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

Optimal. Leaf size=340 \[ -\frac{x \left (-\frac{234 b^2 c x^2}{a}+139 b d+118 b e x\right )}{162 a^4 \left (a+b x^3\right )}-\frac{x \left (-\frac{24 b^2 c x^2}{a}+17 b d+16 b e x\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}+\frac{4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac{4 b c \log (x)}{a^5}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{20 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x} \]

[Out]

-c/(3*a^4*x^3) - d/(2*a^4*x^2) - e/(a^4*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(9*a^2*(a + b*x^3)^3) - (x*(17*
b*d + 16*b*e*x - (24*b^2*c*x^2)/a))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*d + 118*b*e*x - (234*b^2*c*x^2)/a))/(16
2*a^4*(a + b*x^3)) + (20*b^(1/3)*(11*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(81*Sqrt[3]*a^(14/3)) - (4*b*c*Log[x])/a^5 - (20*b^(1/3)*(11*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)
*x])/(243*a^(14/3)) + (10*b^(1/3)*(11*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/(243*a^(14/3)) + (4*b*c*Log[a + b*x^3])/(3*a^5)

________________________________________________________________________________________

Rubi [A]  time = 0.7727, antiderivative size = 340, 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} \[ -\frac{x \left (-\frac{234 b^2 c x^2}{a}+139 b d+118 b e x\right )}{162 a^4 \left (a+b x^3\right )}-\frac{x \left (-\frac{24 b^2 c x^2}{a}+17 b d+16 b e x\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}+\frac{4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac{4 b c \log (x)}{a^5}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{20 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(3*a^4*x^3) - d/(2*a^4*x^2) - e/(a^4*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(9*a^2*(a + b*x^3)^3) - (x*(17*
b*d + 16*b*e*x - (24*b^2*c*x^2)/a))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*d + 118*b*e*x - (234*b^2*c*x^2)/a))/(16
2*a^4*(a + b*x^3)) + (20*b^(1/3)*(11*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(81*Sqrt[3]*a^(14/3)) - (4*b*c*Log[x])/a^5 - (20*b^(1/3)*(11*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)
*x])/(243*a^(14/3)) + (10*b^(1/3)*(11*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/(243*a^(14/3)) + (4*b*c*Log[a + b*x^3])/(3*a^5)

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^4 \left (a+b x^3\right )^4} \, dx &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\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{9 b^2 c x^3}{a}+\frac{8 b^2 d x^4}{a}+\frac{7 b^2 e x^5}{a}-\frac{6 b^3 c x^6}{a^2}}{x^4 \left (a+b x^3\right )^3} \, dx}{9 a b}\\ &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\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{108 b^4 c x^3}{a}-\frac{85 b^4 d x^4}{a}-\frac{64 b^4 e x^5}{a}+\frac{72 b^5 c x^6}{a^2}}{x^4 \left (a+b x^3\right )^2} \, dx}{54 a^2 b^3}\\ &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\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{486 b^6 c x^3}{a}+\frac{278 b^6 d x^4}{a}+\frac{118 b^6 e x^5}{a}}{x^4 \left (a+b x^3\right )} \, dx}{162 a^3 b^5}\\ &=-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{162 b^5 c}{a x^4}-\frac{162 b^5 d}{a x^3}-\frac{162 b^5 e}{a x^2}+\frac{648 b^6 c}{a^2 x}+\frac{8 b^6 \left (55 a d+35 a e x-81 b c x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{162 a^3 b^5}\\ &=-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}-\frac{4 b c \log (x)}{a^5}-\frac{(4 b) \int \frac{55 a d+35 a e x-81 b c x^2}{a+b x^3} \, dx}{81 a^5}\\ &=-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}-\frac{4 b c \log (x)}{a^5}-\frac{(4 b) \int \frac{55 a d+35 a e x}{a+b x^3} \, dx}{81 a^5}+\frac{\left (4 b^2 c\right ) \int \frac{x^2}{a+b x^3} \, dx}{a^5}\\ &=-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}-\frac{4 b c \log (x)}{a^5}+\frac{4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac{\left (4 b^{2/3}\right ) \int \frac{\sqrt [3]{a} \left (110 a \sqrt [3]{b} d+35 a^{4/3} e\right )+\sqrt [3]{b} \left (-55 a \sqrt [3]{b} d+35 a^{4/3} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{17/3}}-\frac{\left (20 b \left (11 d-\frac{7 \sqrt [3]{a} e}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{14/3}}\\ &=-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}-\frac{4 b c \log (x)}{a^5}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{4 b c \log \left (a+b x^3\right )}{3 a^5}+\frac{\left (10 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \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}{243 a^{14/3}}-\frac{\left (10 b^{2/3} \left (11 \sqrt [3]{b} d+7 \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}{81 a^{13/3}}\\ &=-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}-\frac{4 b c \log (x)}{a^5}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\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} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}+\frac{4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac{\left (20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d+7 \sqrt [3]{a} e\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}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left (b d+b e x-\frac{b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac{x \left (17 b d+16 b e x-\frac{24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (139 b d+118 b e x-\frac{234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}+\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d+7 \sqrt [3]{a} e\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{4 b c \log (x)}{a^5}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\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} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}+\frac{4 b c \log \left (a+b x^3\right )}{3 a^5}\\ \end{align*}

