\(\int \frac {c+d x+e x^2}{x^2 (a+b x^3)^4} \, dx\) [362]

Optimal result

Integrand size = 23, antiderivative size = 301 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {10 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4} \]

[Out]

-c/a^4/x+1/9*x*(-b*d*x^2-b*c*x+a*e)/a^2/(b*x^3+a)^3+1/54*x*(-15*b*d*x^2-16*b*c*x+8*a*e)/a^3/(b*x^3+a)^2+1/162*
x*(-99*b*d*x^2-118*b*c*x+40*a*e)/a^4/(b*x^3+a)+d*ln(x)/a^4+20/243*(7*b^(2/3)*c+2*a^(2/3)*e)*ln(a^(1/3)+b^(1/3)
*x)/a^(13/3)/b^(1/3)-10/243*(7*b^(2/3)*c+2*a^(2/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)/b^(1/
3)-1/3*d*ln(b*x^3+a)/a^4+20/243*(7*b^(2/3)*c-2*a^(2/3)*e)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^
(13/3)/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\frac {20 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 b^{2/3} c-2 a^{2/3} e\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}-\frac {10 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac {20 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {c}{a^4 x}+\frac {d \log (x)}{a^4}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

[In]

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

[Out]

-(c/(a^4*x)) + (x*(a*e - b*c*x - b*d*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*e - 16*b*c*x - 15*b*d*x^2))/(54*a^3
*(a + b*x^3)^2) + (x*(40*a*e - 118*b*c*x - 99*b*d*x^2))/(162*a^4*(a + b*x^3)) + (20*(7*b^(2/3)*c - 2*a^(2/3)*e
)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(13/3)*b^(1/3)) + (d*Log[x])/a^4 + (20*(7*b
^(2/3)*c + 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(13/3)*b^(1/3)) - (10*(7*b^(2/3)*c + 2*a^(2/3)*e)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(13/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^4)

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 )}{9 a^2 \left (a+b x^3\right )^3}-\frac {\int \frac {-9 b c-9 b d x-8 b e x^2+\frac {7 b^2 c x^3}{a}+\frac {6 b^2 d x^4}{a}}{x^2 \left (a+b x^3\right )^3} \, dx}{9 a b} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {\int \frac {54 b^3 c+54 b^3 d x+40 b^3 e x^2-\frac {64 b^4 c x^3}{a}-\frac {45 b^4 d x^4}{a}}{x^2 \left (a+b x^3\right )^2} \, dx}{54 a^2 b^3} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac {\int \frac {-162 b^5 c-162 b^5 d x-80 b^5 e x^2+\frac {118 b^6 c x^3}{a}}{x^2 \left (a+b x^3\right )} \, dx}{162 a^3 b^5} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac {\int \left (-\frac {162 b^5 c}{a x^2}-\frac {162 b^5 d}{a x}-\frac {2 b^5 \left (40 a e-140 b c x-81 b d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{162 a^3 b^5} \\ & = -\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {\int \frac {40 a e-140 b c x-81 b d x^2}{a+b x^3} \, dx}{81 a^4} \\ & = -\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {\int \frac {40 a e-140 b c x}{a+b x^3} \, dx}{81 a^4}-\frac {(b d) \int \frac {x^2}{a+b x^3} \, dx}{a^4} \\ & = -\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}-\frac {d \log \left (a+b x^3\right )}{3 a^4}+\frac {\int \frac {\sqrt [3]{a} \left (-140 \sqrt [3]{a} b c+80 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-140 \sqrt [3]{a} b c-40 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{14/3} \sqrt [3]{b}}+\frac {\left (20 \left (7 b^{2/3} c+2 a^{2/3} e\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{13/3}} \\ & = -\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {\left (10 \left (7 b^{2/3} c-2 a^{2/3} e\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^4}-\frac {\left (10 \left (7 b^{2/3} c+2 a^{2/3} 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}{243 a^{13/3} \sqrt [3]{b}} \\ & = -\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {10 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4}-\frac {\left (20 \left (7 b^{2/3} c-2 a^{2/3} 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 )}{81 a^{13/3} \sqrt [3]{b}} \\ & = -\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {10 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\frac {-\frac {486 a c}{x}+\frac {9 a^2 \left (9 a d+8 a e x-16 b c x^2\right )}{\left (a+b x^3\right )^2}+\frac {6 a \left (27 a d+20 a e x-59 b c x^2\right )}{a+b x^3}+\frac {54 a^3 \left (-b c x^2+a (d+e x)\right )}{\left (a+b x^3\right )^3}-\frac {40 \sqrt {3} a^{2/3} \left (-7 b^{2/3} c+2 a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+486 a d \log (x)+\frac {40 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {20 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-162 a d \log \left (a+b x^3\right )}{486 a^5} \]

