\(\int \frac {c+d x+e x^2}{x^3 (a+b x^3)} \, dx\) [343]

Optimal result

Integrand size = 23, antiderivative size = 203 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=-\frac {c}{2 a x^2}-\frac {d}{a x}+\frac {\sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {e \log (x)}{a}-\frac {\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac {e \log \left (a+b x^3\right )}{3 a} \]

[Out]

-1/2*c/a/x^2-d/a/x+e*ln(x)/a-1/3*b^(1/3)*(b^(1/3)*c-a^(1/3)*d)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)+1/6*b^(2/3)*(c-a^
(1/3)*d/b^(1/3))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)-1/3*e*ln(b*x^3+a)/a+1/3*b^(1/3)*(b^(1/3)*c+
a^(1/3)*d)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac {\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}-\frac {e \log \left (a+b x^3\right )}{3 a}-\frac {c}{2 a x^2}-\frac {d}{a x}+\frac {e \log (x)}{a} \]

[In]

Int[(c + d*x + e*x^2)/(x^3*(a + b*x^3)),x]

[Out]

-1/2*c/(a*x^2) - d/(a*x) + (b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(Sqrt[3]*a^(5/3)) + (e*Log[x])/a - (b^(1/3)*(b^(1/3)*c - a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)) + (
b^(2/3)*(c - (a^(1/3)*d)/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(5/3)) - (e*Log[a + b*x
^3])/(3*a)

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 1848

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(Pq/(a + b*x
^n)), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

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]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=\frac {-\frac {3 a c}{x^2}-\frac {6 a d}{x}+2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+6 a e \log (x)+2 \sqrt [3]{b} \left (-\sqrt [3]{a} \sqrt [3]{b} c+a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 a e \log \left (a+b x^3\right )}{6 a^2} \]

[In]

Integrate[(c + d*x + e*x^2)/(x^3*(a + b*x^3)),x]

[Out]

((-3*a*c)/x^2 - (6*a*d)/x + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3
))/Sqrt[3]] + 6*a*e*Log[x] + 2*b^(1/3)*(-(a^(1/3)*b^(1/3)*c) + a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(
a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a*e*Log[a + b*x^3])/(6*a^2)

Maple [C] (verified)

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

Time = 1.68 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {-\frac {x d}{a}-\frac {c}{2 a}}{x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{3}+3 a^{4} e \,\textit {\_Z}^{2}+\left (3 a^{3} e^{2}+3 a^{2} b c d \right ) \textit {\_Z} +a^{2} e^{3}+3 a b c d e -a b \,d^{3}+b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{5}-8 \textit {\_R}^{2} a^{4} e +\left (-4 a^{3} e^{2}-10 a^{2} b c d \right ) \textit {\_R} -6 a b c d e +3 a b \,d^{3}-3 b^{2} c^{3}\right ) x -a^{4} d \,\textit {\_R}^{2}+\left (2 a^{3} d e -a^{2} b \,c^{2}\right ) \textit {\_R} +3 a^{2} d \,e^{2}+3 a b \,c^{2} e \right )\right )}{3}+\frac {e \ln \left (-x \right )}{a}\)	\(209\)
default	\(-\frac {c}{2 a \,x^{2}}-\frac {d}{a x}+\frac {e \ln \left (x \right )}{a}-\frac {\left (c \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+d \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {e \ln \left (b \,x^{3}+a \right )}{3 b}\right ) b}{a}\)	\(232\)

