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

Optimal. Leaf size=310 \[ \frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{20 \sqrt [3]{b} \left (7 \sqrt [3]{a} d+11 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}-\frac{e \log \left (a+b x^3\right )}{3 a^4}-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}+\frac{e \log (x)}{a^4} \]

[Out]

-c/(2*a^4*x^2) - d/(a^4*x) - (x*(b*c + b*d*x + b*e*x^2))/(9*a^2*(a + b*x^3)^3) - (x*(17*b*c + 16*b*d*x + 15*b*
e*x^2))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*c + 118*b*d*x + 99*b*e*x^2))/(162*a^4*(a + b*x^3)) + (20*b^(1/3)*(1
1*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(14/3)) + (e*Log[x
])/a^4 - (20*b^(1/3)*(11*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(14/3)) + (10*b^(1/3)*(11*b
^(1/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(14/3)) - (e*Log[a + b*x^3])/(3
*a^4)

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Rubi [A]  time = 0.65622, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{20 \sqrt [3]{b} \left (7 \sqrt [3]{a} d+11 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}-\frac{e \log \left (a+b x^3\right )}{3 a^4}-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}+\frac{e \log (x)}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*a^4*x^2) - d/(a^4*x) - (x*(b*c + b*d*x + b*e*x^2))/(9*a^2*(a + b*x^3)^3) - (x*(17*b*c + 16*b*d*x + 15*b*
e*x^2))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*c + 118*b*d*x + 99*b*e*x^2))/(162*a^4*(a + b*x^3)) + (20*b^(1/3)*(1
1*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(14/3)) + (e*Log[x
])/a^4 - (20*b^(1/3)*(11*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(14/3)) + (10*b^(1/3)*(11*b
^(1/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(14/3)) - (e*Log[a + b*x^3])/(3
*a^4)

Rule 1829

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)*Coeff[R, x, i]*x^(i - m))/a, {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[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

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 1871

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]

Rule 1860

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

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 617

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 260

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]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2}{x^3 \left (a+b x^3\right )^4} \, dx &=-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{\int \frac{-9 b c-9 b d x-9 b e x^2+\frac{8 b^2 c x^3}{a}+\frac{7 b^2 d x^4}{a}+\frac{6 b^2 e x^5}{a}}{x^3 \left (a+b x^3\right )^3} \, dx}{9 a b}\\ &=-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e 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+54 b^3 e x^2-\frac{85 b^4 c x^3}{a}-\frac{64 b^4 d x^4}{a}-\frac{45 b^4 e x^5}{a}}{x^3 \left (a+b x^3\right )^2} \, dx}{54 a^2 b^3}\\ &=-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{\int \frac{-162 b^5 c-162 b^5 d x-162 b^5 e x^2+\frac{278 b^6 c x^3}{a}+\frac{118 b^6 d x^4}{a}}{x^3 \left (a+b x^3\right )} \, dx}{162 a^3 b^5}\\ &=-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{162 b^5 c}{a x^3}-\frac{162 b^5 d}{a x^2}-\frac{162 b^5 e}{a x}+\frac{2 b^6 \left (220 c+140 d x+81 e x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{162 a^3 b^5}\\ &=-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{e \log (x)}{a^4}-\frac{b \int \frac{220 c+140 d x+81 e x^2}{a+b x^3} \, dx}{81 a^4}\\ &=-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{e \log (x)}{a^4}-\frac{b \int \frac{220 c+140 d x}{a+b x^3} \, dx}{81 a^4}-\frac{(b e) \int \frac{x^2}{a+b x^3} \, dx}{a^4}\\ &=-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{e \log (x)}{a^4}-\frac{e \log \left (a+b x^3\right )}{3 a^4}-\frac{b^{2/3} \int \frac{\sqrt [3]{a} \left (440 \sqrt [3]{b} c+140 \sqrt [3]{a} d\right )+\sqrt [3]{b} \left (-220 \sqrt [3]{b} c+140 \sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{14/3}}-\frac{\left (20 b \left (11 c-\frac{7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{14/3}}\\ &=-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{e \log (x)}{a^4}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}-\frac{e \log \left (a+b x^3\right )}{3 a^4}+\frac{\left (10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right )\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{14/3}}-\frac{\left (10 b^{2/3} \left (11 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^{13/3}}\\ &=-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{e \log (x)}{a^4}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac{e \log \left (a+b x^3\right )}{3 a^4}-\frac{\left (20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{14/3}}\\ &=-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}+\frac{e \log (x)}{a^4}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac{e \log \left (a+b x^3\right )}{3 a^4}\\ \end{align*}