[In]

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

[Out]

((-486*a*c)/x + (9*a^2*(9*a*d + 8*a*e*x - 16*b*c*x^2))/(a + b*x^3)^2 + (6*a*(27*a*d + 20*a*e*x - 59*b*c*x^2))/
(a + b*x^3) + (54*a^3*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3)^3 - (40*Sqrt[3]*a^(2/3)*(-7*b^(2/3)*c + 2*a^(2/3
)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 486*a*d*Log[x] + (40*(7*a^(2/3)*b^(2/3)*c + 2*a^(4
/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (20*(7*a^(2/3)*b^(2/3)*c + 2*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/b^(1/3) - 162*a*d*Log[a + b*x^3])/(486*a^5)

Maple [C] (verified)

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

Time = 1.55 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {-\frac {140 b^{3} c \,x^{9}}{81 a^{4}}+\frac {20 e \,b^{2} x^{8}}{81 a^{3}}+\frac {d \,b^{2} x^{7}}{3 a^{3}}-\frac {385 c \,b^{2} x^{6}}{81 a^{3}}+\frac {52 b e \,x^{5}}{81 a^{2}}+\frac {5 b d \,x^{4}}{6 a^{2}}-\frac {335 b c \,x^{3}}{81 a^{2}}+\frac {41 e \,x^{2}}{81 a}+\frac {11 x d}{18 a}-\frac {c}{a}}{x \left (b \,x^{3}+a \right )^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} b \,\textit {\_Z}^{3}+243 a^{9} b d \,\textit {\_Z}^{2}+\left (-16800 a^{5} b c e +19683 a^{5} b \,d^{2}\right ) \textit {\_Z} -64000 a^{2} e^{3}-1360800 a b c d e +531441 a b \,d^{3}-2744000 b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-\textit {\_R}^{3} a^{13} b -162 \textit {\_R}^{2} a^{9} b d +\left (14000 a^{5} b c e -6561 a^{5} b \,d^{2}\right ) \textit {\_R} +48000 a^{2} e^{3}+680400 a b c d e +2058000 b^{2} c^{3}\right ) x -35 a^{9} b c \,\textit {\_R}^{2}+\left (-400 a^{6} e^{2}+5670 a^{5} b c d \right ) \textit {\_R} +97200 a^{2} d \,e^{2}+688905 a b c \,d^{2}\right )\right )}{243}+\frac {d \ln \left (x \right )}{a^{4}}\)	\(313\)
default	\(-\frac {c}{a^{4} x}+\frac {d \ln \left (x \right )}{a^{4}}+\frac {\frac {-\frac {59}{81} b^{3} c \,x^{8}+\frac {20}{81} a \,b^{2} e \,x^{7}+\frac {1}{3} a \,b^{2} d \,x^{6}-\frac {142}{81} a \,b^{2} c \,x^{5}+\frac {52}{81} a^{2} b e \,x^{4}+\frac {5}{6} a^{2} b d \,x^{3}-\frac {92}{81} a^{2} b c \,x^{2}+\frac {41}{81} a^{3} e x +\frac {11}{18} a^{3} d}{\left (b \,x^{3}+a \right )^{3}}+\frac {40 a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{81}-\frac {140 b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{81}-\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{a^{4}}\)	\(315\)

