Integrand size = 30, antiderivative size = 335 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=-\frac {c}{11 a^2 x^{11}}+\frac {2 b c-a d}{8 a^3 x^8}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}-\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{17/3}}+\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac {b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{17/3}} \] Output:
-1/11*c/a^2/x^11+1/8*(-a*d+2*b*c)/a^3/x^8-1/5*(a^2*e-2*a*b*d+3*b^2*c)/a^4/ x^5+1/2*(-a^3*f+2*a^2*b*e-3*a*b^2*d+4*b^3*c)/a^5/x^2+1/3*b*(-a^3*f+a^2*b*e -a*b^2*d+b^3*c)*x/a^5/(b*x^3+a)-1/9*b^(2/3)*(-5*a^3*f+8*a^2*b*e-11*a*b^2*d +14*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)+1/9*b^(2/3)*(-5*a^3*f+8*a^2*b*e-11*a*b^2*d+14*b^3*c)*ln(a^(1/3)+b^(1/3 )*x)/a^(17/3)-1/18*b^(2/3)*(-5*a^3*f+8*a^2*b*e-11*a*b^2*d+14*b^3*c)*ln(a^( 2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)
Time = 0.15 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=\frac {-\frac {360 a^{11/3} c}{x^{11}}-\frac {495 a^{8/3} (-2 b c+a d)}{x^8}-\frac {792 a^{5/3} \left (3 b^2 c-2 a b d+a^2 e\right )}{x^5}-\frac {1980 a^{2/3} \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{x^2}-\frac {1320 a^{2/3} b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a+b x^3}-440 \sqrt {3} b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+440 b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+220 b^{2/3} \left (-14 b^3 c+11 a b^2 d-8 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{3960 a^{17/3}} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]
Output:
((-360*a^(11/3)*c)/x^11 - (495*a^(8/3)*(-2*b*c + a*d))/x^8 - (792*a^(5/3)* (3*b^2*c - 2*a*b*d + a^2*e))/x^5 - (1980*a^(2/3)*(-4*b^3*c + 3*a*b^2*d - 2 *a^2*b*e + a^3*f))/x^2 - (1320*a^(2/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a ^3*f)*x)/(a + b*x^3) - 440*Sqrt[3]*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2* b*e - 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 440*b^(2/3)*( 14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 22 0*b^(2/3)*(-14*b^3*c + 11*a*b^2*d - 8*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^( 1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3960*a^(17/3))
Time = 1.54 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}-\frac {\int -\frac {\frac {2 b^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{12}}{a^4}-\frac {3 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {3 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c}{x^{12} \left (b x^3+a\right )}dx}{3 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {2 b^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{12}}{a^4}-\frac {3 b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9}{a^3}+\frac {3 b^3 \left (e a^2-b d a+b^2 c\right ) x^6}{a^2}-3 b^3 \left (\frac {b c}{a}-d\right ) x^3+3 b^3 c}{x^{12} \left (b x^3+a\right )}dx}{3 a b^3}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\int \left (-\frac {\left (5 f a^3-8 b e a^2+11 b^2 d a-14 b^3 c\right ) b^4}{a^4 \left (b x^3+a\right )}+\frac {3 \left (f a^3-2 b e a^2+3 b^2 d a-4 b^3 c\right ) b^3}{a^4 x^3}+\frac {3 \left (e a^2-2 b d a+3 b^2 c\right ) b^3}{a^3 x^6}+\frac {3 (a d-2 b c) b^3}{a^2 x^9}+\frac {3 c b^3}{a x^{12}}\right )dx}{3 a b^3}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {\frac {3 b^3 (2 b c-a d)}{8 a^2 x^8}-\frac {3 b^3 \left (a^2 e-2 a b d+3 b^2 c\right )}{5 a^3 x^5}-\frac {b^{11/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{\sqrt {3} a^{14/3}}-\frac {b^{11/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{6 a^{14/3}}+\frac {b^{11/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{3 a^{14/3}}+\frac {3 b^3 \left (a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right )}{2 a^4 x^2}-\frac {3 b^3 c}{11 a x^{11}}}{3 a b^3}\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]
