\(\int \frac {c+d x+e x^2}{x^4 (a+b x^3)^2} \, dx\) [350]

Optimal result

Integrand size = 23, antiderivative size = 262 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {2 b c \log (x)}{a^3}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3} \]

[Out]

-1/3*c/a^2/x^3-1/2*d/a^2/x^2-e/a^2/x-1/3*x*(b*d+b*e*x-b^2*c*x^2/a)/a^2/(b*x^3+a)-2*b*c*ln(x)/a^3-1/9*b^(1/3)*(
5*b^(1/3)*d-4*a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)+1/18*b^(1/3)*(5*b^(1/3)*d-4*a^(1/3)*e)*ln(a^(2/3)-a^(1/
3)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)+2/3*b*c*ln(b*x^3+a)/a^3+1/9*b^(1/3)*(5*b^(1/3)*d+4*a^(1/3)*e)*arctan(1/3*(a^
(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(8/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1843, 1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right )}{3 \sqrt {3} a^{8/3}}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac {2 b c \log (x)}{a^3}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{3 a^2 \left (a+b x^3\right )}-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]

[In]

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

[Out]

-1/3*c/(a^2*x^3) - d/(2*a^2*x^2) - e/(a^2*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(3*a^2*(a + b*x^3)) + (b^(1/3
)*(5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)) - (2*b*c*
Log[x])/a^3 - (b^(1/3)*(5*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)) + (b^(1/3)*(5*b^(1/3)
*d - 4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)) + (2*b*c*Log[a + b*x^3])/(3*a^3
)

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 1843

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x] + S
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]]] /; FreeQ[{a, b}, x] && PolyQ[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

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}& = -\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int \frac {-3 b c-3 b d x-3 b e x^2+\frac {3 b^2 c x^3}{a}+\frac {2 b^2 d x^4}{a}+\frac {b^2 e x^5}{a}}{x^4 \left (a+b x^3\right )} \, dx}{3 a b} \\ & = -\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b c}{a x^4}-\frac {3 b d}{a x^3}-\frac {3 b e}{a x^2}+\frac {6 b^2 c}{a^2 x}+\frac {b^2 \left (5 a d+4 a e x-6 b c x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b} \\ & = -\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {2 b c \log (x)}{a^3}-\frac {b \int \frac {5 a d+4 a e x-6 b c x^2}{a+b x^3} \, dx}{3 a^3} \\ & = -\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {2 b c \log (x)}{a^3}-\frac {b \int \frac {5 a d+4 a e x}{a+b x^3} \, dx}{3 a^3}+\frac {\left (2 b^2 c\right ) \int \frac {x^2}{a+b x^3} \, dx}{a^3} \\ & = -\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {2 b c \log (x)}{a^3}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac {b^{2/3} \int \frac {\sqrt [3]{a} \left (10 a \sqrt [3]{b} d+4 a^{4/3} e\right )+\sqrt [3]{b} \left (-5 a \sqrt [3]{b} d+4 a^{4/3} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{11/3}}-\frac {\left (b \left (5 d-\frac {4 \sqrt [3]{a} e}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}} \\ & = -\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {2 b c \log (x)}{a^3}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3}+\frac {\left (\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\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}{18 a^{8/3}}-\frac {\left (b^{2/3} \left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{7/3}} \\ & = -\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}-\frac {2 b c \log (x)}{a^3}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (\sqrt [3]{b} \left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{8/3}} \\ & = -\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {2 b c \log (x)}{a^3}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}+\frac {2 b c \log \left (a+b x^3\right )}{3 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 a c}{x^3}-\frac {9 a d}{x^2}-\frac {18 a e}{x}-\frac {6 a b (c+x (d+e x))}{a+b x^3}+2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-36 b c \log (x)+2 \sqrt [3]{b} \left (-5 \sqrt [3]{a} \sqrt [3]{b} d+4 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} \left (5 \sqrt [3]{a} \sqrt [3]{b} d-4 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+12 b c \log \left (a+b x^3\right )}{18 a^3} \]

[In]

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

[Out]

