Integrand size = 38, antiderivative size = 311 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {(b c-a f) x}{b^2}+\frac {(b d-a g) x^2}{2 b^2}+\frac {(b e-a h) x^3}{3 b^2}+\frac {f x^4}{4 b}+\frac {g x^5}{5 b}+\frac {h x^6}{6 b}+\frac {\sqrt [3]{a} \left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {\sqrt [3]{a} \left (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac {a (b e-a h) \log \left (a+b x^3\right )}{3 b^3} \] Output:
(-a*f+b*c)*x/b^2+1/2*(-a*g+b*d)*x^2/b^2+1/3*(-a*h+b*e)*x^3/b^2+1/4*f*x^4/b +1/5*g*x^5/b+1/6*h*x^6/b+1/3*a^(1/3)*(b^(4/3)*c+a^(1/3)*b*d-a*b^(1/3)*f-a^ (4/3)*g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b^(8/3) -1/3*a^(1/3)*(b^(1/3)*(-a*f+b*c)-a^(1/3)*(-a*g+b*d))*ln(a^(1/3)+b^(1/3)*x) /b^(8/3)+1/6*a^(1/3)*(b*c-a*f-a^(1/3)*(-a*g+b*d)/b^(1/3))*ln(a^(2/3)-a^(1/ 3)*b^(1/3)*x+b^(2/3)*x^2)/b^(7/3)-1/3*a*(-a*h+b*e)*ln(b*x^3+a)/b^3
Time = 0.16 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {60 b (b c-a f) x+30 b (b d-a g) x^2+20 b (b e-a h) x^3+15 b^2 f x^4+12 b^2 g x^5+10 b^2 h x^6-20 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (-b^{4/3} c-\sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-20 \sqrt [3]{a} \sqrt [3]{b} \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+10 \sqrt [3]{a} \sqrt [3]{b} \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 a (-b e+a h) \log \left (a+b x^3\right )}{60 b^3} \] Input:
Integrate[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]
Output:
(60*b*(b*c - a*f)*x + 30*b*(b*d - a*g)*x^2 + 20*b*(b*e - a*h)*x^3 + 15*b^2 *f*x^4 + 12*b^2*g*x^5 + 10*b^2*h*x^6 - 20*Sqrt[3]*a^(1/3)*b^(1/3)*(-(b^(4/ 3)*c) - a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a ^(1/3))/Sqrt[3]] - 20*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3) *f + a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x] + 10*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*a*(-(b*e) + a*h)*Log[a + b*x^3])/(60*b^3)
Time = 1.80 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2375, 27, 2375, 27, 2375, 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.
| \(\displaystyle \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\int \frac {6 x^3 \left (b g x^4+b f x^3+(b e-a h) x^2+b d x+b c\right )}{b x^3+a}dx}{6 b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^3 \left (b g x^4+b f x^3+(b e-a h) x^2+b d x+b c\right )}{b x^3+a}dx}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {\int \frac {5 x^3 \left (b^2 f x^3+b (b e-a h) x^2+b (b d-a g) x+b^2 c\right )}{b x^3+a}dx}{5 b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^3 \left (b^2 f x^3+b (b e-a h) x^2+b (b d-a g) x+b^2 c\right )}{b x^3+a}dx}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {4 x^3 \left ((b e-a h) x^2 b^2+(b c-a f) b^2+(b d-a g) x b^2\right )}{b x^3+a}dx}{4 b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {x^3 \left ((b e-a h) x^2 b^2+(b c-a f) b^2+(b d-a g) x b^2\right )}{b x^3+a}dx}{b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {\frac {\frac {\int \left (b (b e-a h) x^2+b (b d-a g) x+b (b c-a f)-\frac {a b (b e-a h) x^2+a b (b d-a g) x+a b (b c-a f)}{b x^3+a}\right )dx}{b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt [3]{a} \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt {3}}+\frac {1}{6} \sqrt [3]{a} \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )-\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )+b x (b c-a f)+\frac {1}{2} b x^2 (b d-a g)+\frac {1}{3} b x^3 (b e-a h)-\frac {1}{3} a (b e-a h) \log \left (a+b x^3\right )}{b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
Input:
Int[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]
Output:
(h*x^6)/(6*b) + ((g*x^5)/5 + ((b*f*x^4)/4 + (b*(b*c - a*f)*x + (b*(b*d - a *g)*x^2)/2 + (b*(b*e - a*h)*x^3)/3 + (a^(1/3)*b^(1/3)*(b^(4/3)*c + a^(1/3) *b*d - a*b^(1/3)*f - a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^ (1/3))])/Sqrt[3] - (a^(1/3)*b^(1/3)*(b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*Log[a^(1/3) + b^(1/3)*x])/3 + (a^(1/3)*b^(1/3)*(b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/6 - (a*(b*e - a*h)*Log[a + b*x^3])/3)/b)/b)/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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 = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {h \,x^{6}}{6 b}+\frac {g \,x^{5}}{5 b}+\frac {f \,x^{4}}{4 b}-\frac {a h \,x^{3}}{3 b^{2}}+\frac {e \,x^{3}}{3 b}-\frac {a g \,x^{2}}{2 b^{2}}+\frac {d \,x^{2}}{2 b}-\frac {a f x}{b^{2}}+\frac {c x}{b}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\left (a h -b e \right ) \textit {\_R}^{2}+\left (a g -b d \right ) \textit {\_R} +a f -c b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b^{3}}\) | \(138\) |
| default | \(-\frac {-\frac {1}{6} b h \,x^{6}-\frac {1}{5} b g \,x^{5}-\frac {1}{4} b f \,x^{4}+\frac {1}{3} a h \,x^{3}-\frac {1}{3} b e \,x^{3}+\frac {1}{2} a g \,x^{2}-\frac {1}{2} b d \,x^{2}+a f x -c b x}{b^{2}}+\frac {\left (\left (a f -c b \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a g -b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a h -b e \right ) \ln \left (b \,x^{3}+a \right )}{3 b}\right ) a}{b^{2}}\) | \(291\) |
Input:
int(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/6*h*x^6/b+1/5*g*x^5/b+1/4*f*x^4/b-1/3/b^2*a*h*x^3+1/3*e*x^3/b-1/2/b^2*a* g*x^2+1/2*d*x^2/b-1/b^2*a*f*x+c*x/b+1/3/b^3*a*sum(((a*h-b*e)*_R^2+(a*g-b*d )*_R+a*f-c*b)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 15451, normalized size of antiderivative = 49.68 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="fric as")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (280) = 560\).
Time = 48.47 (sec) , antiderivative size = 845, normalized size of antiderivative = 2.72 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=x^{3} \left (- \frac {a h}{3 b^{2}} + \frac {e}{3 b}\right ) + x^{2} \left (- \frac {a g}{2 b^{2}} + \frac {d}{2 b}\right ) + x \left (- \frac {a f}{b^{2}} + \frac {c}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{9} + t^{2} \left (- 27 a^{2} b^{6} h + 27 a b^{7} e\right ) + t \left (9 a^{4} b^{3} h^{2} - 18 a^{3} b^{4} e h + 9 a^{3} b^{4} f