Output:
(b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a^5*(a + b*x^3)) + ((-3*b^3*c )/(11*a*x^11) + (3*b^3*(2*b*c - a*d))/(8*a^2*x^8) - (3*b^3*(3*b^2*c - 2*a* b*d + a^2*e))/(5*a^3*x^5) + (3*b^3*(4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3* f))/(2*a^4*x^2) - (b^(11/3)*(14*b^3*c - 11*a*b^2*d + 8*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^(14/3)) + (b ^(11/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3 )*x])/(3*a^(14/3)) - (b^(11/3)*(14*b^3*c - 11*a*b^2*d + 8*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^(14/3)))/(3*a*b^3)
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 = 250, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(-\frac {b \left (\frac {\left (\frac {1}{3} f \,a^{3}-\frac {1}{3} e \,a^{2} b +\frac {1}{3} d a \,b^{2}-\frac {1}{3} b^{3} c \right ) x}{b \,x^{3}+a}+\frac {\left (5 f \,a^{3}-8 e \,a^{2} b +11 d a \,b^{2}-14 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}\right )}{a^{5}}-\frac {c}{11 a^{2} x^{11}}-\frac {a d -2 c b}{8 a^{3} x^{8}}-\frac {a^{2} e -2 d a b +3 b^{2} c}{5 a^{4} x^{5}}-\frac {f \,a^{3}-2 e \,a^{2} b +3 d a \,b^{2}-4 b^{3} c}{2 a^{5} x^{2}}\) | \(250\) |
| risch | \(\frac {-\frac {b \left (5 f \,a^{3}-8 e \,a^{2} b +11 d a \,b^{2}-14 b^{3} c \right ) x^{12}}{6 a^{5}}-\frac {\left (5 f \,a^{3}-8 e \,a^{2} b +11 d a \,b^{2}-14 b^{3} c \right ) x^{9}}{10 a^{4}}-\frac {\left (8 a^{2} e -11 d a b +14 b^{2} c \right ) x^{6}}{40 a^{3}}-\frac {\left (11 a d -14 c b \right ) x^{3}}{88 a^{2}}-\frac {c}{11 a}}{x^{11} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{17} \textit {\_Z}^{3}+125 a^{9} b^{2} f^{3}-600 a^{8} b^{3} e \,f^{2}+825 a^{7} b^{4} d \,f^{2}+960 a^{7} b^{4} e^{2} f -1050 a^{6} b^{5} c \,f^{2}-2640 a^{6} b^{5} d e f -512 a^{6} b^{5} e^{3}+3360 a^{5} b^{6} c e f +1815 a^{5} b^{6} d^{2} f +2112 a^{5} b^{6} d \,e^{2}-4620 a^{4} b^{7} c d f -2688 a^{4} b^{7} c \,e^{2}-2904 a^{4} b^{7} d^{2} e +2940 a^{3} b^{8} c^{2} f +7392 a^{3} b^{8} c d e +1331 a^{3} b^{8} d^{3}-4704 a^{2} b^{9} c^{2} e -5082 a^{2} b^{9} c \,d^{2}+6468 a \,b^{10} c^{2} d -2744 b^{11} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{17}-375 a^{9} b^{2} f^{3}+1800 a^{8} b^{3} e \,f^{2}-2475 a^{7} b^{4} d \,f^{2}-2880 a^{7} b^{4} e^{2} f +3150 a^{6} b^{5} c \,f^{2}+7920 a^{6} b^{5} d e f +1536 a^{6} b^{5} e^{3}-10080 a^{5} b^{6} c e f -5445 a^{5} b^{6} d^{2} f -6336 a^{5} b^{6} d \,e^{2}+13860 a^{4} b^{7} c d f +8064 a^{4} b^{7} c \,e^{2}+8712 a^{4} b^{7} d^{2} e -8820 a^{3} b^{8} c^{2} f -22176 a^{3} b^{8} c d e -3993 a^{3} b^{8} d^{3}+14112 a^{2} b^{9} c^{2} e +15246 a^{2} b^{9} c \,d^{2}-19404 a \,b^{10} c^{2} d +8232 b^{11} c^{3}\right ) x +\left (-25 a^{12} b \,f^{2}+80 a^{11} b^{2} e f -110 a^{10} b^{3} d f -64 a^{10} b^{3} e^{2}+140 a^{9} b^{4} c f +176 a^{9} b^{4} d e -224 a^{8} b^{5} c e -121 a^{8} b^{5} d^{2}+308 a^{7} b^{6} c d -196 a^{6} b^{7} c^{2}\right ) \textit {\_R} \right )\right )}{9}\) | \(724\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
-b/a^5*((1/3*f*a^3-1/3*e*a^2*b+1/3*d*a*b^2-1/3*b^3*c)*x/(b*x^3+a)+1/3*(5*a ^3*f-8*a^2*b*e+11*a*b^2*d-14*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))))-1/11*c/a^2/x^11-1/8*(a*d-2*b* c)/a^3/x^8-1/5*(a^2*e-2*a*b*d+3*b^2*c)/a^4/x^5-1/2*(a^3*f-2*a^2*b*e+3*a*b^ 2*d-4*b^3*c)/a^5/x^2
