\(\int \frac {x^{12} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\) [64]

Optimal result

Integrand size = 30, antiderivative size = 416 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x}{b^7}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac {(b e-3 a f) x^{10}}{10 b^4}+\frac {f x^{13}}{13 b^3}+\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac {a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}-\frac {a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{22/3}}+\frac {a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{22/3}}-\frac {a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{22/3}} \] Output:

-a*(-15*a^3*f+10*a^2*b*e-6*a*b^2*d+3*b^3*c)*x/b^7+1/4*(-10*a^3*f+6*a^2*b*e 
-3*a*b^2*d+b^3*c)*x^4/b^6+1/7*(6*a^2*f-3*a*b*e+b^2*d)*x^7/b^5+1/10*(-3*a*f 
+b*e)*x^10/b^4+1/13*f*x^13/b^3+1/6*a^3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^ 
7/(b*x^3+a)^2-1/18*a^2*(-37*a^3*f+31*a^2*b*e-25*a*b^2*d+19*b^3*c)*x/b^7/(b 
*x^3+a)-1/27*a^(4/3)*(-152*a^3*f+104*a^2*b*e-65*a*b^2*d+35*b^3*c)*arctan(1 
/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b^(22/3)+1/27*a^(4/3)*(- 
152*a^3*f+104*a^2*b*e-65*a*b^2*d+35*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/b^(22/3)- 
1/54*a^(4/3)*(-152*a^3*f+104*a^2*b*e-65*a*b^2*d+35*b^3*c)*ln(a^(2/3)-a^(1/ 
3)*b^(1/3)*x+b^(2/3)*x^2)/b^(22/3)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.99 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {a \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right ) x}{b^7}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac {(b e-3 a f) x^{10}}{10 b^4}+\frac {f x^{13}}{13 b^3}+\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}+\frac {a^2 \left (-19 b^3 c+25 a b^2 d-31 a^2 b e+37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac {a^{4/3} \left (-35 b^3 c+65 a b^2 d-104 a^2 b e+152 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{22/3}}-\frac {a^{4/3} \left (-35 b^3 c+65 a b^2 d-104 a^2 b e+152 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{22/3}}+\frac {a^{4/3} \left (-35 b^3 c+65 a b^2 d-104 a^2 b e+152 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{22/3}} \] Input:

Integrate[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
 

Output:

(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b 
^2*d + 6*a^2*b*e - 10*a^3*f)*x^4)/(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x 
^7)/(7*b^5) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3*(b^ 
3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) + (a^2*(-19*b^3* 
c + 25*a*b^2*d - 31*a^2*b*e + 37*a^3*f)*x)/(18*b^7*(a + b*x^3)) + (a^(4/3) 
*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*ArcTan[(1 - (2*b^(1/3) 
*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(22/3)) - (a^(4/3)*(-35*b^3*c + 65*a*b 
^2*d - 104*a^2*b*e + 152*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(22/3)) + 
(a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*Log[a^(2/3) - 
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))
 

Rubi [A] (verified)

Time = 2.57 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2367, 2397, 27, 2426, 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.

\(\displaystyle \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\)

\(\Big \downarrow \) 2367

\(\displaystyle \frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 a b^6 f x^{18}-6 a b^5 (b e-a f) x^{15}-6 a b^4 \left (f a^2-b e a+b^2 d\right ) x^{12}-6 a b^3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^9+6 a^2 b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6-6 a^3 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{\left (b x^3+a\right )^2}dx}{6 a b^7}\)

