\(\int \frac {1}{(d+e x) (a+c x^4)^2} \, dx\) [405]

Optimal result

Integrand size = 17, antiderivative size = 855 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2} \]

[Out]

1/4*(a*e^3+c*x*(d*e^2*x^2-d^2*e*x+d^3))/a/(a*e^4+c*d^4)/(c*x^4+a)+e^7*ln(e*x+d)/(a*e^4+c*d^4)^2-1/4*e^7*ln(c*x
^4+a)/(a*e^4+c*d^4)^2-1/4*d^2*e*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(3/2)/(a*e^4+c*d^4)-1/2*d^2*e^5*arctan(x
^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^2/a^(1/2)-1/8*c^(1/4)*d*e^4*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^
2*c^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/8*c^(1/4)*d*e^4*ln(a^(1/4)*c^(1/4)*x*2
^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/4*c^(1/4)*d*e^4*arcta
n(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/4*c^(1/4)*d*e^4*ar
ctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2*2^(1/2)-1/32*c^(1/4)*d*ln(
-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)*2^(1/2)+1/3
2*c^(1/4)*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^
4)*2^(1/2)+1/16*c^(1/4)*d*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^
4)*2^(1/2)+1/16*c^(1/4)*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^4
)*2^(1/2)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {6874, 1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {\log (d+e x) e^7}{\left (c d^4+a e^4\right )^2}-\frac {\log \left (c x^4+a\right ) e^7}{4 \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac {a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{4 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )} \]

[In]

Int[1/((d + e*x)*(a + c*x^4)^2),x]

[Out]

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

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 1868

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^8}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}-\frac {c e^4 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}\right ) \, dx \\ & = \frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \frac {-d^3+d^2 e x-d e^2 x^2+e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}+\frac {c \int \frac {d^3-d^2 e x+d e^2 x^2-e^3 x^3}{\left (a+c x^4\right )^2} \, dx}{c d^4+a e^4} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \left (\frac {-d^3-d e^2 x^2}{a+c x^4}+\frac {x \left (d^2 e+e^3 x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^2}-\frac {c \int \frac {-3 d^3+2 d^2 e x-d e^2 x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \frac {-d^3-d e^2 x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \int \frac {x \left (d^2 e+e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}-\frac {c \int \left (\frac {2 d^2 e x}{a+c x^4}+\frac {-3 d^3-d e^2 x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c e^4\right ) \text {Subst}\left (\int \frac {d^2 e+e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}-e^2\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 )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\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 )^2}-\frac {c \int \frac {-3 d^3-d e^2 x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )}-\frac {\left (c d^2 e\right ) \int \frac {x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (c d^2 e^5\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 )^2}-\frac {\left (c e^7\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\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 )^2}+\frac {\left (d e^4 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\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 )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\left (c d^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\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}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\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}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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 )^2}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\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 )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\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 )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 558, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {\frac {8 \left (c d^4+a e^4\right ) \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )}{a \left (a+c x^4\right )}-\frac {2 \sqrt [4]{c} d \left (3 \sqrt {2} c^{3/2} d^6-4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt {2} \sqrt {a} c d^4 e^2+7 \sqrt {2} a \sqrt {c} d^2 e^4-12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt {2} a^{3/2} e^6\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \sqrt [4]{c} d \left (3 \sqrt {2} c^{3/2} d^6+4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt {2} \sqrt {a} c d^4 e^2+7 \sqrt {2} a \sqrt {c} d^2 e^4+12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt {2} a^{3/2} e^6\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+32 e^7 \log (d+e x)+\frac {\sqrt {2} \sqrt [4]{c} \left (-3 c^{3/2} d^7+\sqrt {a} c d^5 e^2-7 a \sqrt {c} d^3 e^4+5 a^{3/2} d e^6\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{3/2} d^7-\sqrt {a} c d^5 e^2+7 a \sqrt {c} d^3 e^4-5 a^{3/2} d e^6\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}-8 e^7 \log \left (a+c x^4\right )}{32 \left (c d^4+a e^4\right )^2} \]

[In]