Time = 0.09 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.42 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=\frac {660 \, {\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 396 \, {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 99 \, {\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 360 \, a^{4} c + 45 \, {\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3} - 440 \, \sqrt {3} {\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} + {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 220 \, {\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} + {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 440 \, {\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} + {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{3960 \, {\left (a^{5} b x^{14} + a^{6} x^{11}\right )}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x, algorithm="fricas")
Output:
1/3960*(660*(14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^12 + 396*( 14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^9 - 99*(14*a^2*b^2*c - 11*a^3*b*d + 8*a^4*e)*x^6 - 360*a^4*c + 45*(14*a^3*b*c - 11*a^4*d)*x^3 - 4 40*sqrt(3)*((14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a *b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*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*((14*b^4*c - 11 *a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^2*b^2*d + 8* a^3*b*e - 5*a^4*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*((14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^11) *(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)))/(a^5*b*x^14 + a^6*x^11)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=\frac {220 \, {\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 132 \, {\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 33 \, {\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 120 \, a^{4} c + 15 \, {\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3}}{1320 \, {\left (a^{5} b x^{14} + a^{6} x^{11}\right )}} + \frac {\sqrt {3} {\left (14 \, b^{3} c - 11 \, a b^{2} d + 8 \, a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (14 \, b^{3} c - 11 \, a b^{2} d + 8 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (14 \, b^{3} c - 11 \, a b^{2} d + 8 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/1320*(220*(14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^12 + 132*( 14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^9 - 33*(14*a^2*b^2*c - 11*a^3*b*d + 8*a^4*e)*x^6 - 120*a^4*c + 15*(14*a^3*b*c - 11*a^4*d)*x^3)/(a ^5*b*x^14 + a^6*x^11) + 1/9*sqrt(3)*(14*b^3*c - 11*a*b^2*d + 8*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*(a/b)^(2/ 3)) - 1/18*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*log(x^2 - x*(a/b) ^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(2/3)) + 1/9*(14*b^3*c - 11*a*b^2*d + 8*a ^2*b*e - 5*a^3*f)*log(x + (a/b)^(1/3))/(a^5*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} {\left (14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6}} - \frac {{\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{6}} + \frac {{\left (14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6}} + \frac {b^{4} c x - a b^{3} d x + a^{2} b^{2} e x - a^{3} b f x}{3 \, {\left (b x^{3} + a\right )} a^{5}} + \frac {880 \, b^{3} c x^{9} - 660 \, a b^{2} d x^{9} + 440 \, a^{2} b e x^{9} - 220 \, a^{3} f x^{9} - 264 \, a b^{2} c x^{6} + 176 \, a^{2} b d x^{6} - 88 \, a^{3} e x^{6} + 110 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{5} x^{11}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x, algorithm="giac")
