Integrand size = 30, antiderivative size = 336 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^4}{4 b^4}+\frac {f x^7}{7 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac {\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}-\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{16/3}}+\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac {\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{2/3} b^{16/3}} \] Output:
(6*a^2*f-3*a*b*e+b^2*d)*x/b^5+1/4*(-3*a*f+b*e)*x^4/b^4+1/7*f*x^7/b^3+1/6*a *(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^5/(b*x^3+a)^2-1/18*(-25*a^3*f+19*a^2*b *e-13*a*b^2*d+7*b^3*c)*x/b^5/(b*x^3+a)-1/27*(-65*a^3*f+35*a^2*b*e-14*a*b^2 *d+2*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(2 /3)/b^(16/3)+1/27*(-65*a^3*f+35*a^2*b*e-14*a*b^2*d+2*b^3*c)*ln(a^(1/3)+b^( 1/3)*x)/a^(2/3)/b^(16/3)-1/54*(-65*a^3*f+35*a^2*b*e-14*a*b^2*d+2*b^3*c)*ln (a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(16/3)
Time = 0.25 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.96 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {756 \sqrt [3]{b} \left (b^2 d-3 a b e+6 a^2 f\right ) x+189 b^{4/3} (b e-3 a f) x^4+108 b^{7/3} f x^7+\frac {126 a \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{\left (a+b x^3\right )^2}-\frac {42 \sqrt [3]{b} \left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{a+b x^3}+\frac {28 \sqrt {3} \left (-2 b^3 c+14 a b^2 d-35 a^2 b e+65 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {28 \left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {14 \left (-2 b^3 c+14 a b^2 d-35 a^2 b e+65 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{756 b^{16/3}} \] Input:
Integrate[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
Output:
(756*b^(1/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x + 189*b^(4/3)*(b*e - 3*a*f)*x^4 + 108*b^(7/3)*f*x^7 + (126*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)* x)/(a + b*x^3)^2 - (42*b^(1/3)*(7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3 *f)*x)/(a + b*x^3) + (28*Sqrt[3]*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65* a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*Log[a^(2/3) - a^(1/3) *b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(756*b^(16/3))
Time = 1.51 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.08, 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.
| \(\displaystyle \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 a b^4 f x^{12}-6 a b^3 (b e-a f) x^9-6 a b^2 \left (f a^2-b e a+b^2 d\right ) x^6-6 a b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^3+a^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right )}{\left (b x^3+a\right )^2}dx}{6 a b^5}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {2 \left (9 a^2 b^7 f x^9+9 a^2 b^6 (b e-2 a f) x^6+9 a^2 b^5 \left (3 f a^2-2 b e a+b^2 d\right ) x^3+a^2 b^4 \left (-11 f a^3+8 b e a^2-5 b^2 d a+2 b^3 c\right )\right )}{b x^3+a}dx}{3 a b^4}}{6 a b^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {2 \int \frac {9 a^2 b^7 f x^9+9 a^2 b^6 (b e-2 a f) x^6+9 a^2 b^5 \left (3 f a^2-2 b e a+b^2 d\right ) x^3+a^2 b^4 \left (-11 f a^3+8 b e a^2-5 b^2 d a+2 b^3 c\right )}{b x^3+a}dx}{3 a b^4}}{6 a b^5}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {2 \int \left (9 a^2 b^6 f x^6+9 a^2 b^5 (b e-3 a f) x^3+9 a^2 b^4 \left (6 f a^2-3 b e a+b^2 d\right )+\frac {2 a^2 c b^7-14 a^3 d b^6+35 a^4 e b^5-65 a^5 f b^4}{b x^3+a}\right )dx}{3 a b^4}}{6 a b^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {\frac {a x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{3 \left (a+b x^3\right )}-\frac {2 \left (\frac {9}{7} a^2 b^6 f x^7+\frac {9}{4} a^2 b^5 x^4 (b e-3 a f)+9 a^2 b^4 x \left (6 a^2 f-3 a b e+b^2 d\right )-\frac {a^{4/3} b^{11/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{\sqrt {3}}-\frac {1}{6} a^{4/3} b^{11/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )+\frac {1}{3} a^{4/3} b^{11/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )\right )}{3 a b^4}}{6 a b^5}\) |
