3.4.100 \(\int \frac {1}{(d+e x)^3 (a+c x^4)} \, dx\) [400]

Optimal. Leaf size=680 \[ -\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3} \]

[Out]

-1/2*e^3/(a*e^4+c*d^4)/(e*x+d)^2-4*c*d^3*e^3/(a*e^4+c*d^4)^2/(e*x+d)+2*c*d^2*e^3*(-3*a*e^4+5*c*d^4)*ln(e*x+d)/
(a*e^4+c*d^4)^3-1/2*c*d^2*e^3*(-3*a*e^4+5*c*d^4)*ln(c*x^4+a)/(a*e^4+c*d^4)^3-1/2*e*(a^2*e^8-12*a*c*d^4*e^4+3*c
^2*d^8)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^3/a^(1/2)-1/8*c^(3/4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2
)+a^(1/2)+x^2*c^(1/2))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)
/(a*e^4+c*d^4)^3*2^(1/2)+1/8*c^(3/4)*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(c^2*d^8-12*a*c*d^4*e
^4+3*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(3/4)*d*arcta
n(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8+2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2)
)/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(3/4)*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*
a^2*e^8+2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6857, 1890, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 211, 266} \begin {gather*} -\frac {\sqrt {c} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {e^3}{2 (d+e x)^2 \left (a e^4+c d^4\right )}-\frac {4 c d^3 e^3}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (a e^4+c d^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (a e^4+c d^4\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + c*x^4)),x]

[Out]

-1/2*e^3/((c*d^4 + a*e^4)*(d + e*x)^2) - (4*c*d^3*e^3)/((c*d^4 + a*e^4)^2*(d + e*x)) - (Sqrt[c]*e*(3*c^2*d^8 -
12*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(c^2*d^8
- 12*a*c*d^4*e^4 + 3*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a
^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 + 2*Sqrt[a]*
Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^
4)^3) + (2*c*d^2*e^3*(5*c*d^4 - 3*a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^3 - (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^
4 + 3*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[
c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 - 2*Sqrt[a]*
Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4
)*(c*d^4 + a*e^4)^3) - (c*d^2*e^3*(5*c*d^4 - 3*a*e^4)*Log[a + c*x^4])/(2*(c*d^4 + a*e^4)^3)

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 211

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

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 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1890

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx &=\int \left (\frac {e^4}{\left (c d^4+a e^4\right ) (d+e x)^3}+\frac {4 c d^3 e^4}{\left (c d^4+a e^4\right )^2 (d+e x)^2}+\frac {2 c d^2 e^4 \left (5 c d^4-3 a e^4\right )}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \left (\frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac {x \left (-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {x \left (-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \text {Subst}\left (\int \frac {-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}-\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c^2 d^2 e^3 \left (5 c d^4-3 a e^4\right )\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3}\\ \end {align*}

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Mathematica [A]
time = 0.64, size = 738, normalized size = 1.09 \begin {gather*} \frac {-4 a^{3/4} e^3 \left (c d^4+a e^4\right )^2-32 a^{3/4} c d^3 e^3 \left (c d^4+a e^4\right ) (d+e x)-2 \sqrt {c} \left (\sqrt {2} c^{9/4} d^9-6 \sqrt [4]{a} c^2 d^8 e+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4+24 a^{5/4} c d^4 e^5-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8-2 a^{9/4} e^9\right ) (d+e x)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {c} \left (\sqrt {2} c^{9/4} d^9+6 \sqrt [4]{a} c^2 d^8 e+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4-24 a^{5/4} c d^4 e^5-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8+2 a^{9/4} e^9\right ) (d+e x)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+16 a^{3/4} c d^2 e^3 \left (5 c d^4-3 a e^4\right ) (d+e x)^2 \log (d+e x)-\sqrt {2} c^{3/4} d \left (c^2 d^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8\right ) (d+e x)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} c^{3/4} d \left (c^2 d^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8\right ) (d+e x)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+4 a^{3/4} c d^2 e^3 \left (-5 c d^4+3 a e^4\right ) (d+e x)^2 \log \left (a+c x^4\right )}{8 a^{3/4} \left (c d^4+a e^4\right )^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a + c*x^4)),x]

[Out]