\(\Big \downarrow \) 2397

\(\displaystyle \frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {\frac {a^3 x \left (-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {2 \left (9 a^2 b^{11} f x^{15}+9 a^2 b^{10} (b e-2 a f) x^{12}+9 a^2 b^9 \left (3 f a^2-2 b e a+b^2 d\right ) x^9+9 a^2 b^8 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^6-9 a^3 b^7 \left (-5 f a^3+4 b e a^2-3 b^2 d a+2 b^3 c\right ) x^3+a^4 b^6 \left (-17 f a^3+14 b e a^2-11 b^2 d a+8 b^3 c\right )\right )}{b x^3+a}dx}{3 a b^6}}{6 a b^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {\frac {a^3 x \left (-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {2 \int \frac {9 a^2 b^{11} f x^{15}+9 a^2 b^{10} (b e-2 a f) x^{12}+9 a^2 b^9 \left (3 f a^2-2 b e a+b^2 d\right ) x^9+9 a^2 b^8 \left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^6-9 a^3 b^7 \left (-5 f a^3+4 b e a^2-3 b^2 d a+2 b^3 c\right ) x^3+a^4 b^6 \left (-17 f a^3+14 b e a^2-11 b^2 d a+8 b^3 c\right )}{b x^3+a}dx}{3 a b^6}}{6 a b^7}\)

\(\Big \downarrow \) 2426

\(\displaystyle \frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {\frac {a^3 x \left (-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {2 \int \left (9 a^2 b^{10} f x^{12}+9 a^2 b^9 (b e-3 a f) x^9+9 a^2 b^8 \left (6 f a^2-3 b e a+b^2 d\right ) x^6+9 a^2 b^7 \left (-10 f a^3+6 b e a^2-3 b^2 d a+b^3 c\right ) x^3-9 a^3 b^6 \left (-15 f a^3+10 b e a^2-6 b^2 d a+3 b^3 c\right )+\frac {35 a^4 c b^9-65 a^5 d b^8+104 a^6 e b^7-152 a^7 f b^6}{b x^3+a}\right )dx}{3 a b^6}}{6 a b^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {\frac {a^3 x \left (-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {2 \left (\frac {9}{13} a^2 b^{10} f x^{13}+\frac {9}{10} a^2 b^9 x^{10} (b e-3 a f)+\frac {9}{7} a^2 b^8 x^7 \left (6 a^2 f-3 a b e+b^2 d\right )+\frac {9}{4} a^2 b^7 x^4 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )-9 a^3 b^6 x \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )-\frac {a^{10/3} b^{17/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{\sqrt {3}}-\frac {1}{6} a^{10/3} b^{17/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )+\frac {1}{3} a^{10/3} b^{17/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )\right )}{3 a b^6}}{6 a b^7}\)

Input:

Int[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
 

Output:

(a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) - ((a^3* 
(19*b^3*c - 25*a*b^2*d + 31*a^2*b*e - 37*a^3*f)*x)/(3*(a + b*x^3)) - (2*(- 
9*a^3*b^6*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x + (9*a^2*b^7*(b^ 
3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^4)/4 + (9*a^2*b^8*(b^2*d - 3*a*b 
*e + 6*a^2*f)*x^7)/7 + (9*a^2*b^9*(b*e - 3*a*f)*x^10)/10 + (9*a^2*b^10*f*x 
^13)/13 - (a^(10/3)*b^(17/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^ 
3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/Sqrt[3] + (a^(10/3 
)*b^(17/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*Log[a^(1/3) + 
 b^(1/3)*x])/3 - (a^(10/3)*b^(17/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 
 152*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/6))/(3*a*b^6)) 
/(6*a*b^7)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2367

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = 
 m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) 
*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ 
m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo 
r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))   I 
nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], 
 x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 
] && LtQ[p, -1] && IGtQ[m, 0]
 

rule 2397

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, 
x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, 
 x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S 
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] 
 + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))   Int[(a + b*x^n)^(p + 1)* 
ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, 
n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
 

rule 2426

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a 
+ b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.13 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.68