Input:
Int[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
Output:
(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^5*(a + b*x^3)^2) - ((a*(7*b ^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(3*(a + b*x^3)) - (2*(9*a^2* b^4*(b^2*d - 3*a*b*e + 6*a^2*f)*x + (9*a^2*b^5*(b*e - 3*a*f)*x^4)/4 + (9*a ^2*b^6*f*x^7)/7 - (a^(4/3)*b^(11/3)*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 6 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/Sqrt[3] + (a^( 4/3)*b^(11/3)*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/3 - (a^(4/3)*b^(11/3)*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65 *a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/6))/(3*a*b^4))/(6* a*b^5)
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 = 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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(\frac {f \,x^{7}}{7 b^{3}}-\frac {3 x^{4} a f}{4 b^{4}}+\frac {x^{4} e}{4 b^{3}}+\frac {6 a^{2} x f}{b^{5}}-\frac {3 a x e}{b^{4}}+\frac {d x}{b^{3}}+\frac {\left (\frac {25}{18} a^{3} b f -\frac {19}{18} a^{2} b^{2} e +\frac {13}{18} a \,b^{3} d -\frac {7}{18} b^{4} c \right ) x^{4}+\frac {a \left (11 f \,a^{3}-8 e \,a^{2} b +5 d a \,b^{2}-2 b^{3} c \right ) x}{9}}{b^{5} \left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (-65 f \,a^{3}+35 e \,a^{2} b -14 d a \,b^{2}+2 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{6}}\) | \(186\) |
| default | \(\frac {\frac {1}{7} b^{2} f \,x^{7}-\frac {3}{4} a b f \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+6 f \,a^{2} x -3 a b e x +b^{2} d x}{b^{5}}-\frac {\frac {\left (-\frac {25}{18} a^{3} b f +\frac {19}{18} a^{2} b^{2} e -\frac {13}{18} a \,b^{3} d +\frac {7}{18} b^{4} c \right ) x^{4}-\frac {a \left (11 f \,a^{3}-8 e \,a^{2} b +5 d a \,b^{2}-2 b^{3} c \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (65 f \,a^{3}-35 e \,a^{2} b +14 d a \,b^{2}-2 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}}{b^{5}}\) | \(253\) |
Input:
int(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/7*f*x^7/b^3-3/4/b^4*x^4*a*f+1/4/b^3*x^4*e+6/b^5*a^2*x*f-3/b^4*a*x*e+1/b^ 3*d*x+((25/18*a^3*b*f-19/18*a^2*b^2*e+13/18*a*b^3*d-7/18*b^4*c)*x^4+1/9*a* (11*a^3*f-8*a^2*b*e+5*a*b^2*d-2*b^3*c)*x)/b^5/(b*x^3+a)^2+1/27/b^6*sum((-6 5*a^3*f+35*a^2*b*e-14*a*b^2*d+2*b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (291) = 582\).
Time = 0.11 (sec) , antiderivative size = 1318, normalized size of antiderivative = 3.92 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[1/756*(108*a^2*b^5*f*x^13 + 27*(7*a^2*b^5*e - 13*a^3*b^4*f)*x^10 + 54*(14 *a^2*b^5*d - 35*a^3*b^4*e + 65*a^4*b^3*f)*x^7 - 147*(2*a^2*b^5*c - 14*a^3* b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^4 - 42*sqrt(1/3)*(2*a^3*b^4*c - 14* a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f + (2*a*b^6*c - 14*a^2*b^5*d + 35*a^3 *b^4*e - 65*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^3)*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)) - 14*((2*b^5*c - 14*a*b^4*d + 35* a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*( -a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*((2* b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a ^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^ 2*e - 65*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) - 84*(2* a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f)*x)/(a^2*b^8*x^6 + 2* a^3*b^7*x^3 + a^4*b^6), 1/756*(108*a^2*b^5*f*x^13 + 27*(7*a^2*b^5*e - 13*a ^3*b^4*f)*x^10 + 54*(14*a^2*b^5*d - 35*a^3*b^4*e + 65*a^4*b^3*f)*x^7 - 147 *(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^4 + 84*sqrt( 1/3)*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f + (2*a*b^6*c - 14*a^2*b^5*d + 35*a^3*b^4*e - 65*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c - 14...