(-4*a^(3/4)*e^3*(c*d^4 + a*e^4)^2 - 32*a^(3/4)*c*d^3*e^3*(c*d^4 + a*e^4)*(d + e*x) - 2*Sqrt[c]*(Sqrt[2]*c^(9/4
)*d^9 - 6*a^(1/4)*c^2*d^8*e + 6*Sqrt[2]*Sqrt[a]*c^(7/4)*d^7*e^2 - 12*Sqrt[2]*a*c^(5/4)*d^5*e^4 + 24*a^(5/4)*c*
d^4*e^5 - 10*Sqrt[2]*a^(3/2)*c^(3/4)*d^3*e^6 + 3*Sqrt[2]*a^2*c^(1/4)*d*e^8 - 2*a^(9/4)*e^9)*(d + e*x)^2*ArcTan
[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[c]*(Sqrt[2]*c^(9/4)*d^9 + 6*a^(1/4)*c^2*d^8*e + 6*Sqrt[2]*Sqrt[a]*c
^(7/4)*d^7*e^2 - 12*Sqrt[2]*a*c^(5/4)*d^5*e^4 - 24*a^(5/4)*c*d^4*e^5 - 10*Sqrt[2]*a^(3/2)*c^(3/4)*d^3*e^6 + 3*
Sqrt[2]*a^2*c^(1/4)*d*e^8 + 2*a^(9/4)*e^9)*(d + e*x)^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 16*a^(3/4)*c*
d^2*e^3*(5*c*d^4 - 3*a*e^4)*(d + e*x)^2*Log[d + e*x] - Sqrt[2]*c^(3/4)*d*(c^2*d^8 - 6*Sqrt[a]*c^(3/2)*d^6*e^2
- 12*a*c*d^4*e^4 + 10*a^(3/2)*Sqrt[c]*d^2*e^6 + 3*a^2*e^8)*(d + e*x)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2] + Sqrt[2]*c^(3/4)*d*(c^2*d^8 - 6*Sqrt[a]*c^(3/2)*d^6*e^2 - 12*a*c*d^4*e^4 + 10*a^(3/2)*Sqrt[c]
*d^2*e^6 + 3*a^2*e^8)*(d + e*x)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 4*a^(3/4)*c*d^2*e^3
*(-5*c*d^4 + 3*a*e^4)*(d + e*x)^2*Log[a + c*x^4])/(8*a^(3/4)*(c*d^4 + a*e^4)^3*(d + e*x)^2)

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Maple [A]
time = 0.27, size = 446, normalized size = 0.66

method	result	size
default	\(-\frac {e^{3}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {4 c \,d^{3} e^{3}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}-\frac {2 d^{2} e^{3} c \left (3 e^{4} a -5 d^{4} c \right ) \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{3}}+\frac {c \left (\frac {\left (3 a^{2} d \,e^{8}-12 a c \,d^{5} e^{4}+c^{2} d^{9}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-a^{2} e^{9}+12 a c \,d^{4} e^{5}-3 c^{2} e \,d^{8}\right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (-10 a c \,d^{3} e^{6}+6 c^{2} d^{7} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+\frac {\left (6 a c \,d^{2} e^{7}-10 e^{3} c^{2} d^{6}\right ) \ln \left (c \,x^{4}+a \right )}{4 c}\right )}{\left (e^{4} a +d^{4} c \right )^{3}}\)	\(446\)
risch	\(\frac {-\frac {4 d^{3} e^{4} c x}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}-\frac {\left (e^{4} a +9 d^{4} c \right ) e^{3}}{2 \left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}\right )}}{\left (e x +d \right )^{2}}-\frac {6 d^{2} e^{7} c \ln \left (e x +d \right ) a}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}+\frac {10 d^{6} e^{3} c^{2} \ln \left (e x +d \right )}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{6} e^{12}+3 a^{5} c \,d^{4} e^{8}+3 a^{4} c^{2} d^{8} e^{4}+a^{3} c^{3} d^{12}\right ) \textit {\_Z}^{4}+\left (-24 e^{7} d^{2} c \,a^{4}+40 a^{3} c^{2} d^{6} e^{3}\right ) \textit {\_Z}^{3}+\left (2 a^{3} c \,e^{6}+42 a^{2} c^{2} d^{4} e^{2}\right ) \textit {\_Z}^{2}+12 a \,c^{2} d^{2} e \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{7} e^{22}+17 a^{6} c \,d^{4} e^{18}+18 a^{5} c^{2} d^{8} e^{14}+2 a^{4} c^{3} d^{12} e^{10}-7 a^{3} c^{4} d^{16} e^{6}-3 a^{2} c^{5} d^{20} e^{2}\right ) \textit {\_R}^{4}+\left (-81 a^{5} c \,d^{2} e^{17}-56 a^{4} c^{2} d^{6} e^{13}+126 a^{3} c^{3} d^{10} e^{9}+96 a^{2} c^{4} d^{14} e^{5}-5 a \,c^{5} d^{18} e \right ) \textit {\_R}^{3}+\left (9 a^{4} c \,e^{16}-38 a^{3} c^{2} d^{4} e^{12}+408 a^{2} c^{3} d^{8} e^{8}+198 a \,c^{4} d^{12} e^{4}-c^{5} d^{16}\right ) \textit {\_R}^{2}+\left (72 a^{2} c^{2} d^{2} e^{11}-16 a \,c^{3} d^{6} e^{7}+40 c^{4} d^{10} e^{3}\right ) \textit {\_R} +4 a \,c^{2} e^{10}+4 c^{3} d^{4} e^{6}\right ) x +\left (6 a^{7} d \,e^{21}+22 a^{6} c \,d^{5} e^{17}+28 a^{5} c^{2} d^{9} e^{13}+12 a^{4} c^{3} d^{13} e^{9}-2 a^{3} c^{4} d^{17} e^{5}-2 a^{2} c^{5} d^{21} e \right ) \textit {\_R}^{4}+\left (-61 a^{5} c \,d^{3} e^{16}-56 a^{4} c^{2} d^{7} e^{12}+70 a^{3} c^{3} d^{11} e^{8}+64 a^{2} c^{4} d^{15} e^{4}-a \,c^{5} d^{19}\right ) \textit {\_R}^{3}+\left (4 a^{4} c d \,e^{15}-148 a^{3} c^{2} d^{5} e^{11}+652 a^{2} c^{3} d^{9} e^{7}+36 a \,c^{4} d^{13} e^{3}\right ) \textit {\_R}^{2}+\left (-48 a^{2} c^{2} d^{3} e^{10}+208 a \,c^{3} d^{7} e^{6}\right ) \textit {\_R} +4 a \,c^{2} d \,e^{9}+4 c^{3} d^{5} e^{5}\right )\right )}{4}\)	\(867\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

