{4} b^{7} c \,e^{2}-782628 a^{4} b^{7} d^{2} e +849660 a^{3} b^{8} c^{2} f +2419032 a^{3} b^{8} c d e +456533 a^{3} b^{8} d^{3}-1869252 a^{2} b^{9} c^{2} e -2116653 a^{2} b^{9} c \,d^{2}+3271191 a \,b^{10} c^{2} d -1685159 b^{11} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{20}-24000 a^{9} b^{2} f^{3}+158400 a^{8} b^{3} e \,f^{2}-277200 a^{7} b^{4} d \,f^{2}-348480 a^{7} b^{4} e^{2} f +428400 a^{6} b^{5} c \,f^{2}+1219680 a^{6} b^{5} d e f +255552 a^{6} b^{5} e^{3}-1884960 a^{5} b^{6} c e f -1067220 a^{5} b^{6} d^{2} f -1341648 a^{5} b^{6} d \,e^{2}+3298680 a^{4} b^{7} c d f +2073456 a^{4} b^{7} c \,e^{2}+2347884 a^{4} b^{7} d^{2} e -2548980 a^{3} b^{8} c^{2} f -7257096 a^{3} b^{8} c d e -1369599 a^{3} b^{8} d^{3}+5607756 a^{2} b^{9} c^{2} e +6349959 a^{2} b^{9} c \,d^{2}-9813573 a \,b^{10} c^{2} d +5055477 b^{11} c^{3}\right ) x +\left (-400 a^{13} b \,f^{2}+1760 a^{12} b^{2} e f -3080 a^{11} b^{3} d f -1936 a^{11} b^{3} e^{2}+4760 a^{10} b^{4} c f +6776 a^{10} b^{4} d e -10472 a^{9} b^{5} c e -5929 a^{9} b^{5} d^{2}+18326 a^{8} b^{6} c d -14161 a^{7} b^{7} c^{2}\right ) \textit {\_R} \right )\right )}{27}\) | \(762\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
-b/a^6*(((11/18*a^3*b*f-17/18*a^2*b^2*e+23/18*a*b^3*d-29/18*b^4*c)*x^4+1/9 *a*(7*a^3*f-10*a^2*b*e+13*a*b^2*d-16*b^3*c)*x)/(b*x^3+a)^2+1/9*(20*a^3*f-4 4*a^2*b*e+77*a*b^2*d-119*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)*a rctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))-1/11*c/a^3/x^11-1/8*(a*d-3*b*c)/a ^4/x^8-1/5*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^5-1/2*(a^3*f-3*a^2*b*e+6*a*b^2*d- 10*b^3*c)/a^6/x^2
Time = 0.12 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.72 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x, algorithm="fricas")
Output:
1/11880*(660*(119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^15 + 1056*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^12 + 297* (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^9 - 54*(119*a^3*b ^2*c - 77*a^4*b*d + 44*a^5*e)*x^6 - 1080*a^5*c + 135*(17*a^4*b*c - 11*a^5* d)*x^3 - 440*sqrt(3)*((119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2* f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14 + (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^ (1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) + 220*((1 19*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^14 + (119*a^2*b^3*c - 77*a^ 3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b* x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 440*((119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^17 + 2*(119*a*b^4*c - 77*a^2*b^3*d + 44*a ^3*b^2*e - 20*a^4*b*f)*x^14 + (119*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^11)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)))/(a^6*b^2* x^17 + 2*a^7*b*x^14 + a^8*x^11)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx=\frac {220 \, {\left (119 \, b^{5} c - 77 \, a b^{4} d + 44 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{15} + 352 \, {\left (119 \, a b^{4} c - 77 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{12} + 99 \, {\left (119 \, a^{2} b^{3} c - 77 \, a^{3} b^{2} d + 44 \, a^{4} b e - 20 \, a^{5} f\right )} x^{9} - 18 \, {\left (119 \, a^{3} b^{2} c - 77 \, a^{4} b d + 44 \, a^{5} e\right )} x^{6} - 360 \, a^{5} c + 45 \, {\left (17 \, a^{4} b c - 11 \, a^{5} d\right )} x^{3}}{3960 \, {\left (a^{6} b^{2} x^{17} + 2 \, a^{7} b x^{14} + a^{8} x^{11}\right )}} + \frac {\sqrt {3} {\left (119 \, b^{3} c - 77 \, a b^{2} d + 44 \, a^{2} b e - 20 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (119 \, b^{3} c - 77 \, a b^{2} d + 44 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (119 \, b^{3} c - 77 \, a b^{2} d + 44 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/3960*(220*(119*b^5*c - 77*a*b^4*d + 44*a^2*b^3*e - 20*a^3*b^2*f)*x^15 + 352*(119*a*b^4*c - 77*a^2*b^3*d + 44*a^3*b^2*e - 20*a^4*b*f)*x^12 + 99*(11 9*a^2*b^3*c - 77*a^3*b^2*d + 44*a^4*b*e - 20*a^5*f)*x^9 - 18*(119*a^3*b^2* c - 77*a^4*b*d + 44*a^5*e)*x^6 - 360*a^5*c + 45*(17*a^4*b*c - 11*a^5*d)*x^ 3)/(a^6*b^2*x^17 + 2*a^7*b*x^14 + a^8*x^11) + 1/27*sqrt(3)*(119*b^3*c - 77 *a*b^2*d + 44*a^2*b*e - 20*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/( a/b)^(1/3))/(a^6*(a/b)^(2/3)) - 1/54*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^6*(a/b)^(2/3)) + 1/2 7*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*log(x + (a/b)^(1/3))/(a ^6*(a/b)^(2/3))