g - 9 a^{2} b^{5} c g - 9 a^{2} b^{5} d f + 9 a^{2} b^{5} e^{2} + 9 a b^{6} c d\right ) - a^{6} h^{3} + 3 a^{5} b e h^{2} - 3 a^{5} b f g h + a^{5} b g^{3} + 3 a^{4} b^{2} c g h + 3 a^{4} b^{2} d f h - 3 a^{4} b^{2} d g^{2} - 3 a^{4} b^{2} e^{2} h + 3 a^{4} b^{2} e f g - a^{4} b^{2} f^{3} - 3 a^{3} b^{3} c d h - 3 a^{3} b^{3} c e g + 3 a^{3} b^{3} c f^{2} + 3 a^{3} b^{3} d^{2} g - 3 a^{3} b^{3} d e f + a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + 3 a^{2} b^{4} c d e - a^{2} b^{4} d^{3} + a b^{5} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} a b^{6} g - 9 t^{2} b^{7} d - 6 t a^{3} b^{3} g h + 6 t a^{2} b^{4} d h + 6 t a^{2} b^{4} e g + 3 t a^{2} b^{4} f^{2} - 6 t a b^{5} c f - 6 t a b^{5} d e + 3 t b^{6} c^{2} + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h - a^{4} b f^{2} h + 2 a^{4} b f g^{2} + 2 a^{3} b^{2} c f h - 2 a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h - 4 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g + a^{3} b^{2} e f^{2} - a^{2} b^{3} c^{2} h + 4 a^{2} b^{3} c d g - 2 a^{2} b^{3} c e f + 2 a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} + a b^{4} c^{2} e - 2 a b^{4} c d^{2}}{a^{4} b g^{3} - 3 a^{3} b^{2} d g^{2} + a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g + 3 a b^{4} c^{2} f - a b^{4} d^{3} - b^{5} c^{3}} \right )} \right )\right )} + \frac {f x^{4}}{4 b} + \frac {g x^{5}}{5 b} + \frac {h x^{6}}{6 b} \] Input:
integrate(x**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)
Output:
x**3*(-a*h/(3*b**2) + e/(3*b)) + x**2*(-a*g/(2*b**2) + d/(2*b)) + x*(-a*f/ b**2 + c/b) + RootSum(27*_t**3*b**9 + _t**2*(-27*a**2*b**6*h + 27*a*b**7*e ) + _t*(9*a**4*b**3*h**2 - 18*a**3*b**4*e*h + 9*a**3*b**4*f*g - 9*a**2*b** 5*c*g - 9*a**2*b**5*d*f + 9*a**2*b**5*e**2 + 9*a*b**6*c*d) - a**6*h**3 + 3 *a**5*b*e*h**2 - 3*a**5*b*f*g*h + a**5*b*g**3 + 3*a**4*b**2*c*g*h + 3*a**4 *b**2*d*f*h - 3*a**4*b**2*d*g**2 - 3*a**4*b**2*e**2*h + 3*a**4*b**2*e*f*g - a**4*b**2*f**3 - 3*a**3*b**3*c*d*h - 3*a**3*b**3*c*e*g + 3*a**3*b**3*c*f **2 + 3*a**3*b**3*d**2*g - 3*a**3*b**3*d*e*f + a**3*b**3*e**3 - 3*a**2*b** 4*c**2*f + 3*a**2*b**4*c*d*e - a**2*b**4*d**3 + a*b**5*c**3, Lambda(_t, _t *log(x + (9*_t**2*a*b**6*g - 9*_t**2*b**7*d - 6*_t*a**3*b**3*g*h + 6*_t*a* *2*b**4*d*h + 6*_t*a**2*b**4*e*g + 3*_t*a**2*b**4*f**2 - 6*_t*a*b**5*c*f - 6*_t*a*b**5*d*e + 3*_t*b**6*c**2 + a**5*g*h**2 - a**4*b*d*h**2 - 2*a**4*b *e*g*h - a**4*b*f**2*h + 2*a**4*b*f*g**2 + 2*a**3*b**2*c*f*h - 2*a**3*b**2 *c*g**2 + 2*a**3*b**2*d*e*h - 4*a**3*b**2*d*f*g + a**3*b**2*e**2*g + a**3* b**2*e*f**2 - a**2*b**3*c**2*h + 4*a**2*b**3*c*d*g - 2*a**2*b**3*c*e*f + 2 *a**2*b**3*d**2*f - a**2*b**3*d*e**2 + a*b**4*c**2*e - 2*a*b**4*c*d**2)/(a **4*b*g**3 - 3*a**3*b**2*d*g**2 + a**3*b**2*f**3 - 3*a**2*b**3*c*f**2 + 3* a**2*b**3*d**2*g + 3*a*b**4*c**2*f - a*b**4*d**3 - b**5*c**3)))) + f*x**4/ (4*b) + g*x**5/(5*b) + h*x**6/(6*b)