Mathematica [A]  time = 0.286433, size = 284, normalized size = 0.92 \[ \frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{54 a^3 (a e-b x (c+d x))}{\left (a+b x^3\right )^3}+\frac{9 a^2 (9 a e-b x (17 c+16 d x))}{\left (a+b x^3\right )^2}+40 \sqrt [3]{b} \left (7 a^{2/3} d-11 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a (54 a e-b x (139 c+118 d x))}{a+b x^3}+40 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} d+11 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-162 a e \log \left (a+b x^3\right )-\frac{243 a c}{x^2}-\frac{486 a d}{x}+486 a e \log (x)}{486 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-243*a*c)/x^2 - (486*a*d)/x + (54*a^3*(a*e - b*x*(c + d*x)))/(a + b*x^3)^3 + (9*a^2*(9*a*e - b*x*(17*c + 16*
d*x)))/(a + b*x^3)^2 + (3*a*(54*a*e - b*x*(139*c + 118*d*x)))/(a + b*x^3) + 40*Sqrt[3]*a^(1/3)*b^(1/3)*(11*b^(
1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 486*a*e*Log[x] + 40*b^(1/3)*(-11*a^(1/3)*b
^(1/3)*c + 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + 20*b^(1/3)*(11*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 162*a*e*Log[a + b*x^3])/(486*a^5)