Output:
1/9*sqrt(3)*(14*(-a*b^2)^(1/3)*b^3*c - 11*(-a*b^2)^(1/3)*a*b^2*d + 8*(-a*b ^2)^(1/3)*a^2*b*e - 5*(-a*b^2)^(1/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a/ b)^(1/3))/(-a/b)^(1/3))/a^6 - 1/9*(14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5 *a^3*b*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 + 1/18*(14*(-a*b^2)^ (1/3)*b^3*c - 11*(-a*b^2)^(1/3)*a*b^2*d + 8*(-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 + 1/3*(b^4 *c*x - a*b^3*d*x + a^2*b^2*e*x - a^3*b*f*x)/((b*x^3 + a)*a^5) + 1/440*(880 *b^3*c*x^9 - 660*a*b^2*d*x^9 + 440*a^2*b*e*x^9 - 220*a^3*f*x^9 - 264*a*b^2 *c*x^6 + 176*a^2*b*d*x^6 - 88*a^3*e*x^6 + 110*a^2*b*c*x^3 - 55*a^3*d*x^3 - 40*a^3*c)/(a^5*x^11)
Time = 6.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx=\frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^{17/3}}-\frac {\frac {c}{11\,a}-\frac {x^9\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{10\,a^4}+\frac {x^3\,\left (11\,a\,d-14\,b\,c\right )}{88\,a^2}+\frac {x^6\,\left (8\,e\,a^2-11\,d\,a\,b+14\,c\,b^2\right )}{40\,a^3}-\frac {b\,x^{12}\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{6\,a^5}}{b\,x^{14}+a\,x^{11}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^{17/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+8\,e\,a^2\,b-11\,d\,a\,b^2+14\,c\,b^3\right )}{9\,a^{17/3}} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x)
Output:
(b^(2/3)*log(b^(1/3)*x + a^(1/3))*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2 *b*e))/(9*a^(17/3)) - (c/(11*a) - (x^9*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2*b*e))/(10*a^4) + (x^3*(11*a*d - 14*b*c))/(88*a^2) + (x^6*(14*b^2*c + 8*a^2*e - 11*a*b*d))/(40*a^3) - (b*x^12*(14*b^3*c - 5*a^3*f - 11*a*b^2*d + 8*a^2*b*e))/(6*a^5))/(a*x^11 + 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)*(14*b^3*c - 5*a^3*f - 11* a*b^2*d + 8*a^2*b*e))/(9*a^(17/3)) - (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)*(14*b^3*c - 5*a^3*f - 11*a*b^2* d + 8*a^2*b*e))/(9*a^(17/3))
Time = 0.15 (sec) , antiderivative size = 882, normalized size of antiderivative = 2.63 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x)
Output:
(2200*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 - 3520*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 + 2200*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 + 4840*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 - 3520*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 + 4840*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 - 6160*a**(1/3)*sqrt(3)*a tan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**5*c*x**14 + 1100*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 - 1760*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**11 + 1100*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x* *2)*a**3*b**2*f*x**14 + 2420*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**3*d*x**11 - 1760*a**(1/3)*log(a**(2/3) - b**(1/3)* a**(1/3)*x + b**(2/3)*x**2)*a**2*b**3*e*x**14 - 3080*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**4*c*x**11 + 2420*a**(1/3)*log (a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**4*d*x**14 - 3080*a** (1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**5*c*x**14 - 2 200*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**4*b*f*x**11 + 3520*a**(1/3)*...