-1/2*e^3/(a*e^4+c*d^4)/(e*x+d)^2-4*c*d^3*e^3/(a*e^4+c*d^4)^2/(e*x+d)-2*d^2*e^3*c*(3*a*e^4-5*c*d^4)/(a*e^4+c*d^
4)^3*ln(e*x+d)+c/(a*e^4+c*d^4)^3*(1/8*(3*a^2*d*e^8-12*a*c*d^5*e^4+c^2*d^9)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c
)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*ar
ctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/2*(-a^2*e^9+12*a*c*d^4*e^5-3*c^2*d^8*e)/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+1
/8*(-10*a*c*d^3*e^6+6*c^2*d^7*e^2)/c/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c
)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/4*(6*a*
c*d^2*e^7-10*c^2*d^6*e^3)/c*ln(c*x^4+a))

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Maxima [A]
time = 0.50, size = 779, normalized size = 1.15 \begin {gather*} \frac {c {\left (\frac {\sqrt {2} {\left (c^{3} d^{9} - 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} - 10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 12 \, a c^{2} d^{5} e^{4} + 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} + 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} + 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {\sqrt {2} {\left (c^{3} d^{9} - 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} + 10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 12 \, a c^{2} d^{5} e^{4} + 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} - 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} + 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} + 6 \, \sqrt {a} c^{3} d^{8} e + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} - 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} + 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} - 6 \, \sqrt {a} c^{3} d^{8} e + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} + 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} - 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{8 \, {\left (c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}\right )}} + \frac {2 \, {\left (5 \, c^{2} d^{6} e^{3} - 3 \, a c d^{2} e^{7}\right )} \log \left (x e + d\right )}{c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}} - \frac {8 \, c d^{3} x e^{4} + 9 \, c d^{4} e^{3} + a e^{7}}{2 \, {\left (c^{2} d^{10} + 2 \, a c d^{6} e^{4} + a^{2} d^{2} e^{8} + {\left (c^{2} d^{8} e^{2} + 2 \, a c d^{4} e^{6} + a^{2} e^{10}\right )} x^{2} + 2 \, {\left (c^{2} d^{9} e + 2 \, a c d^{5} e^{5} + a^{2} d e^{9}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^4+a),x, algorithm="maxima")

[Out]