Mathematica [A]  time = 0.47782, size = 284, normalized size = 0.84 \[ -\frac{-20 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{54 a^3 b (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac{9 a^2 b (18 c+x (17 d+16 e x))}{\left (a+b x^3\right )^2}+40 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a b (162 c+x (139 d+118 e x))}{a+b x^3}-648 b c \log \left (a+b x^3\right )-40 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{162 a c}{x^3}+\frac{243 a d}{x^2}+\frac{486 a e}{x}+1944 b c \log (x)}{486 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((162*a*c)/x^3 + (243*a*d)/x^2 + (486*a*e)/x + (54*a^3*b*(c + x*(d + e*x)))/(a + b*x^3)^3 + (9*a^2*b*(18*c +
x*(17*d + 16*e*x)))/(a + b*x^3)^2 + (3*a*b*(162*c + x*(139*d + 118*e*x)))/(a + b*x^3) - 40*Sqrt[3]*a^(1/3)*b^(
1/3)*(11*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 1944*b*c*Log[x] + 40*b^(1/3)*(
11*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] - 20*b^(1/3)*(11*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)
*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 648*b*c*Log[a + b*x^3])/(486*a^5)

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Maple [A]  time = 0.019, size = 415, normalized size = 1.2 \begin{align*} -{\frac{59\,{b}^{3}e{x}^{8}}{81\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{139\,{b}^{3}d{x}^{7}}{162\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{{b}^{3}c{x}^{6}}{{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{142\,{b}^{2}e{x}^{5}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{329\,{b}^{2}d{x}^{4}}{162\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{7\,{b}^{2}c{x}^{3}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{92\,be{x}^{2}}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{104\,bdx}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{13\,bc}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{220\,d}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{110\,d}{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\,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 ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{140\,e}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{70\,e}{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\,e\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{4\,bc\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{5}}}-{\frac{c}{3\,{a}^{4}{x}^{3}}}-{\frac{d}{2\,{a}^{4}{x}^{2}}}-{\frac{e}{{a}^{4}x}}-4\,{\frac{bc\ln \left ( x \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-59/81/a^4*b^3/(b*x^3+a)^3*e*x^8-139/162/a^4*b^3/(b*x^3+a)^3*d*x^7-1/a^4*b^3/(b*x^3+a)^3*c*x^6-142/81/a^3*b^2/
(b*x^3+a)^3*e*x^5-329/162/a^3*b^2/(b*x^3+a)^3*d*x^4-7/3/a^3*b^2/(b*x^3+a)^3*c*x^3-92/81/a^2*b/(b*x^3+a)^3*e*x^
2-104/81/a^2*b/(b*x^3+a)^3*d*x-13/9/a^2*b/(b*x^3+a)^3*c-220/243/a^4*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+110/24
3/a^4*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-220/243/a^4*d/(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*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-70/243/a^4*e/(1/b*a)^(1/3)*ln(x^2-
(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-140/243/a^4*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+4
/3*b*c*ln(b*x^3+a)/a^5-1/3*c/a^4/x^3-1/2*d/a^4/x^2-e/a^4/x-4*b*c*ln(x)/a^5

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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^4/(b*x^3+a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 12.2973, size = 15115, normalized size = 44.46 \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^4/(b*x^3+a)^4,x, algorithm="fricas")

[Out]