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Maple [A]  time = 0.019, size = 400, normalized size = 1.3 \begin{align*} -{\frac{59\,{b}^{3}d{x}^{8}}{81\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{139\,{b}^{3}c{x}^{7}}{162\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{{b}^{2}e{x}^{6}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{142\,{b}^{2}d{x}^{5}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{329\,{b}^{2}c{x}^{4}}{162\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{5\,be{x}^{3}}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{92\,bd{x}^{2}}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{104\,bcx}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{11\,e}{18\,a \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{220\,c}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{110\,c}{243\,{a}^{4}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{220\,c\sqrt{3}}{243\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{140\,d}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{70\,d}{243\,{a}^{4}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{140\,d\sqrt{3}}{243\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{4}}}-{\frac{c}{2\,{a}^{4}{x}^{2}}}-{\frac{d}{{a}^{4}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-59/81*b^3/a^4/(b*x^3+a)^3*d*x^8-139/162*b^3/a^4/(b*x^3+a)^3*c*x^7+1/3*b^2/a^3/(b*x^3+a)^3*e*x^6-142/81*b^2/a^
3/(b*x^3+a)^3*d*x^5-329/162*b^2/a^3/(b*x^3+a)^3*c*x^4+5/6*b/a^2/(b*x^3+a)^3*e*x^3-92/81*b/a^2/(b*x^3+a)^3*d*x^
2-104/81*b/a^2/(b*x^3+a)^3*c*x+11/18/a/(b*x^3+a)^3*e-220/243/a^4*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+110/243/a
^4*c/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-220/243/a^4*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/(1/b*a)^(1/3)*x-1))+140/243/a^4*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-70/243/a^4*d/(1/b*a)^(1/3)*ln(x^2-(1/
b*a)^(1/3)*x+(1/b*a)^(2/3))-140/243/a^4*d*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/3*
e*ln(b*x^3+a)/a^4-1/2*c/a^4/x^2-d/a^4/x+e*ln(x)/a^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 7.40401, size = 15485, normalized size = 49.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/236196*(408240*b^3*d*x^10 + 320760*b^3*c*x^9 - 78732*a*b^2*e*x^8 + 1122660*a*b^2*d*x^7 + 833976*a*b^2*c*x^6
- 196830*a^2*b*e*x^5 + 976860*a^2*b*d*x^4 + 657558*a^2*b*c*x^3 - 144342*a^3*e*x^2 + 236196*a^3*d*x + 118098*a
^3*c + 2*(a^4*b^3*x^11 + 3*a^5*b^2*x^8 + 3*a^6*b*x^5 + a^7*x^2)*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d
+ 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 +
343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1
/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331
*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/
a^14)^(1/3) + 39366*e/a^4)*log(7/236196*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/
27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/2
8697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) +
1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^
14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^
4)^2*a^10*d + 431200*a*b*c*d^2 - 196020*a*b*c^2*e + 45927*a^2*d*e^2 + 1/243*(1210*a^5*b*c^2 - 567*a^6*d*e)*((-
I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*
a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3
- 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d
+ 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*
a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4) + 400*(1331*b^2*c^3 + 343*a*b*d^3)*x) +
(118098*b^3*e*x^11 + 354294*a*b^2*e*x^8 + 354294*a^2*b*e*x^5 + 118098*a^3*e*x^2 - (a^4*b^3*x^11 + 3*a^5*b^2*x^
8 + 3*a^6*b*x^5 + a^7*x^2)*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 +
1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(1064
8000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^
3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/286978
14*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4) - 3*sqrt(1
/3)*(a^4*b^3*x^11 + 3*a^5*b^2*x^8 + 3*a^6*b*x^5 + a^7*x^2)*sqrt(-(((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c
*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3
+ 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^
(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(13
31*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b
)/a^14)^(1/3) + 39366*e/a^4)^2*a^9 - 78732*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(
-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 -
1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3
) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b
/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e
/a^4)*a^5*e + 29099347200*b*c*d + 1549681956*a*e^2)/a^9))*log(-7/236196*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (308
00*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331
*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/
a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/143489
07*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*
e)*a*b)/a^14)^(1/3) + 39366*e/a^4)^2*a^10*d - 431200*a*b*c*d^2 + 196020*a*b*c^2*e - 45927*a^2*d*e^2 - 1/243*(1
210*a^5*b*c^2 - 567*a^6*d*e)*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12
+ 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10
648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*
e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/2869
7814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4) + 800*(1
331*b^2*c^3 + 343*a*b*d^3)*x + 1/78732*sqrt(1/3)*(7*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^
2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*
b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*