[In]

int((e*x^2+d*x+c)/x^3/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

(-1/a*x*d-1/2*c/a)/x^2+1/3*sum(_R*ln((-4*_R^3*a^5-8*_R^2*a^4*e+(-4*a^3*e^2-10*a^2*b*c*d)*_R-6*a*b*c*d*e+3*a*b*
d^3-3*b^2*c^3)*x-a^4*d*_R^2+(2*a^3*d*e-a^2*b*c^2)*_R+3*a^2*d*e^2+3*a*b*c^2*e),_R=RootOf(a^5*_Z^3+3*a^4*e*_Z^2+
(3*a^3*e^2+3*a^2*b*c*d)*_Z+a^2*e^3+3*a*b*c*d*e-a*b*d^3+b^2*c^3))+1/a*e*ln(-x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.07 (sec) , antiderivative size = 4279, normalized size of antiderivative = 21.08 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/36*(2*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*
(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e
^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a
*b)/a^5)^(1/3) + 6*e/a)*a*x^2*log(1/36*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18
*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3
) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3
+ a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)^2*a^4*d + 2*a*b*c*d^2 - a*b*c^2*e + a^2*d*e^2 + 1/6*(a^2
*b*c^2 - 2*a^3*d*e)*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/
a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) +
1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 -
3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a) + (b^2*c^3 + a*b*d^3)*x) - 36*e*x^2*log(x) + 36*d*x - (((-I*sqrt(3) + 1)*(e
^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*
(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)
*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a*x^2
+ 3*sqrt(1/3)*a*x^2*sqrt(-(((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a
*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sq
rt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3
- (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)^2*a^3 - 12*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27
*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)
*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5
- 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a^2*e + 144*b*c*d + 36*a*e^2)/a^3) - 18*e
*x^2)*log(-1/36*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4
+ 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(
-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c
*d*e)*a*b)/a^5)^(1/3) + 6*e/a)^2*a^4*d - 2*a*b*c*d^2 + a*b*c^2*e - a^2*d*e^2 - 1/6*(a^2*b*c^2 - 2*a^3*d*e)*((-
I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d
^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/1
8*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/
3) + 6*e/a) + 2*(b^2*c^3 + a*b*d^3)*x + 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/
27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*
e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^
5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a^4*d - 6*a^2*b*c^2 - 6*a^3*d*e)*sqrt(-
(((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 +
a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 +
1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)
^(1/3) + 6*e/a)^2*a^3 - 12*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*
e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqr
t(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 -
(d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a^2*e + 144*b*c*d + 36*a*e^2)/a^3)) - (((-I*sqrt(3) + 1)*(e^2/a^2 -
(b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3
+ a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 +
1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a*x^2 - 3*sqrt
(1/3)*a*x^2*sqrt(-(((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a
^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1
)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 -
3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)^2*a^3 - 12*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3
+ 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5
)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b
^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a^2*e + 144*b*c*d + 36*a*e^2)/a^3) - 18*e*x^2)*log
(-1/36*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b
*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3
/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b
)/a^5)^(1/3) + 6*e/a)^2*a^4*d - 2*a*b*c*d^2 + a*b*c^2*e - a^2*d*e^2 - 1/6*(a^2*b*c^2 - 2*a^3*d*e)*((-I*sqrt(3)
+ 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5
- 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d
+ a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/
a) + 2*(b^2*c^3 + a*b*d^3)*x - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^
3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a
^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*
(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a^4*d - 6*a^2*b*c^2 - 6*a^3*d*e)*sqrt(-(((-I*sqr
t(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b
/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*e^3/a^3 + 1/18*(b*
c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) +
6*e/a)^2*a^3 - 12*((-I*sqrt(3) + 1)*(e^2/a^2 - (b*c*d + a*e^2)/a^3)/(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^
4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/a^5)^(1/3) + 9*(I*sqrt(3) + 1)
*(-1/27*e^3/a^3 + 1/18*(b*c*d + a*e^2)*e/a^4 + 1/54*(b*c^3 + a*d^3)*b/a^5 - 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3
*c*d*e)*a*b)/a^5)^(1/3) + 6*e/a)*a^2*e + 144*b*c*d + 36*a*e^2)/a^3)) + 18*c)/(a*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((e*x**2+d*x+c)/x**3/(b*x**3+a),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.87 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=\frac {e \log \left (x\right )}{a} - \frac {\sqrt {3} {\left (b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + b c \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {{\left (2 \, e \left (\frac {a}{b}\right )^{\frac {2}{3}} + d \left (\frac {a}{b}\right )^{\frac {1}{3}} - c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (e \left (\frac {a}{b}\right )^{\frac {2}{3}} - d \left (\frac {a}{b}\right )^{\frac {1}{3}} + c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, d x + c}{2 \, a x^{2}} \]