Time = 0.12 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.07 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {10 \, b h x^{6} + 12 \, b g x^{5} + 15 \, b f x^{4} + 20 \, {\left (b e - a h\right )} x^{3} + 30 \, {\left (b d - a g\right )} x^{2} + 60 \, {\left (b c - a f\right )} x}{60 \, b^{2}} - \frac {\sqrt {3} {\left (a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3}} - \frac {{\left (2 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b c + a^{2} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b c - a^{2} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="maxi ma")
Output:
1/60*(10*b*h*x^6 + 12*b*g*x^5 + 15*b*f*x^4 + 20*(b*e - a*h)*x^3 + 30*(b*d - a*g)*x^2 + 60*(b*c - a*f)*x)/b^2 - 1/3*sqrt(3)*(a*b^2*d*(a/b)^(2/3) - a^ 2*b*g*(a/b)^(2/3) + a*b^2*c*(a/b)^(1/3) - a^2*b*f*(a/b)^(1/3))*arctan(1/3* sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^3) - 1/6*(2*a*b*e*(a/b)^(2/3 ) - 2*a^2*h*(a/b)^(2/3) + a*b*d*(a/b)^(1/3) - a^2*g*(a/b)^(1/3) - a*b*c + a^2*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) - 1/3*(a*b *e*(a/b)^(2/3) - a^2*h*(a/b)^(2/3) - a*b*d*(a/b)^(1/3) + a^2*g*(a/b)^(1/3) + a*b*c - a^2*f)*log(x + (a/b)^(1/3))/(b^3*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.13 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=-\frac {{\left (a b e - a^{2} h\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b f - \left (-a b^{2}\right )^{\frac {2}{3}} b d + \left (-a b^{2}\right )^{\frac {2}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b f + \left (-a b^{2}\right )^{\frac {2}{3}} b d - \left (-a b^{2}\right )^{\frac {2}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {10 \, b^{5} h x^{6} + 12 \, b^{5} g x^{5} + 15 \, b^{5} f x^{4} + 20 \, b^{5} e x^{3} - 20 \, a b^{4} h x^{3} + 30 \, b^{5} d x^{2} - 30 \, a b^{4} g x^{2} + 60 \, b^{5} c x - 60 \, a b^{4} f x}{60 \, b^{6}} + \frac {{\left (a b^{12} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{11} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{12} c - a^{2} b^{11} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{13}} \] Input:
integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="giac ")
Output:
-1/3*(a*b*e - a^2*h)*log(abs(b*x^3 + a))/b^3 - 1/3*sqrt(3)*((-a*b^2)^(1/3) *b^2*c - (-a*b^2)^(1/3)*a*b*f - (-a*b^2)^(2/3)*b*d + (-a*b^2)^(2/3)*a*g)*a rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)^( 1/3)*b^2*c - (-a*b^2)^(1/3)*a*b*f + (-a*b^2)^(2/3)*b*d - (-a*b^2)^(2/3)*a* g)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 + 1/60*(10*b^5*h*x^6 + 12* b^5*g*x^5 + 15*b^5*f*x^4 + 20*b^5*e*x^3 - 20*a*b^4*h*x^3 + 30*b^5*d*x^2 - 30*a*b^4*g*x^2 + 60*b^5*c*x - 60*a*b^4*f*x)/b^6 + 1/3*(a*b^12*d*(-a/b)^(1/ 3) - a^2*b^11*g*(-a/b)^(1/3) + a*b^12*c - a^2*b^11*f)*(-a/b)^(1/3)*log(abs (x - (-a/b)^(1/3)))/(a*b^13)
Time = 5.92 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.97 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx =\text {Too large to display} \] Input:
int((x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x)
Output:
x^2*(d/(2*b) - (a*g)/(2*b^2)) + x^3*(e/(3*b) - (a*h)/(3*b^2)) + symsum(log (root(27*b^9*z^3 + 27*a*b^7*e*z^2 - 27*a^2*b^6*h*z^2 + 9*a*b^6*c*d*z - 18* a^3*b^4*e*h*z + 9*a^3*b^4*f*g*z - 9*a^2*b^5*d*f*z - 9*a^2*b^5*c*g*z + 9*a^ 4*b^3*h^2*z + 9*a^2*b^5*e^2*z - 3*a^5*b*f*g*h + 3*a^4*b^2*e*f*g + 3*a^4*b^ 2*d*f*h + 3*a^4*b^2*c*g*h - 3*a^3*b^3*d*e*f - 3*a^3*b^3*c*e*g - 3*a^3*b^3* c*d*h + 3*a^2*b^4*c*d*e + 3*a^5*b*e*h^2 - 3*a^4*b^2*e^2*h - 3*a^4*b^2*d*g^ 2 + 3*a^3*b^3*d^2*g + 3*a^3*b^3*c*f^2 - 3*a^2*b^4*c^2*f + a^3*b^3*e^3 + a^ 5*b*g^3 + a*b^5*c^3 - a^4*b^2*f^3 - a^2*b^4*d^3 - a^6*h^3, z, k)*((6*a^2*b ^4*e - 6*a^3*b^3*h)/b^4 + (x*(3*a^2*b^3*f - 3*a*b^4*c))/b^3 + 9*root(27*b^ 9*z^3 + 27*a*b^7*e*z^2 - 27*a^2*b^6*h*z^2 + 9*a*b^6*c*d*z - 18*a^3*b^4*e*h *z + 9*a^3*b^4*f*g*z - 9*a^2*b^5*d*f*z - 9*a^2*b^5*c*g*z + 9*a^4*b^3*h^2*z + 9*a^2*b^5*e^2*z - 3*a^5*b*f*g*h + 3*a^4*b^2*e*f*g + 3*a^4*b^2*d*f*h + 3 *a^4*b^2*c*g*h - 3*a^3*b^3*d*e*f - 3*a^3*b^3*c*e*g - 3*a^3*b^3*c*d*h + 3*a ^2*b^4*c*d*e + 3*a^5*b*e*h^2 - 3*a^4*b^2*e^2*h - 3*a^4*b^2*d*g^2 + 3*a^3*b ^3*d^2*g + 3*a^3*b^3*c*f^2 - 3*a^2*b^4*c^2*f + a^3*b^3*e^3 + a^5*b*g^3 + a *b^5*c^3 - a^4*b^2*f^3 - a^2*b^4*d^3 - a^6*h^3, z, k)*a*b^2) + (a^5*h^2 + a^3*b^2*e^2 - 2*a^4*b*e*h + a^4*b*f*g + a^2*b^3*c*d - a^3*b^2*c*g - a^3*b^ 2*d*f)/b^4 + (x*(a^4*g^2 + a^2*b^2*d^2 - a^4*f*h + a^3*b*c*h - 2*a^3*b*d*g + a^3*b*e*f - a^2*b^2*c*e))/b^3)*root(27*b^9*z^3 + 27*a*b^7*e*z^2 - 27*a^ 2*b^6*h*z^2 + 9*a*b^6*c*d*z - 18*a^3*b^4*e*h*z + 9*a^3*b^4*f*g*z - 9*a^...
Time = 0.16 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.66 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx =\text {Too large to display} \] Input:
int(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x)
Output:
( - 20*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)* sqrt(3)))*a*b*f + 20*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3) *x)/(a**(1/3)*sqrt(3)))*b**2*c - 20*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x) /(a**(1/3)*sqrt(3)))*a**2*b*g + 20*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/ (a**(1/3)*sqrt(3)))*a*b**2*d - 10*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3 )*a**(1/3)*x + b**(2/3)*x**2)*a*b*f + 10*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**2*c + 20*b**(1/3)*a**(2/3)*log(a** (1/3) + b**(1/3)*x)*a*b*f - 20*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x )*b**2*c + 20*b**(2/3)*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2 /3)*x**2)*a**2*h - 20*b**(2/3)*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b*e + 20*b**(2/3)*a**(1/3)*log(a**(1/3) + b**(1/3)*x)* a**2*h - 20*b**(2/3)*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b*e - 60*b**(2/ 3)*a**(1/3)*a*b*f*x - 30*b**(2/3)*a**(1/3)*a*b*g*x**2 - 20*b**(2/3)*a**(1/ 3)*a*b*h*x**3 + 60*b**(2/3)*a**(1/3)*b**2*c*x + 30*b**(2/3)*a**(1/3)*b**2* d*x**2 + 20*b**(2/3)*a**(1/3)*b**2*e*x**3 + 15*b**(2/3)*a**(1/3)*b**2*f*x* *4 + 12*b**(2/3)*a**(1/3)*b**2*g*x**5 + 10*b**(2/3)*a**(1/3)*b**2*h*x**6 + 10*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*g - 10*log( a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*d - 20*log(a**(1/3) + b**(1/3)*x)*a**2*b*g + 20*log(a**(1/3) + b**(1/3)*x)*a*b**2*d)/(60*b**( 2/3)*a**(1/3)*b**3)