\(\int \frac {c+d x+e x^2}{x^2 (a+b x^3)} \, dx\) [342]

Optimal result

Integrand size = 23, antiderivative size = 192 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=-\frac {c}{a x}+\frac {\left (b^{2/3} c-a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} \sqrt [3]{b}}+\frac {d \log (x)}{a}+\frac {\left (b^{2/3} c+a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt [3]{b}}-\frac {\left (b^{2/3} c+a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a} \]

[Out]

-c/a/x+d*ln(x)/a+1/3*(b^(2/3)*c+a^(2/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(4/3)/b^(1/3)-1/6*(b^(2/3)*c+a^(2/3)*e)*ln(
a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)/b^(1/3)-1/3*d*ln(b*x^3+a)/a+1/3*(b^(2/3)*c-a^(2/3)*e)*arctan(1/
3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(4/3)/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 192, 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} \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (b^{2/3} c-a^{2/3} e\right )}{\sqrt {3} a^{4/3} \sqrt [3]{b}}-\frac {\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt [3]{b}}+\frac {\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a}-\frac {c}{a x}+\frac {d \log (x)}{a} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=-\frac {\frac {6 a c}{x}+\frac {2 \sqrt {3} a^{2/3} \left (-b^{2/3} c+a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-6 a d \log (x)-\frac {2 \left (a^{2/3} b^{2/3} c+a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {\left (a^{2/3} b^{2/3} c+a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+2 a d \log \left (a+b x^3\right )}{6 a^2} \]

[In]

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

[Out]

-1/6*((6*a*c)/x + (2*Sqrt[3]*a^(2/3)*(-(b^(2/3)*c) + a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b
^(1/3) - 6*a*d*Log[x] - (2*(a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((a^(2/3)*b^(2/
3)*c + a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) + 2*a*d*Log[a + b*x^3])/a^2

Maple [C] (verified)

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

Time = 1.54 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {c}{a x}+\frac {d \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} b \,\textit {\_Z}^{3}+3 a^{3} b d \,\textit {\_Z}^{2}+\left (-3 a^{2} b c e +3 a^{2} b \,d^{2}\right ) \textit {\_Z} -a^{2} e^{3}-3 a b c d e +a b \,d^{3}-b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{4} b -8 \textit {\_R}^{2} a^{3} b d +\left (10 a^{2} b c e -4 a^{2} b \,d^{2}\right ) \textit {\_R} +3 a^{2} e^{3}+6 a b c d e +3 b^{2} c^{3}\right ) x -a^{3} b c \,\textit {\_R}^{2}+\left (-a^{3} e^{2}+2 a^{2} b c d \right ) \textit {\_R} +3 a^{2} d \,e^{2}+3 a b c \,d^{2}\right )\right )}{3}\)	\(207\)
default	\(-\frac {c}{a x}+\frac {d \ln \left (x \right )}{a}+\frac {a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{a}\)	\(221\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.17 (sec) , antiderivative size = 4524, normalized size of antiderivative = 23.56 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/36*(2*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3
+ a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27
*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a
^2*e^3)/(a^4*b))^(1/3) + 6*d/a)*a*x*log(-1/36*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1
/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4
*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*
d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)^2*a^3*b*c - a*b*c*d^2 + 2*a*b*c^2*e + a^2
*d*e^2 + 1/6*(2*a^2*b*c*d - a^3*e^2)*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2
- c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3
) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)
/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a) - (b^2*c^3 - a^2*e^3)*x) - 36*d*x*log(x) - (((-I*s
qrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d
^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18
*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b)
)^(1/3) + 6*d/a)*a*x - 3*sqrt(1/3)*a*x*sqrt(-(((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1
/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4
*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*
d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)^2*a^2 - 12*((-I*sqrt(3) + 1)*(d^2/a^2 - (
d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*
b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/5
4*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)*a*d + 3
6*d^2 - 144*c*e)/a^2) - 18*d*x)*log(1/36*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(
d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^
(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*
a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)^2*a^3*b*c + a*b*c*d^2 - 2*a*b*c^2*e - a^2*d*e^
2 - 1/6*(2*a^2*b*c*d - a^3*e^2)*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e
)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9
*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4
*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a) - 2*(b^2*c^3 - a^2*e^3)*x + 1/12*sqrt(1/3)*(((-I*sqrt(3
) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 -
3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2
- c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/
3) + 6*d/a)*a^3*b*c - 6*a^2*b*c*d - 6*a^3*e^2)*sqrt(-(((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3
/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e
^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^
3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)^2*a^2 - 12*((-I*sqrt(3) + 1)*(d^2
/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*
b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a
^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)
*a*d + 36*d^2 - 144*c*e)/a^2)) - (((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c
*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) +
9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a
^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)*a*x + 3*sqrt(1/3)*a*x*sqrt(-(((-I*sqrt(3) + 1)*(d^2/a
^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)
/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3
+ 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)^2
*a^2 - 12*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^
3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/2
7*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 -
a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)*a*d + 36*d^2 - 144*c*e)/a^2) - 18*d*x)*log(1/36*((-I*sqrt(3) + 1)*(d^2/a^2 -
(d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4
*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/
54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)^2*a^3*
b*c + a*b*c*d^2 - 2*a*b*c^2*e - a^2*d*e^2 - 1/6*(2*a^2*b*c*d - a^3*e^2)*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*
e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/5
4*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c
^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a) - 2*(b^2*c^3 -
a^2*e^3)*x - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d
/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I
*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b)
- 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)*a^3*b*c - 6*a^2*b*c*d - 6*a^3*e^2)*sqrt(-(((-I*sqrt(3) + 1
)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d
*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*
e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) +
6*d/a)^2*a^2 - 12*((-I*sqrt(3) + 1)*(d^2/a^2 - (d^2 - c*e)/a^2)/(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54
*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 9*(I*sqrt(3) +
1)*(-1/27*d^3/a^3 + 1/18*(d^2 - c*e)*d/a^3 + 1/54*(b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^4*b) - 1/54*(b^
2*c^3 - a^2*e^3)/(a^4*b))^(1/3) + 6*d/a)*a*d + 36*d^2 - 144*c*e)/a^2)) + 36*c)/(a*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=\frac {d \log \left (x\right )}{a} - \frac {\sqrt {3} {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {c}{a x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=-\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac {d \log \left ({\left | x \right |}\right )}{a} + \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} a e + \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} - \frac {c}{a x} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} a e - \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} + \frac {{\left (a b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3} b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.93 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.77 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {b^4\,c^3\,x+a^2\,b^2\,d\,e^2-{\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )}^3\,a^4\,b^3\,x\,36+a^2\,b^2\,e^3\,x+a\,b^3\,c\,d^2-{\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )}^2\,a^3\,b^3\,c\,3-\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\,a^3\,b^2\,e^2-\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\,a^2\,b^3\,d^2\,x\,4-{\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )}^2\,a^3\,b^3\,d\,x\,24+\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\,a^2\,b^3\,c\,d\,2+2\,a\,b^3\,c\,d\,e\,x+\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\,a^2\,b^3\,c\,e\,x\,10}{a^2}\right )\,\mathrm {root}\left (27\,a^4\,b\,z^3+27\,a^3\,b\,d\,z^2-9\,a^2\,b\,c\,e\,z+9\,a^2\,b\,d^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\right )-\frac {c}{a\,x}+\frac {d\,\ln \left (x\right )}{a} \]

[In]

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

[Out]

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