[In]

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

[Out]

e*log(x)/a - 1/3*sqrt(3)*(b*d*(a/b)^(2/3) + b*c*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3
))/a^2 - 1/6*(2*e*(a/b)^(2/3) + d*(a/b)^(1/3) - c)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^(2/3)) - 1/
3*(e*(a/b)^(2/3) - d*(a/b)^(1/3) + c)*log(x + (a/b)^(1/3))/(a*(a/b)^(2/3)) - 1/2*(2*d*x + c)/(a*x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=-\frac {e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac {e \log \left ({\left | x \right |}\right )}{a} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {2}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {2}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} + \frac {{\left (a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3} b} - \frac {2 \, d x + c}{2 \, a x^{2}} \]

[In]

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

[Out]

-1/3*e*log(abs(b*x^3 + a))/a + e*log(abs(x))/a - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b*c - (-a*b^2)^(2/3)*d)*arctan(1/
3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b) - 1/6*((-a*b^2)^(1/3)*b*c + (-a*b^2)^(2/3)*d)*log(x^2 + x
*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b) + 1/3*(a*b^2*d*(-a/b)^(1/3) + a*b^2*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(
1/3)))/(a^3*b) - 1/2*(2*d*x + c)/(a*x^2)

Mupad [B] (verification not implemented)

Time = 8.99 (sec) , antiderivative size = 701, normalized size of antiderivative = 3.45 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {b^5\,c^3\,x-a^2\,b^3\,d\,e^2+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )}^3\,a^5\,b^3\,x\,36-a\,b^4\,c^2\,e-a\,b^4\,d^3\,x+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a^2\,b^4\,c^2+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )}^2\,a^4\,b^3\,d\,3+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a^3\,b^3\,e^2\,x\,4+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )}^2\,a^4\,b^3\,e\,x\,24-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a^3\,b^3\,d\,e\,2+2\,a\,b^4\,c\,d\,e\,x+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a^2\,b^4\,c\,d\,x\,10}{a^3}\right )\,\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,e\,z^2+9\,a^2\,b\,c\,d\,z+9\,a^3\,e^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\right )-\frac {c}{2\,a\,x^2}-\frac {d}{a\,x}+\frac {e\,\ln \left (x\right )}{a} \]

[In]

int((c + d*x + e*x^2)/(x^3*(a + b*x^3)),x)

[Out]

symsum(log(-(b^5*c^3*x - a^2*b^3*d*e^2 + 36*root(27*a^5*z^3 + 27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a
*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)^3*a^5*b^3*x - a*b^4*c^2*e - a*b^4*d^3*x + root(27*a^5*z^3 + 27*a
^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)*a^2*b^4*c^2 + 3*root
(27*a^5*z^3 + 27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)^2*
a^4*b^3*d + 4*root(27*a^5*z^3 + 27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 +
b^2*c^3, z, k)*a^3*b^3*e^2*x + 24*root(27*a^5*z^3 + 27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a*b*c*d*e
- a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)^2*a^4*b^3*e*x - 2*root(27*a^5*z^3 + 27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*
e^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)*a^3*b^3*d*e + 2*a*b^4*c*d*e*x + 10*root(27*a^5*z^3 +
27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)*a^2*b^4*c*d*x)/a
^3)*root(27*a^5*z^3 + 27*a^4*e*z^2 + 9*a^2*b*c*d*z + 9*a^3*e^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3,
z, k), k, 1, 3) - c/(2*a*x^2) - d/(a*x) + (e*log(x))/a