\(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx\) [407]

Optimal result

Integrand size = 35, antiderivative size = 259 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}-\frac {\left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{5/3}}+\frac {(b e-a h) \log \left (a+b x^3\right )}{3 b^2} \]

[Out]

f*x/b+1/2*g*x^2/b+1/3*h*x^3/b+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^(1/3)*(-a*f+b*c)-a^(1/3)*(-a*g+b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(5/3)+1/3*(-
a*h+b*e)*ln(b*x^3+a)/b^2-1/3*(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)/a^
(1/3)*3^(1/2))/a^(2/3)/b^(5/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt {3} a^{2/3} b^{5/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}-a f+b c\right )}{6 a^{2/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{2/3} b^{5/3}}+\frac {(b e-a h) \log \left (a+b x^3\right )}{3 b^2}+\frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b} \]

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x]

[Out]

(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))])/(Sqrt[3]*a^(2/3)*b^(5/3)) + ((b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*L
og[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)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1901

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f}{b}+\frac {g x}{b}+\frac {h x^2}{b}+\frac {b c-a f+(b d-a g) x+(b e-a h) x^2}{b \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}+\frac {\int \frac {b c-a f+(b d-a g) x+(b e-a h) x^2}{a+b x^3} \, dx}{b} \\ & = \frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}+\frac {\int \frac {b c-a f+(b d-a g) x}{a+b x^3} \, dx}{b}+\frac {(b e-a h) \int \frac {x^2}{a+b x^3} \, dx}{b} \\ & = \frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}+\frac {(b e-a h) \log \left (a+b x^3\right )}{3 b^2}+\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} (b c-a f)+\sqrt [3]{a} (b d-a g)\right )+\sqrt [3]{b} \left (-\sqrt [3]{b} (b c-a f)+\sqrt [3]{a} (b d-a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^{4/3}}+\frac {\left (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b} \\ & = \frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}+\frac {\left (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}+\frac {(b e-a h) \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^{4/3}}-\frac {\left (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{2/3} b^{4/3}} \\ & = \frac {f x}{b}+\frac {g x^2}{2 b}+\frac {h x^3}{3 b}+\frac {\left (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\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}+\frac {\left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{5/3}} \\ & = \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 ) \tan ^{-1}\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 (b c-a f-\frac {\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {6 b^{2/3} f x+3 b^{2/3} g x^2+2 b^{2/3} h x^3+\frac {2 \sqrt {3} \left (-b^{4/3} c-\sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {\left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac {2 (b e-a h) \log \left (a+b x^3\right )}{\sqrt [3]{b}}}{6 b^{5/3}} \]

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x]

[Out]

(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))

Maple [C] (verified)

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

Time = 1.56 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.32

method	result	size
risch	\(\frac {h \,x^{3}}{3 b}+\frac {g \,x^{2}}{2 b}+\frac {f x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (b c -a f +\left (-a g +b d \right ) \textit {\_R} +\left (-a h +b e \right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{2}}\)	\(82\)
default	\(\frac {\frac {1}{3} h \,x^{3}+\frac {1}{2} g \,x^{2}+f x}{b}+\frac {\left (-a f +b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-a g +b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a h +b e \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b}\)	\(246\)

[In]

int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

1/3*h*x^3/b+1/2*g*x^2/b+f*x/b+1/3/b^2*sum((b*c-a*f+(-a*g+b*d)*_R+(-a*h+b*e)*_R^2)/_R^2*ln(x-_R),_R=RootOf(_Z^3
*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.78 (sec) , antiderivative size = 15235, normalized size of antiderivative = 58.82 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\text {Too large to display} \]

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\text {Timed out} \]

[In]

integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\frac {2 \, h x^{3} + 3 \, g x^{2} + 6 \, f x}{6 \, b} + \frac {\sqrt {3} {\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} + \frac {{\left (2 \, b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} + b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - b c + a f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + b c - a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (b^{2} c - a b f - \left (-a b^{2}\right )^{\frac {1}{3}} b d + \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{2} c - a b f + \left (-a b^{2}\right )^{\frac {1}{3}} b d - \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} + \frac {{\left (b e - a h\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac {2 \, b^{2} h x^{3} + 3 \, b^{2} g x^{2} + 6 \, b^{2} f x}{6 \, b^{3}} - \frac {{\left (b^{7} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{6} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b^{7} c - a b^{6} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{7}} \]

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*(b^2*c - a*b*f - (-a*b^2)^(1/3)*b*d + (-a*b^2)^(1/3)*a*g)*arctan(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)

Mupad [B] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 1150, normalized size of antiderivative = 4.44 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {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}+\mathrm {root}\left (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\right )\,\left (\frac {6\,a^2\,b^2\,h-6\,a\,b^3\,e}{b^2}+\frac {x\,\left (3\,b^3\,c-3\,a\,b^2\,f\right )}{b}+\mathrm {root}\left (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\right )\,a\,b^2\,9\right )+\frac {x\,\left (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\right )}{b}\right )\,\mathrm {root}\left (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\right )\right )+\frac {g\,x^2}{2\,b}+\frac {h\,x^3}{3\,b}+\frac {f\,x}{b} \]

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x)

[Out]

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*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), k, 1, 3) + (g*x^2)/(2*b) + (h*x^3)/(3*b) + (f*x)/b