\(\int \frac {c+d x+e x^2}{x^2 (a+b x^3)^3} \, dx\) [355]

Optimal result

Integrand size = 23, antiderivative size = 267 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3} \]

[Out]

-c/a^3/x+1/6*x*(-b*d*x^2-b*c*x+a*e)/a^2/(b*x^3+a)^2+1/18*x*(-9*b*d*x^2-10*b*c*x+5*a*e)/a^3/(b*x^3+a)+d*ln(x)/a
^3+1/27*(14*b^(2/3)*c+5*a^(2/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^(1/3)-1/54*(14*b^(2/3)*c+5*a^(2/3)*e)*ln(a
^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/b^(1/3)-1/3*d*ln(b*x^3+a)/a^3+1/27*(14*b^(2/3)*c-5*a^(2/3)*e)*a
rctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(10/3)/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (14 b^{2/3} c-5 a^{2/3} e\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}-\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {c}{a^3 x}+\frac {d \log (x)}{a^3}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[In]

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

[Out]

-(c/(a^3*x)) + (x*(a*e - b*c*x - b*d*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*e - 10*b*c*x - 9*b*d*x^2))/(18*a^3*
(a + b*x^3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(
10/3)*b^(1/3)) + (d*Log[x])/a^3 + ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(1/3)
) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*b^(1/3)) - (d*L
og[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 (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b c-6 b d x-5 b e x^2+\frac {4 b^2 c x^3}{a}+\frac {3 b^2 d x^4}{a}}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^3 c+18 b^3 d x+10 b^3 e x^2-\frac {10 b^4 c x^3}{a}}{x^2 \left (a+b x^3\right )} \, dx}{18 a^2 b^3} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^3 c}{a x^2}+\frac {18 b^3 d}{a x}+\frac {2 b^3 \left (5 a e-14 b c x-9 b d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {5 a e-14 b c x-9 b d x^2}{a+b x^3} \, dx}{9 a^3} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {5 a e-14 b c x}{a+b x^3} \, dx}{9 a^3}-\frac {(b d) \int \frac {x^2}{a+b x^3} \, dx}{a^3} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}-\frac {d \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (-14 \sqrt [3]{a} b c+10 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-14 \sqrt [3]{a} b c-5 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3}} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^3}-\frac {\left (14 b^{2/3} c+5 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}{54 a^{10/3} \sqrt [3]{b}} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (14 b^{2/3} c-5 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 )}{9 a^{10/3} \sqrt [3]{b}} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {54 a c}{x}+\frac {3 a \left (6 a d+5 a e x-10 b c x^2\right )}{a+b x^3}+\frac {9 a^2 \left (-b c x^2+a (d+e x)\right )}{\left (a+b x^3\right )^2}-\frac {2 \sqrt {3} a^{2/3} \left (-14 b^{2/3} c+5 a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+54 a d \log (x)+\frac {2 \left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-18 a d \log \left (a+b x^3\right )}{54 a^4} \]

[In]

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

[Out]

((-54*a*c)/x + (3*a*(6*a*d + 5*a*e*x - 10*b*c*x^2))/(a + b*x^3) + (9*a^2*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^
3)^2 - (2*Sqrt[3]*a^(2/3)*(-14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) +
54*a*d*Log[x] + (2*(14*a^(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - ((14*a^(2/3)*b^(2
/3)*c + 5*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 18*a*d*Log[a + b*x^3])/(54*a^4)

Maple [C] (verified)

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

Time = 1.55 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {-\frac {14 c \,b^{2} x^{6}}{9 a^{3}}+\frac {5 b e \,x^{5}}{18 a^{2}}+\frac {b d \,x^{4}}{3 a^{2}}-\frac {49 b c \,x^{3}}{18 a^{2}}+\frac {4 e \,x^{2}}{9 a}+\frac {x d}{2 a}-\frac {c}{a}}{x \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} b \,\textit {\_Z}^{3}+27 a^{7} b d \,\textit {\_Z}^{2}+\left (-210 a^{4} b c e +243 a^{4} b \,d^{2}\right ) \textit {\_Z} -125 a^{2} e^{3}-1890 a b c d e +729 a b \,d^{3}-2744 b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10} b -72 \textit {\_R}^{2} a^{7} b d +\left (700 a^{4} b c e -324 a^{4} b \,d^{2}\right ) \textit {\_R} +375 a^{2} e^{3}+3780 a b c d e +8232 b^{2} c^{3}\right ) x -14 a^{7} b c \,\textit {\_R}^{2}+\left (-25 a^{5} e^{2}+252 a^{4} b c d \right ) \textit {\_R} +675 a^{2} d \,e^{2}+3402 a b c \,d^{2}\right )\right )}{27}+\frac {d \ln \left (x \right )}{a^{3}}\)	\(277\)
default	\(-\frac {c}{a^{3} x}+\frac {d \ln \left (x \right )}{a^{3}}+\frac {\frac {-\frac {5}{9} b^{2} c \,x^{5}+\frac {5}{18} a b e \,x^{4}+\frac {1}{3} x^{3} a b d -\frac {13}{18} a b c \,x^{2}+\frac {4}{9} a^{2} e x +\frac {1}{2} a^{2} d}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}-\frac {14 b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9}-\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{a^{3}}\)	\(279\)

[In]

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

[Out]