Time = 0.14 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (119 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{7}} - \frac {{\left (119 \, b^{4} c - 77 \, a b^{3} d + 44 \, a^{2} b^{2} e - 20 \, a^{3} b f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7}} + \frac {{\left (119 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{7}} + \frac {29 \, b^{5} c x^{4} - 23 \, a b^{4} d x^{4} + 17 \, a^{2} b^{3} e x^{4} - 11 \, a^{3} b^{2} f x^{4} + 32 \, a b^{4} c x - 26 \, a^{2} b^{3} d x + 20 \, a^{3} b^{2} e x - 14 \, a^{4} b f x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{6}} + \frac {2200 \, b^{3} c x^{9} - 1320 \, a b^{2} d x^{9} + 660 \, a^{2} b e x^{9} - 220 \, a^{3} f x^{9} - 528 \, a b^{2} c x^{6} + 264 \, a^{2} b d x^{6} - 88 \, a^{3} e x^{6} + 165 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{6} x^{11}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x, algorithm="giac")
Output:
1/27*sqrt(3)*(119*(-a*b^2)^(1/3)*b^3*c - 77*(-a*b^2)^(1/3)*a*b^2*d + 44*(- a*b^2)^(1/3)*a^2*b*e - 20*(-a*b^2)^(1/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^7 - 1/27*(119*b^4*c - 77*a*b^3*d + 44*a^2*b^ 2*e - 20*a^3*b*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^7 + 1/54*(119* (-a*b^2)^(1/3)*b^3*c - 77*(-a*b^2)^(1/3)*a*b^2*d + 44*(-a*b^2)^(1/3)*a^2*b *e - 20*(-a*b^2)^(1/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^7 + 1/18*(29*b^5*c*x^4 - 23*a*b^4*d*x^4 + 17*a^2*b^3*e*x^4 - 11*a^3*b^2*f*x ^4 + 32*a*b^4*c*x - 26*a^2*b^3*d*x + 20*a^3*b^2*e*x - 14*a^4*b*f*x)/((b*x^ 3 + a)^2*a^6) + 1/440*(2200*b^3*c*x^9 - 1320*a*b^2*d*x^9 + 660*a^2*b*e*x^9 - 220*a^3*f*x^9 - 528*a*b^2*c*x^6 + 264*a^2*b*d*x^6 - 88*a^3*e*x^6 + 165* a^2*b*c*x^3 - 55*a^3*d*x^3 - 40*a^3*c)/(a^6*x^11)
Time = 6.37 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx=\frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{27\,a^{20/3}}-\frac {\frac {c}{11\,a}-\frac {x^9\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{40\,a^4}+\frac {x^3\,\left (11\,a\,d-17\,b\,c\right )}{88\,a^2}+\frac {x^6\,\left (44\,e\,a^2-77\,d\,a\,b+119\,c\,b^2\right )}{220\,a^3}-\frac {4\,b\,x^{12}\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{45\,a^5}-\frac {b^2\,x^{15}\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{18\,a^6}}{a^2\,x^{11}+2\,a\,b\,x^{14}+b^2\,x^{17}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{27\,a^{20/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{27\,a^{20/3}} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x)
Output:
(b^(2/3)*log(b^(1/3)*x + a^(1/3))*(119*b^3*c - 20*a^3*f - 77*a*b^2*d + 44* a^2*b*e))/(27*a^(20/3)) - (c/(11*a) - (x^9*(119*b^3*c - 20*a^3*f - 77*a*b^ 2*d + 44*a^2*b*e))/(40*a^4) + (x^3*(11*a*d - 17*b*c))/(88*a^2) + (x^6*(119 *b^2*c + 44*a^2*e - 77*a*b*d))/(220*a^3) - (4*b*x^12*(119*b^3*c - 20*a^3*f - 77*a*b^2*d + 44*a^2*b*e))/(45*a^5) - (b^2*x^15*(119*b^3*c - 20*a^3*f - 77*a*b^2*d + 44*a^2*b*e))/(18*a^6))/(a^2*x^11 + b^2*x^17 + 2*a*b*x^14) + ( 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)*(119*b^3*c - 20*a^3*f - 77*a*b^2*d + 44*a^2*b*e))/(27*a^(20/3)) - (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)*(119*b^3*c - 20*a^3*f - 77*a*b^2*d + 44*a^2*b*e))/(27*a^(20/3))
Time = 0.16 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.40 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^3,x)
Output:
(8800*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* a**5*b*f*x**11 - 19360*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a* *(1/3)*sqrt(3)))*a**4*b**2*e*x**11 + 17600*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4*b**2*f*x**14 + 33880*a**(1/3)*sq rt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**3*d*x**11 - 38720*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3) ))*a**3*b**3*e*x**14 + 8800*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x )/(a**(1/3)*sqrt(3)))*a**3*b**3*f*x**17 - 52360*a**(1/3)*sqrt(3)*atan((a** (1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**4*c*x**11 + 67760*a**(1/ 3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**4*d* x**14 - 19360*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sq rt(3)))*a**2*b**4*e*x**17 - 104720*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b** (1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**5*c*x**14 + 33880*a**(1/3)*sqrt(3)*atan( (a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**5*d*x**17 - 52360*a**(1 /3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**6*c*x**1 7 + 4400*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**5 *b*f*x**11 - 9680*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x **2)*a**4*b**2*e*x**11 + 8800*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*b**2*f*x**14 + 16940*a**(1/3)*log(a**(2/3) - b**(1/3 )*a**(1/3)*x + b**(2/3)*x**2)*a**3*b**3*d*x**11 - 19360*a**(1/3)*log(a*...