Integrand size = 30, antiderivative size = 274 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac {(b e-a f) x^7}{7 b^2}+\frac {f x^{10}}{10 b}+\frac {\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{13/3}}-\frac {\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{13/3}}+\frac {\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{13/3}} \] Output:
(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^4+1/4*(a^2*f-a*b*e+b^2*d)*x^4/b^3+1/7*( -a*f+b*e)*x^7/b^2+1/10*f*x^10/b+1/3*a^(1/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c) *arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b^(13/3)-1/3*a^ (1/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/b^(13/3)+1/6*a^ (1/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)* x^2)/b^(13/3)
Time = 0.07 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {420 \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x+105 b^{4/3} \left (b^2 d-a b e+a^2 f\right ) x^4+60 b^{7/3} (b e-a f) x^7+42 b^{10/3} f x^{10}-140 \sqrt {3} \sqrt [3]{a} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+140 \sqrt [3]{a} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-70 \sqrt [3]{a} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{420 b^{13/3}} \] Input:
Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
Output:
(420*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x + 105*b^(4/3)*(b^2*d - a*b*e + a^2*f)*x^4 + 60*b^(7/3)*(b*e - a*f)*x^7 + 42*b^(10/3)*f*x^10 - 140 *Sqrt[3]*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^( 1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^ 3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(420*b^(13/3))
Time = 0.84 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2375, 27, 1812, 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.
| \(\displaystyle \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\int \frac {10 x^3 \left ((b e-a f) x^6+b d x^3+b c\right )}{b x^3+a}dx}{10 b}+\frac {f x^{10}}{10 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^3 \left ((b e-a f) x^6+b d x^3+b c\right )}{b x^3+a}dx}{b}+\frac {f x^{10}}{10 b}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\int \left (\frac {(b e-a f) x^6}{b}+\frac {\left (f a^2-b e a+b^2 d\right ) x^3}{b^2}+c-\frac {a \left (f a^2-b e a+b^2 d\right )}{b^3}+\frac {f a^4-b e a^3+b^2 d a^2-b^3 c a}{b^3 \left (b x^3+a\right )}\right )dx}{b}+\frac {f x^{10}}{10 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^2}+x \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} b^{10/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{10/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{10/3}}+\frac {x^7 (b e-a f)}{7 b}}{b}+\frac {f x^{10}}{10 b}\) |
Input:
Int[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
Output:
(f*x^10)/(10*b) + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x + ((b^2*d - a*b *e + a^2*f)*x^4)/(4*b^2) + ((b*e - a*f)*x^7)/(7*b) + (a^(1/3)*(b^3*c - a*b ^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]) /(Sqrt[3]*b^(10/3)) - (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^( 1/3) + b^(1/3)*x])/(3*b^(10/3)) + (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^ 3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3)))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( (d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {f \,x^{10}}{10 b}-\frac {x^{7} f a}{7 b^{2}}+\frac {x^{7} e}{7 b}+\frac {x^{4} f \,a^{2}}{4 b^{3}}-\frac {x^{4} e a}{4 b^{2}}+\frac {x^{4} d}{4 b}-\frac {x f \,a^{3}}{b^{4}}+\frac {x e \,a^{2}}{b^{3}}-\frac {a d x}{b^{2}}+\frac {c x}{b}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b^{5}}\) | \(146\) |
| default | \(-\frac {-\frac {1}{10} b^{3} f \,x^{10}+\frac {1}{7} x^{7} a \,b^{2} f -\frac {1}{7} x^{7} b^{3} e -\frac {1}{4} x^{4} a^{2} b f +\frac {1}{4} x^{4} a \,b^{2} e -\frac {1}{4} b^{3} d \,x^{4}+f \,a^{3} x -e \,a^{2} b x +d a \,b^{2} x -b^{3} c x}{b^{4}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a \left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right )}{b^{4}}\) | \(213\) |
Input:
int(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/10*f*x^10/b-1/7/b^2*x^7*f*a+1/7/b*x^7*e+1/4/b^3*x^4*f*a^2-1/4/b^2*x^4*e* a+1/4/b*x^4*d-1/b^4*x*f*a^3+1/b^3*x*e*a^2-1/b^2*a*d*x+c*x/b+1/3/b^5*a*sum( (a^3*f-a^2*b*e+a*b^2*d-b^3*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {42 \, b^{3} f x^{10} + 60 \, {\left (b^{3} e - a b^{2} f\right )} x^{7} + 105 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{4} - 140 \, \sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 70 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 140 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 420 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{420 \, b^{4}} \] Input:
integrate(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="fricas")
Output:
1/420*(42*b^3*f*x^10 + 60*(b^3*e - a*b^2*f)*x^7 + 105*(b^3*d - a*b^2*e + a ^2*b*f)*x^4 - 140*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a/b)^(1/3)* arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3)*a)/a) + 70*(b^3*c - a*b^2* d + a^2*b*e - a^3*f)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 420*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4
Time = 0.87 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.37 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=x^{7} \left (- \frac {a f}{7 b^{2}} + \frac {e}{7 b}\right ) + x^{4} \left (\frac {a^{2} f}{4 b^{3}} - \frac {a e}{4 b^{2}} + \frac {d}{4 b}\right ) + x \left (- \frac {a^{3} f}{b^{4}} + \frac {a^{2} e}{b^{3}} - \frac {a d}{b^{2}} + \frac {c}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{13} - a^{10} f^{3} + 3 a^{9} b e f^{2} - 3 a^{8} b^{2} d f^{2} - 3 a^{8} b^{2} e^{2} f + 3 a^{7} b^{3} c f^{2} + 6 a^{7} b^{3} d e f + a^{7} b^{3} e^{3} - 6 a^{6} b^{4} c e f - 3 a^{6} b^{4} d^{2} f - 3 a^{6} b^{4} d e^{2} + 6 a^{5} b^{5} c d f + 3 a^{5} b^{5} c e^{2} + 3 a^{5} b^{5} d^{2} e - 3 a^{4} b^{6} c^{2} f - 6 a^{4} b^{6} c d e - a^{4} b^{6} d^{3} + 3 a^{3} b^{7} c^{2} e + 3 a^{3} b^{7} c d^{2} - 3 a^{2} b^{8} c^{2} d + a b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {3 t b^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac {f x^{10}}{10 b} \] Input:
integrate(x**3*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
Output:
x**7*(-a*f/(7*b**2) + e/(7*b)) + x**4*(a**2*f/(4*b**3) - a*e/(4*b**2) + d/ (4*b)) + x*(-a**3*f/b**4 + a**2*e/b**3 - a*d/b**2 + c/b) + RootSum(27*_t** 3*b**13 - a**10*f**3 + 3*a**9*b*e*f**2 - 3*a**8*b**2*d*f**2 - 3*a**8*b**2* e**2*f + 3*a**7*b**3*c*f**2 + 6*a**7*b**3*d*e*f + a**7*b**3*e**3 - 6*a**6* b**4*c*e*f - 3*a**6*b**4*d**2*f - 3*a**6*b**4*d*e**2 + 6*a**5*b**5*c*d*f + 3*a**5*b**5*c*e**2 + 3*a**5*b**5*d**2*e - 3*a**4*b**6*c**2*f - 6*a**4*b** 6*c*d*e - a**4*b**6*d**3 + 3*a**3*b**7*c**2*e + 3*a**3*b**7*c*d**2 - 3*a** 2*b**8*c**2*d + a*b**9*c**3, Lambda(_t, _t*log(3*_t*b**4/(a**3*f - a**2*b* e + a*b**2*d - b**3*c) + x))) + f*x**10/(10*b)
Time = 0.12 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {14 \, b^{3} f x^{10} + 20 \, {\left (b^{3} e - a b^{2} f\right )} x^{7} + 35 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{4} + 140 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{140 \, b^{4}} - \frac {\sqrt {3} {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")
Output:
1/140*(14*b^3*f*x^10 + 20*(b^3*e - a*b^2*f)*x^7 + 35*(b^3*d - a*b^2*e + a^ 2*b*f)*x^4 + 140*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4 - 1/3*sqrt(3)* (a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1 /3))/(a/b)^(1/3))/(b^5*(a/b)^(2/3)) + 1/6*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^5*(a/b)^(2/3)) - 1/3*(a* b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(x + (a/b)^(1/3))/(b^5*(a/b)^(2/3) )
Time = 0.13 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{5}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{5}} + \frac {{\left (a b^{9} c - a^{2} b^{8} d + a^{3} b^{7} e - a^{4} b^{6} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac {14 \, b^{9} f x^{10} + 20 \, b^{9} e x^{7} - 20 \, a b^{8} f x^{7} + 35 \, b^{9} d x^{4} - 35 \, a b^{8} e x^{4} + 35 \, a^{2} b^{7} f x^{4} + 140 \, b^{9} c x - 140 \, a b^{8} d x + 140 \, a^{2} b^{7} e x - 140 \, a^{3} b^{6} f x}{140 \, b^{10}} \] Input:
integrate(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="giac")
Output:
-1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d + (-a*b^2)^(1/ 3)*a^2*b*e - (-a*b^2)^(1/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3)) /(-a/b)^(1/3))/b^5 - 1/6*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d + (-a*b^2)^(1/3)*a^2*b*e - (-a*b^2)^(1/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 + 1/3*(a*b^9*c - a^2*b^8*d + a^3*b^7*e - a^4*b^6*f)*(-a/ b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^10) + 1/140*(14*b^9*f*x^10 + 20*b ^9*e*x^7 - 20*a*b^8*f*x^7 + 35*b^9*d*x^4 - 35*a*b^8*e*x^4 + 35*a^2*b^7*f*x ^4 + 140*b^9*c*x - 140*a*b^8*d*x + 140*a^2*b^7*e*x - 140*a^3*b^6*f*x)/b^10
Time = 6.01 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=x^7\,\left (\frac {e}{7\,b}-\frac {a\,f}{7\,b^2}\right )+x^4\,\left (\frac {d}{4\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{4\,b}\right )+x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )+\frac {f\,x^{10}}{10\,b}-\frac {a^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{13/3}}-\frac {a^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{13/3}}+\frac {a^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{13/3}} \] Input:
int((x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x)
Output:
x^7*(e/(7*b) - (a*f)/(7*b^2)) + x^4*(d/(4*b) - (a*(e/b - (a*f)/b^2))/(4*b) ) + x*(c/b - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/b) + (f*x^10)/(10*b) - (a ^(1/3)*log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^ (13/3)) - (a^(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)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^(13/3)) + (a^(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)* (b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^(13/3))
Time = 0.16 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.46 \[ \int \frac {x^3 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {-140 a^{\frac {10}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) f +140 a^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b e -140 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} d +140 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{3} c -70 a^{\frac {10}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) f +70 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b e -70 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} d +70 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{3} c +140 a^{\frac {10}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) f -140 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b e +140 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} d -140 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{3} c -420 b^{\frac {1}{3}} a^{3} f x +420 b^{\frac {4}{3}} a^{2} e x +105 b^{\frac {4}{3}} a^{2} f \,x^{4}-420 b^{\frac {7}{3}} a d x -105 b^{\frac {7}{3}} a e \,x^{4}-60 b^{\frac {7}{3}} a f \,x^{7}+420 b^{\frac {10}{3}} c x +105 b^{\frac {10}{3}} d \,x^{4}+60 b^{\frac {10}{3}} e \,x^{7}+42 b^{\frac {10}{3}} f \,x^{10}}{420 b^{\frac {13}{3}}} \] Input:
int(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)
Output:
( - 140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)) )*a**3*f + 140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a**2*b*e - 140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a **(1/3)*sqrt(3)))*a*b**2*d + 140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1 /3)*x)/(a**(1/3)*sqrt(3)))*b**3*c - 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a* *(1/3)*x + b**(2/3)*x**2)*a**3*f + 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a** (1/3)*x + b**(2/3)*x**2)*a**2*b*e - 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a* *(1/3)*x + b**(2/3)*x**2)*a*b**2*d + 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a **(1/3)*x + b**(2/3)*x**2)*b**3*c + 140*a**(1/3)*log(a**(1/3) + b**(1/3)*x )*a**3*f - 140*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**2*b*e + 140*a**(1/3) *log(a**(1/3) + b**(1/3)*x)*a*b**2*d - 140*a**(1/3)*log(a**(1/3) + b**(1/3 )*x)*b**3*c - 420*b**(1/3)*a**3*f*x + 420*b**(1/3)*a**2*b*e*x + 105*b**(1/ 3)*a**2*b*f*x**4 - 420*b**(1/3)*a*b**2*d*x - 105*b**(1/3)*a*b**2*e*x**4 - 60*b**(1/3)*a*b**2*f*x**7 + 420*b**(1/3)*b**3*c*x + 105*b**(1/3)*b**3*d*x* *4 + 60*b**(1/3)*b**3*e*x**7 + 42*b**(1/3)*b**3*f*x**10)/(420*b**(1/3)*b** 4)