(-14/9*c/a^3*b^2*x^6+5/18*b*e/a^2*x^5+1/3*b*d/a^2*x^4-49/18*b*c/a^2*x^3+4/9/a*e*x^2+1/2/a*x*d-c/a)/x/(b*x^3+a)
^2+1/27*sum(_R*ln((-4*_R^3*a^10*b-72*_R^2*a^7*b*d+(700*a^4*b*c*e-324*a^4*b*d^2)*_R+375*a^2*e^3+3780*a*b*c*d*e+
8232*b^2*c^3)*x-14*a^7*b*c*_R^2+(-25*a^5*e^2+252*a^4*b*c*d)*_R+675*a^2*d*e^2+3402*a*b*c*d^2),_R=RootOf(a^10*b*
_Z^3+27*a^7*b*d*_Z^2+(-210*a^4*b*c*e+243*a^4*b*d^2)*_Z-125*a^2*e^3-1890*a*b*c*d*e+729*a*b*d^3-2744*b^2*c^3))+d
*ln(x)/a^3

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 5112, normalized size of antiderivative = 19.15 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {28 \, b^{2} c x^{6} - 5 \, a b e x^{5} - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} e x^{2} - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} + \frac {d \log \left (x\right )}{a^{3}} - \frac {\sqrt {3} {\left (14 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4}} - \frac {{\left (18 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/18*(28*b^2*c*x^6 - 5*a*b*e*x^5 - 6*a*b*d*x^4 + 49*a*b*c*x^3 - 8*a^2*e*x^2 - 9*a^2*d*x + 18*a^2*c)/(a^3*b^2*
x^7 + 2*a^4*b*x^4 + a^5*x) + d*log(x)/a^3 - 1/27*sqrt(3)*(14*b*c*(a/b)^(2/3) - 5*a*e*(a/b)^(1/3))*arctan(1/3*s
qrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^4 - 1/54*(18*b*d*(a/b)^(2/3) + 14*b*c*(a/b)^(1/3) + 5*a*e)*log(x^2 -
x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(2/3)) - 1/27*(9*b*d*(a/b)^(2/3) - 14*b*c*(a/b)^(1/3) - 5*a*e)*log(
x + (a/b)^(1/3))/(a^3*b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e + 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} + \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b} - \frac {28 \, b^{2} c x^{6} - 5 \, a b e x^{5} - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} e x^{2} - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} x} + \frac {{\left (14 \, a^{3} b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a^{4} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b} \]

[In]

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

[Out]

-1/3*d*log(abs(b*x^3 + a))/a^3 + d*log(abs(x))/a^3 + 1/27*sqrt(3)*(5*(-a*b^2)^(1/3)*a*e + 14*(-a*b^2)^(2/3)*c)
*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b) + 1/54*(5*(-a*b^2)^(1/3)*a*e - 14*(-a*b^2)^(2/3
)*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/18*(28*b^2*c*x^6 - 5*a*b*e*x^5 - 6*a*b*d*x^4 + 49*a*
b*c*x^3 - 8*a^2*e*x^2 - 9*a^2*d*x + 18*a^2*c)/((b*x^3 + a)^2*a^3*x) + 1/27*(14*a^3*b^2*c*(-a/b)^(1/3) - 5*a^4*
b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b)

Mupad [B] (verification not implemented)

Time = 9.35 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.97 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {\frac {4\,e\,x^2}{9\,a}-\frac {c}{a}+\frac {d\,x}{2\,a}-\frac {14\,b^2\,c\,x^6}{9\,a^3}-\frac {49\,b\,c\,x^3}{18\,a^2}+\frac {b\,d\,x^4}{3\,a^2}+\frac {5\,b\,e\,x^5}{18\,a^2}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}+\left (\sum _{k=1}^3\ln \left (\frac {b^2\,\left (-\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^5\,e^2\,225+225\,a^2\,d\,e^2+2744\,b^2\,c^3\,x+125\,a^2\,e^3\,x+1134\,a\,b\,c\,d^2-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b\,c\,3402-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^3\,a^{10}\,b\,x\,26244-\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,d^2\,x\,2916-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b\,d\,x\,17496+\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,c\,d\,2268+\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,c\,e\,x\,6300+1260\,a\,b\,c\,d\,e\,x\right )}{a^8\,729}\right )\,\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\right )+\frac {d\,\ln \left (x\right )}{a^3} \]

[In]

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

[Out]

((4*e*x^2)/(9*a) - c/a + (d*x)/(2*a) - (14*b^2*c*x^6)/(9*a^3) - (49*b*c*x^3)/(18*a^2) + (b*d*x^4)/(3*a^2) + (5
*b*e*x^5)/(18*a^2))/(a^2*x + b^2*x^7 + 2*a*b*x^4) + symsum(log((b^2*(225*a^2*d*e^2 - 225*root(19683*a^10*b*z^3
+ 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744
*b^2*c^3, z, k)*a^5*e^2 + 2744*b^2*c^3*x + 125*a^2*e^3*x + 1134*a*b*c*d^2 - 3402*root(19683*a^10*b*z^3 + 19683
*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3
, z, k)^2*a^7*b*c - 26244*root(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 18
90*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3, z, k)^3*a^10*b*x - 2916*root(19683*a^10*b*z^3 + 19683
*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3
, z, k)*a^4*b*d^2*x - 17496*root(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z -
1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3, z, k)^2*a^7*b*d*x + 2268*root(19683*a^10*b*z^3 + 19
683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*
c^3, z, k)*a^4*b*c*d + 6300*root(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z -
1890*a*b*c*d*e + 729*a*b*d^3 - 125*a^2*e^3 - 2744*b^2*c^3, z, k)*a^4*b*c*e*x + 1260*a*b*c*d*e*x))/(729*a^8))*r
oot(19683*a^10*b*z^3 + 19683*a^7*b*d*z^2 - 5670*a^4*b*c*e*z + 6561*a^4*b*d^2*z - 1890*a*b*c*d*e + 729*a*b*d^3
- 125*a^2*e^3 - 2744*b^2*c^3, z, k), k, 1, 3) + (d*log(x))/a^3