((-6*a*c)/x^3 - (9*a*d)/x^2 - (18*a*e)/x - (6*a*b*(c + x*(d + e*x)))/(a + b*x^3) + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(
5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 36*b*c*Log[x] + 2*b^(1/3)*(-5*a^(1/3)
*b^(1/3)*d + 4*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*d - 4*a^(2/3)*e)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 12*b*c*Log[a + b*x^3])/(18*a^3)

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.03

method	result	size
default	\(-\frac {e}{a^{2} x}-\frac {c}{3 a^{2} x^{3}}-\frac {d}{2 a^{2} x^{2}}-\frac {2 b c \ln \left (x \right )}{a^{3}}-\frac {b \left (\frac {\frac {1}{3} a e \,x^{2}+\frac {1}{3} a d x +\frac {1}{3} a c}{b \,x^{3}+a}+\frac {5 a d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {4 a e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 c \ln \left (b \,x^{3}+a \right )}{3}\right )}{a^{3}}\)	\(271\)
risch	\(\frac {-\frac {4 b e \,x^{5}}{3 a^{2}}-\frac {5 b d \,x^{4}}{6 a^{2}}-\frac {2 b c \,x^{3}}{3 a^{2}}-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3} \left (b \,x^{3}+a \right )}-\frac {2 b c \ln \left (x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} \textit {\_Z}^{3}-18 a^{6} b c \,\textit {\_Z}^{2}+\left (60 a^{4} b d e +108 a^{3} b^{2} c^{2}\right ) \textit {\_Z} -64 a^{2} b \,e^{3}-360 a \,b^{2} c d e +125 a \,b^{2} d^{3}-216 b^{3} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8}+48 a^{5} b c \,\textit {\_R}^{2}+\left (-200 a^{3} b d e -144 a^{2} b^{2} c^{2}\right ) \textit {\_R} +192 a b \,e^{3}+720 b^{2} d c e -375 b^{2} d^{3}\right ) x -4 \textit {\_R}^{2} a^{6} e +\left (-48 a^{3} b c e -25 a^{3} b \,d^{2}\right ) \textit {\_R} +432 b^{2} c^{2} e -450 b^{2} c \,d^{2}\right )\right )}{9}\)	\(275\)

[In]

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

[Out]

-e/a^2/x-1/3*c/a^2/x^3-1/2*d/a^2/x^2-2*b*c*ln(x)/a^3-1/a^3*b*((1/3*a*e*x^2+1/3*a*d*x+1/3*a*c)/(b*x^3+a)+5/3*a*
d*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3
^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))+4/3*a*e*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3
)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-2/3*c*l
n(b*x^3+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.19 (sec) , antiderivative size = 5373, normalized size of antiderivative = 20.51 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {8 \, b e x^{5} + 5 \, b d x^{4} + 4 \, b c x^{3} + 6 \, a e x^{2} + 3 \, a d x + 2 \, a c}{6 \, {\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} - \frac {2 \, b c \log \left (x\right )}{a^{3}} - \frac {\sqrt {3} {\left (4 \, a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} + \frac {{\left (12 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (6 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/6*(8*b*e*x^5 + 5*b*d*x^4 + 4*b*c*x^3 + 6*a*e*x^2 + 3*a*d*x + 2*a*c)/(a^2*b*x^6 + a^3*x^3) - 2*b*c*log(x)/a^
3 - 1/9*sqrt(3)*(4*a*e*(a/b)^(2/3) + 5*a*d*(a/b)^(1/3))*b*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/
a^4 + 1/18*(12*b*c*(a/b)^(2/3) - 4*a*e*(a/b)^(1/3) + 5*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^
(2/3)) + 1/9*(6*b*c*(a/b)^(2/3) + 4*a*e*(a/b)^(1/3) - 5*a*d)*log(x + (a/b)^(1/3))/(a^3*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {2 \, b c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac {2 \, b c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b} + \frac {{\left (4 \, a^{4} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{4} b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{7} b} - \frac {8 \, a b e x^{5} + 5 \, a b d x^{4} + 4 \, a b c x^{3} + 6 \, a^{2} e x^{2} + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, {\left (b x^{3} + a\right )} a^{3} x^{3}} \]

[In]

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

[Out]