[In]

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

[Out]

(-140/81/a^4*b^3*c*x^9+20/81*e/a^3*b^2*x^8+1/3*d/a^3*b^2*x^7-385/81*c/a^3*b^2*x^6+52/81*b*e/a^2*x^5+5/6*b*d/a^
2*x^4-335/81*b*c/a^2*x^3+41/81/a*e*x^2+11/18/a*x*d-c/a)/x/(b*x^3+a)^3+1/243*sum(_R*ln((-_R^3*a^13*b-162*_R^2*a
^9*b*d+(14000*a^5*b*c*e-6561*a^5*b*d^2)*_R+48000*a^2*e^3+680400*a*b*c*d*e+2058000*b^2*c^3)*x-35*a^9*b*c*_R^2+(
-400*a^6*e^2+5670*a^5*b*c*d)*_R+97200*a^2*d*e^2+688905*a*b*c*d^2),_R=RootOf(a^13*b*_Z^3+243*a^9*b*d*_Z^2+(-168
00*a^5*b*c*e+19683*a^5*b*d^2)*_Z-64000*a^2*e^3-1360800*a*b*c*d*e+531441*a*b*d^3-2744000*b^2*c^3))+d*ln(x)/a^4

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 5250, normalized size of antiderivative = 17.44 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=-\frac {280 \, b^{3} c x^{9} - 40 \, a b^{2} e x^{8} - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b e x^{5} - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} e x^{2} - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \, {\left (a^{4} b^{3} x^{10} + 3 \, a^{5} b^{2} x^{7} + 3 \, a^{6} b x^{4} + a^{7} x\right )}} + \frac {d \log \left (x\right )}{a^{4}} - \frac {20 \, \sqrt {3} {\left (7 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5}} - \frac {{\left (81 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 70 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (81 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 140 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 40 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/162*(280*b^3*c*x^9 - 40*a*b^2*e*x^8 - 54*a*b^2*d*x^7 + 770*a*b^2*c*x^6 - 104*a^2*b*e*x^5 - 135*a^2*b*d*x^4
+ 670*a^2*b*c*x^3 - 82*a^3*e*x^2 - 99*a^3*d*x + 162*a^3*c)/(a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x
) + d*log(x)/a^4 - 20/243*sqrt(3)*(7*b*c*(a/b)^(2/3) - 2*a*e*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3
))/(a/b)^(1/3))/a^5 - 1/243*(81*b*d*(a/b)^(2/3) + 70*b*c*(a/b)^(1/3) + 20*a*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)
^(2/3))/(a^4*b*(a/b)^(2/3)) - 1/243*(81*b*d*(a/b)^(2/3) - 140*b*c*(a/b)^(1/3) - 40*a*e)*log(x + (a/b)^(1/3))/(
a^4*b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=-\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {20 \, \sqrt {3} {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5} b} + \frac {10 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} b} - \frac {280 \, b^{3} c x^{9} - 40 \, a b^{2} e x^{8} - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b e x^{5} - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} e x^{2} - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \, {\left (b x^{3} + a\right )}^{3} a^{4} x} + \frac {20 \, {\left (7 \, a^{4} b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{5} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{9} b} \]

[In]

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

[Out]

-1/3*d*log(abs(b*x^3 + a))/a^4 + d*log(abs(x))/a^4 + 20/243*sqrt(3)*(2*(-a*b^2)^(1/3)*a*e + 7*(-a*b^2)^(2/3)*c
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) + 10/243*(2*(-a*b^2)^(1/3)*a*e - 7*(-a*b^2)^(2
/3)*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/162*(280*b^3*c*x^9 - 40*a*b^2*e*x^8 - 54*a*b^2*d*x
^7 + 770*a*b^2*c*x^6 - 104*a^2*b*e*x^5 - 135*a^2*b*d*x^4 + 670*a^2*b*c*x^3 - 82*a^3*e*x^2 - 99*a^3*d*x + 162*a
^3*c)/((b*x^3 + a)^3*a^4*x) + 20/243*(7*a^4*b^2*c*(-a/b)^(1/3) - 2*a^5*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1
/3)))/(a^9*b)