Timed out. \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**6*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.97 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {{\left (7 \, b^{4} c - 13 \, a b^{3} d + 19 \, a^{2} b^{2} e - 25 \, a^{3} b f\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f\right )} x}{18 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {4 \, b^{2} f x^{7} + 7 \, {\left (b^{2} e - 3 \, a b f\right )} x^{4} + 28 \, {\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x}{28 \, b^{5}} + \frac {\sqrt {3} {\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/18*((7*b^4*c - 13*a*b^3*d + 19*a^2*b^2*e - 25*a^3*b*f)*x^4 + 2*(2*a*b^3 *c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f)*x)/(b^7*x^6 + 2*a*b^6*x^3 + a^2*b ^5) + 1/28*(4*b^2*f*x^7 + 7*(b^2*e - 3*a*b*f)*x^4 + 28*(b^2*d - 3*a*b*e + 6*a^2*f)*x)/b^5 + 1/27*sqrt(3)*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3 *f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(2/3)) - 1/54*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*log(x^2 - x*(a/b)^(1 /3) + (a/b)^(2/3))/(b^6*(a/b)^(2/3)) + 1/27*(2*b^3*c - 14*a*b^2*d + 35*a^2 *b*e - 65*a^3*f)*log(x + (a/b)^(1/3))/(b^6*(a/b)^(2/3))
Time = 0.14 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.01 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{4}} - \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{4}} - \frac {{\left (2 \, b^{3} c - 14 \, a b^{2} d + 35 \, a^{2} b e - 65 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{5}} - \frac {7 \, b^{4} c x^{4} - 13 \, a b^{3} d x^{4} + 19 \, a^{2} b^{2} e x^{4} - 25 \, a^{3} b f x^{4} + 4 \, a b^{3} c x - 10 \, a^{2} b^{2} d x + 16 \, a^{3} b e x - 22 \, a^{4} f x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{5}} + \frac {4 \, b^{18} f x^{7} + 7 \, b^{18} e x^{4} - 21 \, a b^{17} f x^{4} + 28 \, b^{18} d x - 84 \, a b^{17} e x + 168 \, a^{2} b^{16} f x}{28 \, b^{21}} \] Input:
integrate(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")
Output:
-1/27*sqrt(3)*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*arctan(1/3*sq rt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*b^4) - 1/54*(2*b^ 3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b )^(2/3))/((-a*b^2)^(2/3)*b^4) - 1/27*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/18*(7*b^4*c* x^4 - 13*a*b^3*d*x^4 + 19*a^2*b^2*e*x^4 - 25*a^3*b*f*x^4 + 4*a*b^3*c*x - 1 0*a^2*b^2*d*x + 16*a^3*b*e*x - 22*a^4*f*x)/((b*x^3 + a)^2*b^5) + 1/28*(4*b ^18*f*x^7 + 7*b^18*e*x^4 - 21*a*b^17*f*x^4 + 28*b^18*d*x - 84*a*b^17*e*x + 168*a^2*b^16*f*x)/b^21
Time = 6.45 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^4\,\left (\frac {e}{4\,b^3}-\frac {3\,a\,f}{4\,b^4}\right )-x\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )-\frac {x^4\,\left (-\frac {25\,f\,a^3\,b}{18}+\frac {19\,e\,a^2\,b^2}{18}-\frac {13\,d\,a\,b^3}{18}+\frac {7\,c\,b^4}{18}\right )-x\,\left (\frac {11\,f\,a^4}{9}-\frac {8\,e\,a^3\,b}{9}+\frac {5\,d\,a^2\,b^2}{9}-\frac {2\,c\,a\,b^3}{9}\right )}{a^2\,b^5+2\,a\,b^6\,x^3+b^7\,x^6}+\frac {f\,x^7}{7\,b^3}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-65\,f\,a^3+35\,e\,a^2\,b-14\,d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{2/3}\,b^{16/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-65\,f\,a^3+35\,e\,a^2\,b-14\,d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{2/3}\,b^{16/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-65\,f\,a^3+35\,e\,a^2\,b-14\,d\,a\,b^2+2\,c\,b^3\right )}{27\,a^{2/3}\,b^{16/3}} \] Input:
int((x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x)
Output:
x^4*(e/(4*b^3) - (3*a*f)/(4*b^4)) - x*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b) - (x^4*((7*b^4*c)/18 + (19*a^2*b^2*e)/18 - (13*a*b^3*d )/18 - (25*a^3*b*f)/18) - x*((11*a^4*f)/9 + (5*a^2*b^2*d)/9 - (2*a*b^3*c)/ 9 - (8*a^3*b*e)/9))/(a^2*b^5 + b^7*x^6 + 2*a*b^6*x^3) + (f*x^7)/(7*b^3) + (log(b^(1/3)*x + a^(1/3))*(2*b^3*c - 65*a^3*f - 14*a*b^2*d + 35*a^2*b*e))/ (27*a^(2/3)*b^(16/3)) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*( (3^(1/2)*1i)/2 - 1/2)*(2*b^3*c - 65*a^3*f - 14*a*b^2*d + 35*a^2*b*e))/(27* a^(2/3)*b^(16/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^( 1/2)*1i)/2 + 1/2)*(2*b^3*c - 65*a^3*f - 14*a*b^2*d + 35*a^2*b*e))/(27*a^(2 /3)*b^(16/3))
Time = 0.15 (sec) , antiderivative size = 1180, normalized size of antiderivative = 3.51 \[ \int \frac {x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)
Output:
(1820*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* a**5*f - 980*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqr t(3)))*a**4*b*e + 3640*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a* *(1/3)*sqrt(3)))*a**4*b*f*x**3 + 392*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b **(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2*d - 1960*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**3 + 1820*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**6 - 56*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3 )))*a**2*b**3*c + 784*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**3 - 980*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**6 - 112*a**(1/3)*sqrt(3)*a tan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*c*x**3 + 392*a**( 1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*d*x **6 - 56*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3) ))*b**5*c*x**6 + 910*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3 )*x**2)*a**5*f - 490*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3 )*x**2)*a**4*b*e + 1820*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**( 2/3)*x**2)*a**4*b*f*x**3 + 196*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b**2*d - 980*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**3 + 910*a**(1/3)*log(a**(2/3) - b...