method	result	size
risch	\(\frac {f \,x^{13}}{13 b^{3}}-\frac {3 x^{10} a f}{10 b^{4}}+\frac {x^{10} e}{10 b^{3}}+\frac {6 x^{7} a^{2} f}{7 b^{5}}-\frac {3 x^{7} a e}{7 b^{4}}+\frac {x^{7} d}{7 b^{3}}-\frac {5 x^{4} a^{3} f}{2 b^{6}}+\frac {3 x^{4} a^{2} e}{2 b^{5}}-\frac {3 x^{4} a d}{4 b^{4}}+\frac {x^{4} c}{4 b^{3}}+\frac {15 a^{4} f x}{b^{7}}-\frac {10 a^{3} e x}{b^{6}}+\frac {6 a^{2} d x}{b^{5}}-\frac {3 a c x}{b^{4}}+\frac {\left (\frac {37}{18} a^{5} b f -\frac {31}{18} e \,a^{4} b^{2}+\frac {25}{18} a^{3} d \,b^{3}-\frac {19}{18} a^{2} c \,b^{4}\right ) x^{4}+\frac {a^{3} \left (17 f \,a^{3}-14 e \,a^{2} b +11 d a \,b^{2}-8 b^{3} c \right ) x}{9}}{b^{7} \left (b \,x^{3}+a \right )^{2}}-\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (152 f \,a^{3}-104 e \,a^{2} b +65 d a \,b^{2}-35 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 b^{8}}\)	\(284\)
default	\(\frac {\frac {1}{13} f \,x^{13} b^{4}-\frac {3}{10} x^{10} a \,b^{3} f +\frac {1}{10} x^{10} b^{4} e +\frac {6}{7} x^{7} a^{2} b^{2} f -\frac {3}{7} x^{7} a \,b^{3} e +\frac {1}{7} x^{7} b^{4} d -\frac {5}{2} x^{4} a^{3} b f +\frac {3}{2} x^{4} a^{2} b^{2} e -\frac {3}{4} x^{4} a \,b^{3} d +\frac {1}{4} x^{4} b^{4} c +15 a^{4} f x -10 a^{3} b e x +6 a^{2} b^{2} d x -3 a \,b^{3} c x}{b^{7}}-\frac {a^{2} \left (\frac {\left (-\frac {37}{18} a^{3} b f +\frac {31}{18} a^{2} b^{2} e -\frac {25}{18} a \,b^{3} d +\frac {19}{18} b^{4} c \right ) x^{4}-\frac {a \left (17 f \,a^{3}-14 e \,a^{2} b +11 d a \,b^{2}-8 b^{3} c \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (152 f \,a^{3}-104 e \,a^{2} b +65 d a \,b^{2}-35 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}\right )}{b^{7}}\)	\(344\)

Input:

int(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/13*f*x^13/b^3-3/10/b^4*x^10*a*f+1/10/b^3*x^10*e+6/7/b^5*x^7*a^2*f-3/7/b^ 
4*x^7*a*e+1/7/b^3*x^7*d-5/2/b^6*x^4*a^3*f+3/2/b^5*x^4*a^2*e-3/4/b^4*x^4*a* 
d+1/4/b^3*x^4*c+15/b^7*a^4*f*x-10/b^6*a^3*e*x+6/b^5*a^2*d*x-3/b^4*a*c*x+(( 
37/18*a^5*b*f-31/18*e*a^4*b^2+25/18*a^3*d*b^3-19/18*a^2*c*b^4)*x^4+1/9*a^3 
*(17*a^3*f-14*a^2*b*e+11*a*b^2*d-8*b^3*c)*x)/b^7/(b*x^3+a)^2-1/27/b^8*a^2* 
sum((152*a^3*f-104*a^2*b*e+65*a*b^2*d-35*b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z 
^3*b+a))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.60 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")
 

Output:

1/49140*(3780*b^6*f*x^19 + 378*(13*b^6*e - 19*a*b^5*f)*x^16 + 108*(65*b^6* 
d - 104*a*b^5*e + 152*a^2*b^4*f)*x^13 + 351*(35*b^6*c - 65*a*b^5*d + 104*a 
^2*b^4*e - 152*a^3*b^3*f)*x^10 - 3510*(35*a*b^5*c - 65*a^2*b^4*d + 104*a^3 
*b^3*e - 152*a^4*b^2*f)*x^7 - 9555*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4* 
b^2*e - 152*a^5*b*f)*x^4 - 1820*sqrt(3)*(35*a^3*b^3*c - 65*a^4*b^2*d + 104 
*a^5*b*e - 152*a^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^ 
4*b^2*f)*x^6 + 2*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b* 
f)*x^3)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a 
) + 910*(35*a^3*b^3*c - 65*a^4*b^2*d + 104*a^5*b*e - 152*a^6*f + (35*a*b^5 
*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b^2*f)*x^6 + 2*(35*a^2*b^4*c - 
 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^3)*(-a/b)^(1/3)*log(x^2 + x 
*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*(35*a^3*b^3*c - 65*a^4*b^2*d + 104*a^ 
5*b*e - 152*a^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b 
^2*f)*x^6 + 2*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)* 
x^3)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) - 5460*(35*a^3*b^3*c - 65*a^4*b^2* 
d + 104*a^5*b*e - 152*a^6*f)*x)/(b^9*x^6 + 2*a*b^8*x^3 + a^2*b^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**12*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.02 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {{\left (19 \, a^{2} b^{4} c - 25 \, a^{3} b^{3} d + 31 \, a^{4} b^{2} e - 37 \, a^{5} b f\right )} x^{4} + 2 \, {\left (8 \, a^{3} b^{3} c - 11 \, a^{4} b^{2} d + 14 \, a^{5} b e - 17 \, a^{6} f\right )} x}{18 \, {\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} + \frac {140 \, b^{4} f x^{13} + 182 \, {\left (b^{4} e - 3 \, a b^{3} f\right )} x^{10} + 260 \, {\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{7} + 455 \, {\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{4} - 1820 \, {\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x}{1820 \, b^{7}} + \frac {\sqrt {3} {\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{8} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{8} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{8} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:

integrate(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")
 

Output:

-1/18*((19*a^2*b^4*c - 25*a^3*b^3*d + 31*a^4*b^2*e - 37*a^5*b*f)*x^4 + 2*( 
8*a^3*b^3*c - 11*a^4*b^2*d + 14*a^5*b*e - 17*a^6*f)*x)/(b^9*x^6 + 2*a*b^8* 
x^3 + a^2*b^7) + 1/1820*(140*b^4*f*x^13 + 182*(b^4*e - 3*a*b^3*f)*x^10 + 2 
60*(b^4*d - 3*a*b^3*e + 6*a^2*b^2*f)*x^7 + 455*(b^4*c - 3*a*b^3*d + 6*a^2* 
b^2*e - 10*a^3*b*f)*x^4 - 1820*(3*a*b^3*c - 6*a^2*b^2*d + 10*a^3*b*e - 15* 
a^4*f)*x)/b^7 + 1/27*sqrt(3)*(35*a^2*b^3*c - 65*a^3*b^2*d + 104*a^4*b*e - 
152*a^5*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^8*(a/b)^ 
(2/3)) - 1/54*(35*a^2*b^3*c - 65*a^3*b^2*d + 104*a^4*b*e - 152*a^5*f)*log( 
x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^8*(a/b)^(2/3)) + 1/27*(35*a^2*b^3*c 
- 65*a^3*b^2*d + 104*a^4*b*e - 152*a^5*f)*log(x + (a/b)^(1/3))/(b^8*(a/b)^ 
(2/3))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.18 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (35 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d + 104 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e - 152 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{8}} - \frac {{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{7}} + \frac {{\left (35 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d + 104 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e - 152 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{8}} - \frac {19 \, a^{2} b^{4} c x^{4} - 25 \, a^{3} b^{3} d x^{4} + 31 \, a^{4} b^{2} e x^{4} - 37 \, a^{5} b f x^{4} + 16 \, a^{3} b^{3} c x - 22 \, a^{4} b^{2} d x + 28 \, a^{5} b e x - 34 \, a^{6} f x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{7}} + \frac {140 \, b^{36} f x^{13} + 182 \, b^{36} e x^{10} - 546 \, a b^{35} f x^{10} + 260 \, b^{36} d x^{7} - 780 \, a b^{35} e x^{7} + 1560 \, a^{2} b^{34} f x^{7} + 455 \, b^{36} c x^{4} - 1365 \, a b^{35} d x^{4} + 2730 \, a^{2} b^{34} e x^{4} - 4550 \, a^{3} b^{33} f x^{4} - 5460 \, a b^{35} c x + 10920 \, a^{2} b^{34} d x - 18200 \, a^{3} b^{33} e x + 27300 \, a^{4} b^{32} f x}{1820 \, b^{39}} \] Input:

integrate(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")
 

Output:

1/27*sqrt(3)*(35*(-a*b^2)^(1/3)*a*b^3*c - 65*(-a*b^2)^(1/3)*a^2*b^2*d + 10 
4*(-a*b^2)^(1/3)*a^3*b*e - 152*(-a*b^2)^(1/3)*a^4*f)*arctan(1/3*sqrt(3)*(2 
*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^8 - 1/27*(35*a^2*b^3*c - 65*a^3*b^2*d + 
 104*a^4*b*e - 152*a^5*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^7) 
+ 1/54*(35*(-a*b^2)^(1/3)*a*b^3*c - 65*(-a*b^2)^(1/3)*a^2*b^2*d + 104*(-a* 
b^2)^(1/3)*a^3*b*e - 152*(-a*b^2)^(1/3)*a^4*f)*log(x^2 + x*(-a/b)^(1/3) + 
(-a/b)^(2/3))/b^8 - 1/18*(19*a^2*b^4*c*x^4 - 25*a^3*b^3*d*x^4 + 31*a^4*b^2 
*e*x^4 - 37*a^5*b*f*x^4 + 16*a^3*b^3*c*x - 22*a^4*b^2*d*x + 28*a^5*b*e*x - 
 34*a^6*f*x)/((b*x^3 + a)^2*b^7) + 1/1820*(140*b^36*f*x^13 + 182*b^36*e*x^ 
10 - 546*a*b^35*f*x^10 + 260*b^36*d*x^7 - 780*a*b^35*e*x^7 + 1560*a^2*b^34 
*f*x^7 + 455*b^36*c*x^4 - 1365*a*b^35*d*x^4 + 2730*a^2*b^34*e*x^4 - 4550*a 
^3*b^33*f*x^4 - 5460*a*b^35*c*x + 10920*a^2*b^34*d*x - 18200*a^3*b^33*e*x 
+ 27300*a^4*b^32*f*x)/b^39
 

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.38 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^{10}\,\left (\frac {e}{10\,b^3}-\frac {3\,a\,f}{10\,b^4}\right )+x^4\,\left (\frac {c}{4\,b^3}-\frac {a^3\,f}{4\,b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{4\,b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{4\,b}\right )+\frac {x\,\left (\frac {17\,f\,a^6}{9}-\frac {14\,e\,a^5\,b}{9}+\frac {11\,d\,a^4\,b^2}{9}-\frac {8\,c\,a^3\,b^3}{9}\right )-x^4\,\left (-\frac {37\,f\,a^5\,b}{18}+\frac {31\,e\,a^4\,b^2}{18}-\frac {25\,d\,a^3\,b^3}{18}+\frac {19\,c\,a^2\,b^4}{18}\right )}{a^2\,b^7+2\,a\,b^8\,x^3+b^9\,x^6}-x\,\left (\frac {3\,a\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )}{b}-\frac {3\,a^2\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b^2}+\frac {a^3\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^3}\right )-x^7\,\left (\frac {3\,a^2\,f}{7\,b^5}-\frac {d}{7\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{7\,b}\right )+\frac {f\,x^{13}}{13\,b^3}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-152\,f\,a^3+104\,e\,a^2\,b-65\,d\,a\,b^2+35\,c\,b^3\right )}{27\,b^{22/3}}+\frac {a^{4/3}\,\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 (-152\,f\,a^3+104\,e\,a^2\,b-65\,d\,a\,b^2+35\,c\,b^3\right )}{27\,b^{22/3}}-\frac {a^{4/3}\,\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 (-152\,f\,a^3+104\,e\,a^2\,b-65\,d\,a\,b^2+35\,c\,b^3\right )}{27\,b^{22/3}} \] Input:

int((x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x)
 

Output:

x^10*(e/(10*b^3) - (3*a*f)/(10*b^4)) + x^4*(c/(4*b^3) - (a^3*f)/(4*b^6) - 
(3*a^2*(e/b^3 - (3*a*f)/b^4))/(4*b^2) + (3*a*((3*a^2*f)/b^5 - d/b^3 + (3*a 
*(e/b^3 - (3*a*f)/b^4))/b))/(4*b)) + (x*((17*a^6*f)/9 - (8*a^3*b^3*c)/9 + 
(11*a^4*b^2*d)/9 - (14*a^5*b*e)/9) - x^4*((19*a^2*b^4*c)/18 - (25*a^3*b^3* 
d)/18 + (31*a^4*b^2*e)/18 - (37*a^5*b*f)/18))/(a^2*b^7 + b^9*x^6 + 2*a*b^8 
*x^3) - x*((3*a*(c/b^3 - (a^3*f)/b^6 - (3*a^2*(e/b^3 - (3*a*f)/b^4))/b^2 + 
 (3*a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b))/b - (3* 
a^2*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b^2 + (a^3*(e 
/b^3 - (3*a*f)/b^4))/b^3) - x^7*((3*a^2*f)/(7*b^5) - d/(7*b^3) + (3*a*(e/b 
^3 - (3*a*f)/b^4))/(7*b)) + (f*x^13)/(13*b^3) + (a^(4/3)*log(b^(1/3)*x + a 
^(1/3))*(35*b^3*c - 152*a^3*f - 65*a*b^2*d + 104*a^2*b*e))/(27*b^(22/3)) + 
 (a^(4/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 
- 1/2)*(35*b^3*c - 152*a^3*f - 65*a*b^2*d + 104*a^2*b*e))/(27*b^(22/3)) - 
(a^(4/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 
 1/2)*(35*b^3*c - 152*a^3*f - 65*a*b^2*d + 104*a^2*b*e))/(27*b^(22/3))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1261, normalized size of antiderivative = 3.03 \[ \int \frac {x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)
 

Output:

(276640*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)) 
)*a**6*f - 189280*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3 
)*sqrt(3)))*a**5*b*e + 553280*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3) 
*x)/(a**(1/3)*sqrt(3)))*a**5*b*f*x**3 + 118300*a**(1/3)*sqrt(3)*atan((a**( 
1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4*b**2*d - 378560*a**(1/3)*sqr 
t(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4*b**2*e*x**3 + 
 276640*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**6 - 63700*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x) 
/(a**(1/3)*sqrt(3)))*a**3*b**3*c + 236600*a**(1/3)*sqrt(3)*atan((a**(1/3) 
- 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**3*d*x**3 - 189280*a**(1/3)*sqr 
t(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**3*e*x**6 - 
 127400*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**3 + 118300*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**6 - 63700*a**(1/3)*sqrt(3)*atan((a**( 
1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**5*c*x**6 + 138320*a**(1/3)*l 
og(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**6*f - 94640*a**(1/3) 
*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**5*b*e + 276640*a** 
(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**5*b*f*x**3 + 
59150*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*b* 
*2*d - 189280*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x*...