1/8*c*(sqrt(2)*(c^3*d^9 - 6*sqrt(a)*c^(5/2)*d^7*e^2 - 10*sqrt(2)*a^(3/4)*c^(9/4)*d^6*e^3 - 12*a*c^2*d^5*e^4 +
10*a^(3/2)*c^(3/2)*d^3*e^6 + 6*sqrt(2)*a^(7/4)*c^(5/4)*d^2*e^7 + 3*a^2*c*d*e^8)*log(sqrt(c)*x^2 + sqrt(2)*a^(1
/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) - sqrt(2)*(c^3*d^9 - 6*sqrt(a)*c^(5/2)*d^7*e^2 + 10*sqrt(2)*a^(3/4)
*c^(9/4)*d^6*e^3 - 12*a*c^2*d^5*e^4 + 10*a^(3/2)*c^(3/2)*d^3*e^6 - 6*sqrt(2)*a^(7/4)*c^(5/4)*d^2*e^7 + 3*a^2*c
*d*e^8)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + 2*(sqrt(2)*a^(1/4)*c^(13/4)
*d^9 + 6*sqrt(a)*c^3*d^8*e + 6*sqrt(2)*a^(3/4)*c^(11/4)*d^7*e^2 - 12*sqrt(2)*a^(5/4)*c^(9/4)*d^5*e^4 - 24*a^(3
/2)*c^2*d^4*e^5 - 10*sqrt(2)*a^(7/4)*c^(7/4)*d^3*e^6 + 3*sqrt(2)*a^(9/4)*c^(5/4)*d*e^8 + 2*a^(5/2)*c*e^9)*arct
an(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c
^(5/4)) + 2*(sqrt(2)*a^(1/4)*c^(13/4)*d^9 - 6*sqrt(a)*c^3*d^8*e + 6*sqrt(2)*a^(3/4)*c^(11/4)*d^7*e^2 - 12*sqrt
(2)*a^(5/4)*c^(9/4)*d^5*e^4 + 24*a^(3/2)*c^2*d^4*e^5 - 10*sqrt(2)*a^(7/4)*c^(7/4)*d^3*e^6 + 3*sqrt(2)*a^(9/4)*
c^(5/4)*d*e^8 - 2*a^(5/2)*c*e^9)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(
c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)))/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + 2*(5
*c^2*d^6*e^3 - 3*a*c*d^2*e^7)*log(x*e + d)/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) - 1/2*(8*
c*d^3*x*e^4 + 9*c*d^4*e^3 + a*e^7)/(c^2*d^10 + 2*a*c*d^6*e^4 + a^2*d^2*e^8 + (c^2*d^8*e^2 + 2*a*c*d^4*e^6 + a^
2*e^10)*x^2 + 2*(c^2*d^9*e + 2*a*c*d^5*e^5 + a^2*d*e^9)*x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^4+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**4+a),x)

[Out]

Timed out

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Giac [A]
time = 4.03, size = 901, normalized size = 1.32 \begin {gather*} \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 2 \, a c^{2} e^{3} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (a c^{3} d^{6} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e + 9 \, \sqrt {a c} a c^{2} d^{4} e^{2} + 9 \, a^{2} c^{2} d^{2} e^{4} - 8 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{3} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{5} + \sqrt {a c} a^{2} c e^{6}\right )}} + \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - 2 \, a c^{2} e^{3} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (a c^{3} d^{6} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e - 9 \, \sqrt {a c} a c^{2} d^{4} e^{2} + 9 \, a^{2} c^{2} d^{2} e^{4} + 8 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{3} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{5} + \sqrt {a c} a^{2} c e^{6}\right )}} + \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{9} - 6 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{7} e^{2} - 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e^{4} + 10 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{6} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{8}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, {\left (a c^{4} d^{12} + 3 \, a^{2} c^{3} d^{8} e^{4} + 3 \, a^{3} c^{2} d^{4} e^{8} + a^{4} c e^{12}\right )}} - \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{9} - 6 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{7} e^{2} - 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e^{4} + 10 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{6} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{8}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, {\left (a c^{4} d^{12} + 3 \, a^{2} c^{3} d^{8} e^{4} + 3 \, a^{3} c^{2} d^{4} e^{8} + a^{4} c e^{12}\right )}} - \frac {{\left (5 \, c^{2} d^{6} e^{3} - 3 \, a c d^{2} e^{7}\right )} \log \left ({\left | c x^{4} + a \right |}\right )}{2 \, {\left (c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}\right )}} + \frac {2 \, {\left (5 \, c^{2} d^{6} e^{4} - 3 \, a c d^{2} e^{8}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{12} e + 3 \, a c^{2} d^{8} e^{5} + 3 \, a^{2} c d^{4} e^{9} + a^{3} e^{13}} - \frac {9 \, c^{2} d^{8} e^{3} + 10 \, a c d^{4} e^{7} + a^{2} e^{11} + 8 \, {\left (c^{2} d^{7} e^{4} + a c d^{3} e^{8}\right )} x}{2 \, {\left (c d^{4} + a e^{4}\right )}^{3} {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^4+a),x, algorithm="giac")