(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343
*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3)
+ 39366*e/a^4)*a^10*d - 1176120*a^5*b*c^2 - 275562*a^6*d*e)*sqrt(-(((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*
c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^
3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)
^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1
331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*
b)/a^14)^(1/3) + 39366*e/a^4)^2*a^9 - 78732*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/
(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 -
1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(
3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*
b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*
e/a^4)*a^5*e + 29099347200*b*c*d + 1549681956*a*e^2)/a^9)) + (118098*b^3*e*x^11 + 354294*a*b^2*e*x^8 + 354294*
a^2*b*e*x^5 + 118098*a^3*e*x^2 - (a^4*b^3*x^11 + 3*a^5*b^2*x^8 + 3*a^6*b*x^5 + a^7*x^2)*((-I*sqrt(3) + 1)*(656
1*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 400
0/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2
673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^
13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980
*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4) + 3*sqrt(1/3)*(a^4*b^3*x^11 + 3*a^5*b^2*x^8 + 3*a^6*b*x^5 +
a^7*x^2)*sqrt(-(((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*
(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c
^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1
/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(106480
00*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4)^2*a^9 - 78732*((-I*s
qrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e
^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2
800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d +
6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2
*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4)*a^5*e + 29099347200*b*c*d + 1549681956*a*e^
2)/a^9))*log(-7/236196*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/1
18098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000
*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^
12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(
10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4)^2*a^10*d - 431
200*a*b*c*d^2 + 196020*a*b*c^2*e - 45927*a^2*d*e^2 - 1/243*(1210*a^5*b*c^2 - 567*a^6*d*e)*((-I*sqrt(3) + 1)*(6
561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4
000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 -
2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*e^2)*e/
a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 - 2800*(9
80*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4) + 800*(1331*b^2*c^3 + 343*a*b*d^3)*x - 1/78732*sqrt(1/3)*
(7*((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d +
6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^
2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800
*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 5
31441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4)*a^10*d - 1176120*a^5*b*c^2 - 27556
2*a^6*d*e)*sqrt(-(((-I*sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098
*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*
c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 +
1/118098*(30800*b*c*d + 6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648
000*b^2*c^3 + 531441*a^2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4)^2*a^9 - 78732*((-I*
sqrt(3) + 1)*(6561*e^2/a^8 - (30800*b*c*d + 6561*a*e^2)/a^9)/(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d + 6561*a*
e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^2*e^3 -
2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 59049*(I*sqrt(3) + 1)*(-1/27*e^3/a^12 + 1/118098*(30800*b*c*d +
6561*a*e^2)*e/a^13 + 4000/14348907*(1331*b*c^3 + 343*a*d^3)*b/a^14 - 1/28697814*(10648000*b^2*c^3 + 531441*a^
2*e^3 - 2800*(980*d^3 - 2673*c*d*e)*a*b)/a^14)^(1/3) + 39366*e/a^4)*a^5*e + 29099347200*b*c*d + 1549681956*a*e
^2)/a^9)) - 236196*(b^3*e*x^11 + 3*a*b^2*e*x^8 + 3*a^2*b*e*x^5 + a^3*e*x^2)*log(x))/(a^4*b^3*x^11 + 3*a^5*b^2*
x^8 + 3*a^6*b*x^5 + a^7*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.07115, size = 441, normalized size = 1.42 \begin{align*} -\frac{e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac{e \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{10 \,{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{243 \, a^{5} b} - \frac{20 \, \sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{6} b^{3}} + \frac{20 \,{\left (7 \, a^{4} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 11 \, a^{4} b^{2} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{9} b} - \frac{280 \, b^{3} d x^{10} + 220 \, b^{3} c x^{9} - 54 \, a b^{2} x^{8} e + 770 \, a b^{2} d x^{7} + 572 \, a b^{2} c x^{6} - 135 \, a^{2} b x^{5} e + 670 \, a^{2} b d x^{4} + 451 \, a^{2} b c x^{3} - 99 \, a^{3} x^{2} e + 162 \, a^{3} d x + 81 \, a^{3} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e*log(abs(b*x^3 + a))/a^4 + e*log(abs(x))/a^4 - 10/243*(11*(-a*b^2)^(1/3)*b*c + 7*(-a*b^2)^(2/3)*d)*log(x
^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 20/243*sqrt(3)*(11*(-a*b^2)^(1/3)*a*b^3*c - 7*(-a*b^2)^(2/3)*a*b
^2*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b^3) + 20/243*(7*a^4*b^2*d*(-a/b)^(1/3) + 11*
a^4*b^2*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^9*b) - 1/162*(280*b^3*d*x^10 + 220*b^3*c*x^9 - 54*a*b^2*
x^8*e + 770*a*b^2*d*x^7 + 572*a*b^2*c*x^6 - 135*a^2*b*x^5*e + 670*a^2*b*d*x^4 + 451*a^2*b*c*x^3 - 99*a^3*x^2*e
+ 162*a^3*d*x + 81*a^3*c)/((b*x^3 + a)^3*a^4*x^2)