Integrand size = 35, antiderivative size = 257 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}-\frac {\left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{5/3}}-\frac {\left (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}+\frac {(b e-a h) \log \left (a+b x^3\right )}{3 b^2} \] Output:
f*x/b+1/2*g*x^2/b+1/3*h*x^3/b-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)/a^(2/3)/b^ (5/3)+1/3*(b^(1/3)*(-a*f+b*c)-a^(1/3)*(-a*g+b*d))*ln(a^(1/3)+b^(1/3)*x)/a^ (2/3)/b^(5/3)-1/6*(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)/a^(2/3)/b^(4/3)+1/3*(-a*h+b*e)*ln(b*x^3+a)/b^2
Time = 0.18 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {6 b^{2/3} f x+3 b^{2/3} g x^2+2 b^{2/3} h x^3+\frac {2 \sqrt {3} \left (-b^{4/3} c-\sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {\left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac {2 (b e-a h) \log \left (a+b x^3\right )}{\sqrt [3]{b}}}{6 b^{5/3}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x]
Output:
(6*b^(2/3)*f*x + 3*b^(2/3)*g*x^2 + 2*b^(2/3)*h*x^3 + (2*Sqrt[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]])/a^(2/3) + (2*(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])/a^(2/3) - ((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])/a^(2 /3) + (2*(b*e - a*h)*Log[a + b*x^3])/b^(1/3))/(6*b^(5/3))
Time = 0.92 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \int \left (\frac {x (b d-a g)+x^2 (b e-a h)-a f+b c}{b \left (a+b x^3\right )}+\frac {f}{b}+\frac {g x}{b}+\frac {h x^2}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt {3} a^{2/3} b^{5/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}-a f+b c\right )}{6 a^{2/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{2/3} b^{5/3}}+\frac {(b e-a h) \log \left (a+b x^3\right )}{3 b^2}+\frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x]
Output:
(f*x)/b + (g*x^2)/(2*b) + (h*x^3)/(3*b) - ((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))])/(S qrt[3]*a^(2/3)*b^(5/3)) + ((b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*Log [a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(5/3)) - ((b*c - a*f - (a^(1/3)*(b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3) *b^(4/3)) + ((b*e - a*h)*Log[a + b*x^3])/(3*b^2)
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 = 82, normalized size of antiderivative = 0.32
| method | result | size |
| risch | \(\frac {h \,x^{3}}{3 b}+\frac {g \,x^{2}}{2 b}+\frac {f x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (c b -a f +\left (-a g +b d \right ) \textit {\_R} +\left (-a h +b e \right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{2}}\) | \(82\) |
| default | \(\frac {\frac {1}{3} h \,x^{3}+\frac {1}{2} g \,x^{2}+f x}{b}+\frac {\left (-a f +c b \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-a g +b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a h +b e \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b}\) | \(246\) |
Input:
int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/3/b*h*x^3+1/2*g*x^2/b+f*x/b+1/3/b^2*sum((c*b-a*f+(-a*g+b*d)*_R+(-a*h+b*e )*_R^2)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 15235, normalized size of antiderivative = 59.28 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\text {Too large to display} \] Input:
integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (241) = 482\).
Time = 33.36 (sec) , antiderivative size = 804, normalized size of antiderivative = 3.13 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{6} + t^{2} \cdot \left (27 a^{3} b^{4} h - 27 a^{2} b^{5} e\right ) + t \left (9 a^{4} b^{2} h^{2} - 18 a^{3} b^{3} e h + 9 a^{3} b^{3} f g - 9 a^{2} b^{4} c g - 9 a^{2} b^{4} d f + 9 a^{2} b^{4} e^{2} + 9 a b^{5} c d\right ) + a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{4} b f g h - a^{4} b g^{3} - 3 a^{3} b^{2} c g h - 3 a^{3} b^{2} d f h + 3 a^{3} b^{2} d g^{2} + 3 a^{3} b^{2} e^{2} h - 3 a^{3} b^{2} e f g + a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c d h + 3 a^{2} b^{3} c e g - 3 a^{2} b^{3} c f^{2} - 3 a^{2} b^{3} d^{2} g + 3 a^{2} b^{3} d e f - a^{2} b^{3} e^{3} + 3 a b^{4} c^{2} f - 3 a b^{4} c d e + a b^{4} d^{3} - b^{5} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} a^{3} b^{4} g - 9 t^{2} a^{2} b^{5} d + 6 t a^{4} b^{2} g h - 6 t a^{3} b^{3} d h - 6 t a^{3} b^{3} e g - 3 t a^{3} b^{3} f^{2} + 6 t a^{2} b^{4} c f + 6 t a^{2} b^{4} d e - 3 t a b^{5} c^{2} + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h - a^{4} b f^{2} h + 2 a^{4} b f g^{2} + 2 a^{3} b^{2} c f h - 2 a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h - 4 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g + a^{3} b^{2} e f^{2} - a^{2} b^{3} c^{2} h + 4 a^{2} b^{3} c d g - 2 a^{2} b^{3} c e f + 2 a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} + a b^{4} c^{2} e - 2 a b^{4} c d^{2}}{a^{4} b g^{3} - 3 a^{3} b^{2} d g^{2} + a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g + 3 a b^{4} c^{2} f - a b^{4} d^{3} - b^{5} c^{3}} \right )} \right )\right )} + \frac {f x}{b} + \frac {g x^{2}}{2 b} + \frac {h x^{3}}{3 b} \] Input:
integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)
Output:
RootSum(27*_t**3*a**2*b**6 + _t**2*(27*a**3*b**4*h - 27*a**2*b**5*e) + _t* (9*a**4*b**2*h**2 - 18*a**3*b**3*e*h + 9*a**3*b**3*f*g - 9*a**2*b**4*c*g - 9*a**2*b**4*d*f + 9*a**2*b**4*e**2 + 9*a*b**5*c*d) + a**5*h**3 - 3*a**4*b *e*h**2 + 3*a**4*b*f*g*h - a**4*b*g**3 - 3*a**3*b**2*c*g*h - 3*a**3*b**2*d *f*h + 3*a**3*b**2*d*g**2 + 3*a**3*b**2*e**2*h - 3*a**3*b**2*e*f*g + a**3* b**2*f**3 + 3*a**2*b**3*c*d*h + 3*a**2*b**3*c*e*g - 3*a**2*b**3*c*f**2 - 3 *a**2*b**3*d**2*g + 3*a**2*b**3*d*e*f - a**2*b**3*e**3 + 3*a*b**4*c**2*f - 3*a*b**4*c*d*e + a*b**4*d**3 - b**5*c**3, Lambda(_t, _t*log(x + (9*_t**2* a**3*b**4*g - 9*_t**2*a**2*b**5*d + 6*_t*a**4*b**2*g*h - 6*_t*a**3*b**3*d* h - 6*_t*a**3*b**3*e*g - 3*_t*a**3*b**3*f**2 + 6*_t*a**2*b**4*c*f + 6*_t*a **2*b**4*d*e - 3*_t*a*b**5*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/b + g *x**2/(2*b) + h*x**3/(3*b)
Time = 0.12 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {2 \, h x^{3} + 3 \, g x^{2} + 6 \, f x}{6 \, b} + \frac {\sqrt {3} {\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} + \frac {{\left (2 \, b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} + b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - b c + a f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + b c - a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="maxima")
Output:
1/6*(2*h*x^3 + 3*g*x^2 + 6*f*x)/b + 1/3*sqrt(3)*(b^2*d*(a/b)^(2/3) - a*b*g *(a/b)^(2/3) + b^2*c*(a/b)^(1/3) - a*b*f*(a/b)^(1/3))*arctan(1/3*sqrt(3)*( 2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2) + 1/6*(2*b*e*(a/b)^(2/3) - 2*a*h*( a/b)^(2/3) + b*d*(a/b)^(1/3) - a*g*(a/b)^(1/3) - b*c + a*f)*log(x^2 - x*(a /b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) + 1/3*(b*e*(a/b)^(2/3) - a*h*(a /b)^(2/3) - b*d*(a/b)^(1/3) + a*g*(a/b)^(1/3) + b*c - a*f)*log(x + (a/b)^( 1/3))/(b^2*(a/b)^(2/3))
Time = 0.14 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (b^{2} c - a b f - \left (-a b^{2}\right )^{\frac {1}{3}} b d + \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{2} c - a b f + \left (-a b^{2}\right )^{\frac {1}{3}} b d - \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} + \frac {{\left (b e - a h\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac {2 \, b^{2} h x^{3} + 3 \, b^{2} g x^{2} + 6 \, b^{2} f x}{6 \, b^{3}} - \frac {{\left (b^{7} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{6} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b^{7} c - a b^{6} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{7}} \] Input:
integrate((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*sqrt(3)*(b^2*c - a*b*f - (-a*b^2)^(1/3)*b*d + (-a*b^2)^(1/3)*a*g)*arc tan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*b) - 1/ 6*(b^2*c - a*b*f + (-a*b^2)^(1/3)*b*d - (-a*b^2)^(1/3)*a*g)*log(x^2 + x*(- a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b) + 1/3*(b*e - a*h)*log(abs(b* x^3 + a))/b^2 + 1/6*(2*b^2*h*x^3 + 3*b^2*g*x^2 + 6*b^2*f*x)/b^3 - 1/3*(b^7 *d*(-a/b)^(1/3) - a*b^6*g*(-a/b)^(1/3) + b^7*c - a*b^6*f)*(-a/b)^(1/3)*log (abs(x - (-a/b)^(1/3)))/(a*b^7)
Time = 6.04 (sec) , antiderivative size = 1150, normalized size of antiderivative = 4.47 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx =\text {Too large to display} \] Input:
int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x)
Output:
symsum(log((a^3*h^2 + a*b^2*e^2 + b^3*c*d - a*b^2*c*g - a*b^2*d*f - 2*a^2* b*e*h + a^2*b*f*g)/b^2 + root(27*a^2*b^6*z^3 + 27*a^3*b^4*h*z^2 - 27*a^2*b ^5*e*z^2 + 9*a*b^5*c*d*z - 18*a^3*b^3*e*h*z + 9*a^3*b^3*f*g*z - 9*a^2*b^4* d*f*z - 9*a^2*b^4*c*g*z + 9*a^4*b^2*h^2*z + 9*a^2*b^4*e^2*z + 3*a^4*b*f*g* h - 3*a*b^4*c*d*e - 3*a^3*b^2*e*f*g - 3*a^3*b^2*d*f*h - 3*a^3*b^2*c*g*h + 3*a^2*b^3*d*e*f + 3*a^2*b^3*c*e*g + 3*a^2*b^3*c*d*h - 3*a^4*b*e*h^2 + 3*a* b^4*c^2*f + 3*a^3*b^2*e^2*h + 3*a^3*b^2*d*g^2 - 3*a^2*b^3*d^2*g - 3*a^2*b^ 3*c*f^2 + a^3*b^2*f^3 + a*b^4*d^3 + a^5*h^3 - a^2*b^3*e^3 - a^4*b*g^3 - b^ 5*c^3, z, k)*((6*a^2*b^2*h - 6*a*b^3*e)/b^2 + (x*(3*b^3*c - 3*a*b^2*f))/b + 9*root(27*a^2*b^6*z^3 + 27*a^3*b^4*h*z^2 - 27*a^2*b^5*e*z^2 + 9*a*b^5*c* d*z - 18*a^3*b^3*e*h*z + 9*a^3*b^3*f*g*z - 9*a^2*b^4*d*f*z - 9*a^2*b^4*c*g *z + 9*a^4*b^2*h^2*z + 9*a^2*b^4*e^2*z + 3*a^4*b*f*g*h - 3*a*b^4*c*d*e - 3 *a^3*b^2*e*f*g - 3*a^3*b^2*d*f*h - 3*a^3*b^2*c*g*h + 3*a^2*b^3*d*e*f + 3*a ^2*b^3*c*e*g + 3*a^2*b^3*c*d*h - 3*a^4*b*e*h^2 + 3*a*b^4*c^2*f + 3*a^3*b^2 *e^2*h + 3*a^3*b^2*d*g^2 - 3*a^2*b^3*d^2*g - 3*a^2*b^3*c*f^2 + a^3*b^2*f^3 + a*b^4*d^3 + a^5*h^3 - a^2*b^3*e^3 - a^4*b*g^3 - b^5*c^3, z, k)*a*b^2) + (x*(b^2*d^2 + a^2*g^2 - b^2*c*e - a^2*f*h + a*b*c*h - 2*a*b*d*g + a*b*e*f ))/b)*root(27*a^2*b^6*z^3 + 27*a^3*b^4*h*z^2 - 27*a^2*b^5*e*z^2 + 9*a*b^5* c*d*z - 18*a^3*b^3*e*h*z + 9*a^3*b^3*f*g*z - 9*a^2*b^4*d*f*z - 9*a^2*b^4*c *g*z + 9*a^4*b^2*h^2*z + 9*a^2*b^4*e^2*z + 3*a^4*b*f*g*h - 3*a*b^4*c*d*...
Time = 0.16 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.72 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {2 b^{\frac {4}{3}} a^{\frac {5}{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 -2 b^{\frac {7}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) c +2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b g -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{2} d +b^{\frac {4}{3}} a^{\frac {5}{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 -b^{\frac {7}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) c -2 b^{\frac {4}{3}} a^{\frac {5}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) f +2 b^{\frac {7}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) c -2 b^{\frac {2}{3}} 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 ) h +2 b^{\frac {5}{3}} 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 ) e -2 b^{\frac {2}{3}} a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) h +2 b^{\frac {5}{3}} a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) e +6 b^{\frac {5}{3}} a^{\frac {4}{3}} f x +3 b^{\frac {5}{3}} a^{\frac {4}{3}} g \,x^{2}+2 b^{\frac {5}{3}} a^{\frac {4}{3}} h \,x^{3}-\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b g +\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{2} d +2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b g -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{2} d}{6 b^{\frac {8}{3}} a^{\frac {4}{3}}} \] Input:
int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x)
Output:
(2*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 - 2*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 + 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**( 1/3)*sqrt(3)))*a**2*b*g - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/ 3)*sqrt(3)))*a*b**2*d + 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 - 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 - 2*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/ 3)*x)*a*b*f + 2*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*b**2*c - 2*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 + 2*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 - 2*b**(2/3)*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**2*h + 2*b**(2/3 )*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b*e + 6*b**(2/3)*a**(1/3)*a*b*f*x + 3*b**(2/3)*a**(1/3)*a*b*g*x**2 + 2*b**(2/3)*a**(1/3)*a*b*h*x**3 - log(a* *(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*g + log(a**(2/3) - b* *(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*d + 2*log(a**(1/3) + b**(1/3)*x) *a**2*b*g - 2*log(a**(1/3) + b**(1/3)*x)*a*b**2*d)/(6*b**(2/3)*a**(1/3)*a* b**2)