[Out]

1/4*(sqrt(2)*(a*c^3)^(1/4)*c^2*d^3 + 2*a*c^2*e^3 - 3*sqrt(2)*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(2*x + sq
rt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3*d^6 - 3*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e + 9*sqrt(a*c)*a*c^2*d^4*e^2 +
9*a^2*c^2*d^2*e^4 - 8*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^3 - 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^5 + sqrt(a*c)*a^2*c
*e^6) + 1/4*(sqrt(2)*(a*c^3)^(1/4)*c^2*d^3 - 2*a*c^2*e^3 - 3*sqrt(2)*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(
2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3*d^6 + 3*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e - 9*sqrt(a*c)*a*c^2*d
^4*e^2 + 9*a^2*c^2*d^2*e^4 + 8*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^3 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^5 + sqrt(a*
c)*a^2*c*e^6) + 1/8*(sqrt(2)*(a*c^3)^(1/4)*c^3*d^9 - 6*sqrt(2)*(a*c^3)^(3/4)*c*d^7*e^2 - 12*sqrt(2)*(a*c^3)^(1
/4)*a*c^2*d^5*e^4 + 10*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^6 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^8)*log(x^2 + sqrt(2
)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4*d^12 + 3*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^4*e^8 + a^4*c*e^12) - 1/8*(sqrt(2)*
(a*c^3)^(1/4)*c^3*d^9 - 6*sqrt(2)*(a*c^3)^(3/4)*c*d^7*e^2 - 12*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e^4 + 10*sqrt(2
)*(a*c^3)^(3/4)*a*d^3*e^6 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^8)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/
(a*c^4*d^12 + 3*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^4*e^8 + a^4*c*e^12) - 1/2*(5*c^2*d^6*e^3 - 3*a*c*d^2*e^7)*log(ab
s(c*x^4 + a))/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + 2*(5*c^2*d^6*e^4 - 3*a*c*d^2*e^8)*lo
g(abs(x*e + d))/(c^3*d^12*e + 3*a*c^2*d^8*e^5 + 3*a^2*c*d^4*e^9 + a^3*e^13) - 1/2*(9*c^2*d^8*e^3 + 10*a*c*d^4*
e^7 + a^2*e^11 + 8*(c^2*d^7*e^4 + a*c*d^3*e^8)*x)/((c*d^4 + a*e^4)^3*(x*e + d)^2)

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Mupad [B]
time = 3.67, size = 1955, normalized size = 2.88 \begin {gather*} \left (\sum _{k=1}^4\ln \left (\frac {c^7\,d^5\,e^6+a\,c^6\,d\,e^{10}}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (\frac {208\,a\,c^7\,d^7\,e^7-48\,a^2\,c^6\,d^3\,e^{11}}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (-\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (\frac {976\,a^5\,c^5\,d^3\,e^{17}+896\,a^4\,c^6\,d^7\,e^{13}-1120\,a^3\,c^7\,d^{11}\,e^9-1024\,a^2\,c^8\,d^{15}\,e^5+16\,a\,c^9\,d^{19}\,e}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}-\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (\frac {384\,a^7\,c^4\,d\,e^{22}+1408\,a^6\,c^5\,d^5\,e^{18}+1792\,a^5\,c^6\,d^9\,e^{14}+768\,a^4\,c^7\,d^{13}\,e^{10}-128\,a^3\,c^8\,d^{17}\,e^6-128\,a^2\,c^9\,d^{21}\,e^2}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\frac {x\,\left (320\,a^7\,c^4\,e^{23}+1088\,a^6\,c^5\,d^4\,e^{19}+1152\,a^5\,c^6\,d^8\,e^{15}+128\,a^4\,c^7\,d^{12}\,e^{11}-448\,a^3\,c^8\,d^{16}\,e^7-192\,a^2\,c^9\,d^{20}\,e^3\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {x\,\left (1296\,a^5\,c^5\,d^2\,e^{18}+896\,a^4\,c^6\,d^6\,e^{14}-2016\,a^3\,c^7\,d^{10}\,e^{10}-1536\,a^2\,c^8\,d^{14}\,e^6+80\,a\,c^9\,d^{18}\,e^2\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {16\,a^4\,c^5\,d\,e^{16}-592\,a^3\,c^6\,d^5\,e^{12}+2608\,a^2\,c^7\,d^9\,e^8+144\,a\,c^8\,d^{13}\,e^4}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\frac {x\,\left (36\,a^4\,c^5\,e^{17}-152\,a^3\,c^6\,d^4\,e^{13}+1632\,a^2\,c^7\,d^8\,e^9+792\,a\,c^8\,d^{12}\,e^5-4\,c^9\,d^{16}\,e\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {x\,\left (72\,a^2\,c^6\,d^2\,e^{12}-16\,a\,c^7\,d^6\,e^8+40\,c^8\,d^{10}\,e^4\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {x\,\left (c^7\,d^4\,e^7+a\,c^6\,e^{11}\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )\,\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\right )-\frac {\frac {9\,c\,d^4\,e^3+a\,e^7}{2\,\left (a^2\,e^8+2\,a\,c\,d^4\,e^4+c^2\,d^8\right )}+\frac {4\,c\,d^3\,e^4\,x}{a^2\,e^8+2\,a\,c\,d^4\,e^4+c^2\,d^8}}{d^2+2\,d\,e\,x+e^2\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (10\,c^2\,d^6\,e^3-6\,a\,c\,d^2\,e^7\right )}{a^3\,e^{12}+3\,a^2\,c\,d^4\,e^8+3\,a\,c^2\,d^8\,e^4+c^3\,d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^4)*(d + e*x)^3),x)