-1/486*(840*a*b^3*e*x^11 + 660*a*b^3*d*x^10 + 648*a*b^3*c*x^9 + 2310*a^2*b^2*e*x^8 + 1716*a^2*b^2*d*x^7 + 1620
*a^2*b^2*c*x^6 + 2010*a^3*b*e*x^5 + 1353*a^3*b*d*x^4 + 1188*a^3*b*c*x^3 + 486*a^4*e*x^2 + 243*a^4*d*x + 162*a^
4*c + 2*(a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 -
(6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^
2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(
1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2
+ 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3)
- 324*b*c/a^5)*log(7*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(10628
82*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b
^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^
3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^
3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10*e + 784080*b^2*c*d^2
+ 734832*b^2*c^2*e + 431200*a*b*d*e^2 + 4*(605*a^5*b*d^2 + 1134*a^5*b*c*e)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^
2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 -
243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a
*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*
(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)
/a^15)^(1/3) - 324*b*c/a^5) + 400*(1331*b^2*d^3 + 343*a*b*e^3)*x) - (972*b^4*c*x^12 + 2916*a*b^3*c*x^9 + 2916*
a^2*b^2*c*x^6 + 972*a^3*b*c*x^3 + (a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*(4^(2/3)*(-I*sqrt(3)
+ 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*
e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 -
1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*
b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*
c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) + 3*sqrt(1/3)*(a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*
sqrt(-((4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^1
5 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875
*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 1
25*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*
b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6
561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/
a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*
d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14
- 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*
a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^5*b*c + 104976*b^2*c^2 + 123200*a*b*d*e)/a^10))*log(-7*(4^(2/3)*(-I*sqrt(3
) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*
a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3
- 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3
)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 170
1*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10*e - 784080*b^2*c*d^2 - 734832*b^2*c^2*e - 431200*a*b*d*e^2 -
4*(605*a^5*b*d^2 + 1134*a^5*b*c*e)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*
e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a
^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) +
1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 +
(531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) + 800*(1331*
b^2*d^3 + 343*a*b*e^3)*x + 3*sqrt(1/3)*(7*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*
a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)
*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqr
t(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/
a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^10*e
- 2420*a^5*b*d^2 + 2268*a^5*b*c*e)*sqrt(-((4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925
*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e
)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sq
rt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c
/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^1
0 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^
15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 4287
5*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 +
125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2
*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^5*b*c + 104976*b^2*c^2 + 123200*a*b*d*
e)/a^10)) - (972*b^4*c*x^12 + 2916*a*b^3*c*x^9 + 2916*a^2*b^2*c*x^6 + 972*a^3*b*c*x^3 + (a^5*b^3*x^12 + 3*a^6*
b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/
a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15
+ (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)
*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (5
31441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) - 3*sqrt(1/3)*(
a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*sqrt(-((4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6
561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*
c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/
3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 +
1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) -
324*b*c/a^5)^2*a^10 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(
1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531
441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(10628
82*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b
^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^5*b*c + 104976*b^2*c
^2 + 123200*a*b*d*e)/a^10))*log(-7*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e
)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^
15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) +
1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 +
(531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10*e - 78
4080*b^2*c*d^2 - 734832*b^2*c^2*e - 431200*a*b*d*e^2 - 4*(605*a^5*b*d^2 + 1134*a^5*b*c*e)*(4^(2/3)*(-I*sqrt(3)
+ 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a
*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 -
1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)
*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701
*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) + 800*(1331*b^2*d^3 + 343*a*b*e^3)*x - 3*sqrt(1/3)*(7*(4^(2/3)*(-I*s
qrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 +
343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605
*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*
a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3
- 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^10*e - 2420*a^5*b*d^2 + 2268*a^5*b*c*e)*sqrt(-((4^(2/3)*(-I*
sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3
+ 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(60
5*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343
*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3
- 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (
6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2
*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1
/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2
+ 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) -
324*b*c/a^5)*a^5*b*c + 104976*b^2*c^2 + 123200*a*b*d*e)/a^10)) + 1944*(b^4*c*x^12 + 3*a*b^3*c*x^9 + 3*a^2*b^2
*c*x^6 + a^3*b*c*x^3)*log(x))/(a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)

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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**4/(b*x**3+a)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.06862, size = 459, normalized size = 1.35 \begin{align*} \frac{4 \, b c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5}} - \frac{4 \, b c \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{10 \,{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d + 7 \, \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 )}{243 \, a^{5} b} - \frac{280 \, b^{3} x^{11} e + 220 \, b^{3} d x^{10} + 216 \, b^{3} c x^{9} + 770 \, a b^{2} x^{8} e + 572 \, a b^{2} d x^{7} + 540 \, a b^{2} c x^{6} + 670 \, a^{2} b x^{5} e + 451 \, a^{2} b d x^{4} + 396 \, a^{2} b c x^{3} + 162 \, a^{3} x^{2} e + 81 \, a^{3} d x + 54 \, a^{3} c}{162 \,{\left (b x^{4} + a x\right )}^{3} a^{4}} - \frac{20 \, \sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} 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 )}{243 \, a^{6} b^{3}} + \frac{20 \,{\left (7 \, a^{6} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 11 \, a^{6} 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 )}{243 \, a^{11} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*b*c*log(abs(b*x^3 + a))/a^5 - 4*b*c*log(abs(x))/a^5 - 10/243*(11*(-a*b^2)^(1/3)*b*d + 7*(-a*b^2)^(2/3)*e)*
log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/162*(280*b^3*x^11*e + 220*b^3*d*x^10 + 216*b^3*c*x^9 + 77
0*a*b^2*x^8*e + 572*a*b^2*d*x^7 + 540*a*b^2*c*x^6 + 670*a^2*b*x^5*e + 451*a^2*b*d*x^4 + 396*a^2*b*c*x^3 + 162*
a^3*x^2*e + 81*a^3*d*x + 54*a^3*c)/((b*x^4 + a*x)^3*a^4) - 20/243*sqrt(3)*(11*(-a*b^2)^(1/3)*a*b^3*d - 7*(-a*b
^2)^(2/3)*a*b^2*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b^3) + 20/243*(7*a^6*b^2*(-a/b)^
(1/3)*e + 11*a^6*b^2*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^11*b)