Mupad [B] (verification not implemented)

Time = 11.85 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.79 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\frac {\frac {41\,e\,x^2}{81\,a}-\frac {c}{a}+\frac {11\,d\,x}{18\,a}-\frac {385\,b^2\,c\,x^6}{81\,a^3}-\frac {140\,b^3\,c\,x^9}{81\,a^4}+\frac {b^2\,d\,x^7}{3\,a^3}+\frac {20\,b^2\,e\,x^8}{81\,a^3}-\frac {335\,b\,c\,x^3}{81\,a^2}+\frac {5\,b\,d\,x^4}{6\,a^2}+\frac {52\,b\,e\,x^5}{81\,a^2}}{a^3\,x+3\,a^2\,b\,x^4+3\,a\,b^2\,x^7+b^3\,x^{10}}+\left (\sum _{k=1}^3\ln \left (\frac {b^2\,\left (-\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^6\,e^2\,32400+32400\,a^2\,d\,e^2+686000\,b^2\,c^3\,x+16000\,a^2\,e^3\,x+229635\,a\,b\,c\,d^2-{\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )}^2\,a^9\,b\,c\,688905-{\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )}^3\,a^{13}\,b\,x\,4782969-\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^5\,b\,d^2\,x\,531441-{\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )}^2\,a^9\,b\,d\,x\,3188646+\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^5\,b\,c\,d\,459270+\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\,a^5\,b\,c\,e\,x\,1134000+226800\,a\,b\,c\,d\,e\,x\right )\,4}{a^{11}\,531441}\right )\,\mathrm {root}\left (14348907\,a^{13}\,b\,z^3+14348907\,a^9\,b\,d\,z^2-4082400\,a^5\,b\,c\,e\,z+4782969\,a^5\,b\,d^2\,z-1360800\,a\,b\,c\,d\,e+531441\,a\,b\,d^3-64000\,a^2\,e^3-2744000\,b^2\,c^3,z,k\right )\right )+\frac {d\,\ln \left (x\right )}{a^4} \]

[In]

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

[Out]

((41*e*x^2)/(81*a) - c/a + (11*d*x)/(18*a) - (385*b^2*c*x^6)/(81*a^3) - (140*b^3*c*x^9)/(81*a^4) + (b^2*d*x^7)
/(3*a^3) + (20*b^2*e*x^8)/(81*a^3) - (335*b*c*x^3)/(81*a^2) + (5*b*d*x^4)/(6*a^2) + (52*b*e*x^5)/(81*a^2))/(a^
3*x + b^3*x^10 + 3*a^2*b*x^4 + 3*a*b^2*x^7) + symsum(log((4*b^2*(32400*a^2*d*e^2 - 32400*root(14348907*a^13*b*
z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 -
64000*a^2*e^3 - 2744000*b^2*c^3, z, k)*a^6*e^2 + 686000*b^2*c^3*x + 16000*a^2*e^3*x + 229635*a*b*c*d^2 - 68890
5*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*
d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)^2*a^9*b*c - 4782969*root(14348907*a^13*b*z^3 + 1
4348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a
^2*e^3 - 2744000*b^2*c^3, z, k)^3*a^13*b*x - 531441*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*
a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k
)*a^5*b*d^2*x - 3188646*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*
d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)^2*a^9*b*d*x + 459270*root(
14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 5
31441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)*a^5*b*c*d + 1134000*root(14348907*a^13*b*z^3 + 14348907
*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3
- 2744000*b^2*c^3, z, k)*a^5*b*c*e*x + 226800*a*b*c*d*e*x))/(531441*a^11))*root(14348907*a^13*b*z^3 + 14348907
*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3
- 2744000*b^2*c^3, z, k), k, 1, 3) + (d*log(x))/a^4