[Out]

symsum(log((c^7*d^5*e^6 + a*c^6*d*e^10)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2
*d^8*e^8) + root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1
536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e
*z + c^2, z, k)*((208*a*c^7*d^7*e^7 - 48*a^2*c^6*d^3*e^11)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d
^4*e^12 + 6*a^2*c^2*d^8*e^8) + root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 2
56*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z
^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((144*a*c^8*d^13*e^4 + 16*a^4*c^5*d*e^16 + 2608*a^2*c^7*d^9*e^8 - 592*a^3*c
^6*d^5*e^12)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) - root(768*a^5*c*
d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 256
0*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((896*a^4*c
^6*d^7*e^13 - 1120*a^3*c^7*d^11*e^9 - 1024*a^2*c^8*d^15*e^5 + 976*a^5*c^5*d^3*e^17 + 16*a*c^9*d^19*e)/(a^4*e^1
6 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) - root(768*a^5*c*d^4*e^8*z^4 + 768*a^4
*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3
+ 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((384*a^7*c^4*d*e^22 - 128*a^2*c
^9*d^21*e^2 - 128*a^3*c^8*d^17*e^6 + 768*a^4*c^7*d^13*e^10 + 1792*a^5*c^6*d^9*e^14 + 1408*a^6*c^5*d^5*e^18)/(a
^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) + (x*(320*a^7*c^4*e^23 - 192*a^2
*c^9*d^20*e^3 - 448*a^3*c^8*d^16*e^7 + 128*a^4*c^7*d^12*e^11 + 1152*a^5*c^6*d^8*e^15 + 1088*a^6*c^5*d^4*e^19))
/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(80*a*c^9*d^18*e^2 - 15
36*a^2*c^8*d^14*e^6 - 2016*a^3*c^7*d^10*e^10 + 896*a^4*c^6*d^6*e^14 + 1296*a^5*c^5*d^2*e^18))/(a^4*e^16 + c^4*
d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(36*a^4*c^5*e^17 - 4*c^9*d^16*e + 792*a*
c^8*d^12*e^5 + 1632*a^2*c^7*d^8*e^9 - 152*a^3*c^6*d^4*e^13))/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c
*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(40*c^8*d^10*e^4 - 16*a*c^7*d^6*e^8 + 72*a^2*c^6*d^2*e^12))/(a^4*e^16 + c
^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(a*c^6*e^11 + c^7*d^4*e^7))/(a^4*e^16
+ c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8))*root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c
^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 +
672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k), k, 1, 4) - ((a*e^7 + 9*c*d^4*e^3)
/(2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (4*c*d^3*e^4*x)/(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4))/(d^2 + e^2*x^2
+ 2*d*e*x) + (log(d + e*x)*(10*c^2*d^6*e^3 - 6*a*c*d^2*e^7))/(a^3*e^12 + c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c
*d^4*e^8)

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