2/3*b*c*log(abs(b*x^3 + a))/a^3 - 2*b*c*log(abs(x))/a^3 - 1/9*sqrt(3)*(5*(-a*b^2)^(1/3)*b*d - 4*(-a*b^2)^(2/3)
*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) - 1/18*(5*(-a*b^2)^(1/3)*b*d + 4*(-a*b^2)^(2
/3)*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) + 1/9*(4*a^4*b^2*e*(-a/b)^(1/3) + 5*a^4*b^2*d)*(-a/b)^
(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b) - 1/6*(8*a*b*e*x^5 + 5*a*b*d*x^4 + 4*a*b*c*x^3 + 6*a^2*e*x^2 + 3*a^2*
d*x + 2*a^2*c)/((b*x^3 + a)*a^3*x^3)

Mupad [B] (verification not implemented)

Time = 9.29 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.05 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {50\,b^5\,c\,d^2-48\,b^5\,c^2\,e}{9\,a^6}-\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\,\left (\frac {25\,a^3\,b^4\,d^2+48\,c\,e\,a^3\,b^4}{9\,a^6}+\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\,\left (4\,b^3\,e+\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\,a^2\,b^3\,x\,36-\frac {48\,b^4\,c\,x}{a}\right )+\frac {x\,\left (600\,d\,e\,a^3\,b^4+432\,a^2\,b^5\,c^2\right )}{27\,a^6}\right )+\frac {x\,\left (-125\,b^5\,d^3+240\,c\,b^5\,d\,e+64\,a\,b^4\,e^3\right )}{27\,a^6}\right )\,\mathrm {root}\left (729\,a^9\,z^3-1458\,a^6\,b\,c\,z^2+540\,a^4\,b\,d\,e\,z+972\,a^3\,b^2\,c^2\,z-360\,a\,b^2\,c\,d\,e-64\,a^2\,b\,e^3+125\,a\,b^2\,d^3-216\,b^3\,c^3,z,k\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {d\,x}{2\,a}+\frac {2\,b\,c\,x^3}{3\,a^2}+\frac {5\,b\,d\,x^4}{6\,a^2}+\frac {4\,b\,e\,x^5}{3\,a^2}}{b\,x^6+a\,x^3}-\frac {2\,b\,c\,\ln \left (x\right )}{a^3} \]

[In]

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

[Out]

symsum(log((x*(64*a*b^4*e^3 - 125*b^5*d^3 + 240*b^5*c*d*e))/(27*a^6) - root(729*a^9*z^3 - 1458*a^6*b*c*z^2 + 5
40*a^4*b*d*e*z + 972*a^3*b^2*c^2*z - 360*a*b^2*c*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3*c^3, z, k)*((25*
a^3*b^4*d^2 + 48*a^3*b^4*c*e)/(9*a^6) + root(729*a^9*z^3 - 1458*a^6*b*c*z^2 + 540*a^4*b*d*e*z + 972*a^3*b^2*c^
2*z - 360*a*b^2*c*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3*c^3, z, k)*(4*b^3*e + 36*root(729*a^9*z^3 - 145
8*a^6*b*c*z^2 + 540*a^4*b*d*e*z + 972*a^3*b^2*c^2*z - 360*a*b^2*c*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3
*c^3, z, k)*a^2*b^3*x - (48*b^4*c*x)/a) + (x*(432*a^2*b^5*c^2 + 600*a^3*b^4*d*e))/(27*a^6)) - (50*b^5*c*d^2 -
48*b^5*c^2*e)/(9*a^6))*root(729*a^9*z^3 - 1458*a^6*b*c*z^2 + 540*a^4*b*d*e*z + 972*a^3*b^2*c^2*z - 360*a*b^2*c
*d*e - 64*a^2*b*e^3 + 125*a*b^2*d^3 - 216*b^3*c^3, z, k), k, 1, 3) - (c/(3*a) + (e*x^2)/a + (d*x)/(2*a) + (2*b
*c*x^3)/(3*a^2) + (5*b*d*x^4)/(6*a^2) + (4*b*e*x^5)/(3*a^2))/(a*x^3 + b*x^6) - (2*b*c*log(x))/a^3