Integrate[1/((d + e*x)*(a + c*x^4)^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.50

method	result	size
default	\(\frac {c \left (\frac {\frac {d \,e^{2} \left (e^{4} a +d^{4} c \right ) x^{3}}{4 a}-\frac {d^{2} e \left (e^{4} a +d^{4} c \right ) x^{2}}{4 a}+\frac {d^{3} \left (e^{4} a +d^{4} c \right ) x}{4 a}+\frac {e^{3} \left (e^{4} a +d^{4} c \right )}{4 c}}{c \,x^{4}+a}+\frac {\frac {\left (7 a \,d^{3} e^{4}+3 c \,d^{7}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-6 a \,d^{2} e^{5}-2 c \,d^{6} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (5 a d \,e^{6}+c \,d^{5} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}-\frac {a \,e^{7} \ln \left (c \,x^{4}+a \right )}{c}}{4 a}\right )}{\left (e^{4} a +d^{4} c \right )^{2}}+\frac {e^{7} \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{2}}\)	\(430\)
risch	\(\frac {\frac {c d \,e^{2} x^{3}}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {d^{2} c e \,x^{2}}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {d^{3} c x}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {e^{3}}{4 e^{4} a +4 d^{4} c}}{c \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{9} e^{8}+2 a^{8} c \,d^{4} e^{4}+a^{7} c^{2} d^{8}\right ) \textit {\_Z}^{4}+16 a^{7} e^{7} \textit {\_Z}^{3}+\left (96 a^{5} e^{6}+20 a^{4} c \,d^{4} e^{2}\right ) \textit {\_Z}^{2}+\left (256 a^{3} e^{5}+72 a^{2} c \,d^{4} e \right ) \textit {\_Z} +256 e^{4} a +81 d^{4} c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{9} e^{14}+7 a^{8} c \,d^{4} e^{10}-a^{7} c^{2} d^{8} e^{6}-3 a^{6} c^{3} d^{12} e^{2}\right ) \textit {\_R}^{4}+\left (60 a^{7} e^{13}+64 a^{6} c \,d^{4} e^{9}-4 a^{5} c^{2} d^{8} e^{5}-8 a^{4} c^{3} d^{12} e \right ) \textit {\_R}^{3}+\left (240 a^{5} e^{12}+295 a^{4} c \,d^{4} e^{8}+30 a^{3} c^{2} d^{8} e^{4}-9 a^{2} c^{3} d^{12}\right ) \textit {\_R}^{2}+\left (320 a^{3} e^{11}+560 a^{2} c \,d^{4} e^{7}+64 a \,c^{2} d^{8} e^{3}\right ) \textit {\_R} +324 c \,d^{4} e^{6}\right ) x +\left (6 a^{9} d \,e^{13}+10 a^{8} c \,d^{5} e^{9}+2 a^{7} c^{2} d^{9} e^{5}-2 a^{6} c^{3} d^{13} e \right ) \textit {\_R}^{4}+\left (51 a^{7} d \,e^{12}+67 a^{6} c \,d^{5} e^{8}+13 a^{5} c^{2} d^{9} e^{4}-3 a^{4} c^{3} d^{13}\right ) \textit {\_R}^{3}+\left (136 a^{5} d \,e^{11}+240 a^{4} c \,d^{5} e^{7}+56 a^{3} c^{2} d^{9} e^{3}\right ) \textit {\_R}^{2}+\left (176 a^{3} d \,e^{10}+384 a^{2} c \,d^{5} e^{6}+48 a \,c^{2} d^{9} e^{2}\right ) \textit {\_R} +256 a d \,e^{9}+324 c \,d^{5} e^{5}\right )\right )}{16}+\frac {e^{7} \ln \left (e x +d \right )}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}\)	\(653\)

[In]

int(1/(e*x+d)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x**4+a)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {e^{7} \log \left (e x + d\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} - \frac {c {\left (\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {1}{4}} e^{7} - 3 \, c^{2} d^{7} + \sqrt {a} c^{\frac {3}{2}} d^{5} e^{2} - 7 \, a c d^{3} e^{4} + 5 \, a^{\frac {3}{2}} \sqrt {c} d e^{6}\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 (4 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {1}{4}} e^{7} + 3 \, c^{2} d^{7} - \sqrt {a} c^{\frac {3}{2}} d^{5} e^{2} + 7 \, a c d^{3} e^{4} - 5 \, a^{\frac {3}{2}} \sqrt {c} d e^{6}\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 (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{7} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{5} e^{2} + 7 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{3} e^{4} + 5 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} d e^{6} + 4 \, \sqrt {a} c^{2} d^{6} e + 12 \, a^{\frac {3}{2}} c d^{2} e^{5}\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 (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{7} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{5} e^{2} + 7 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{3} e^{4} + 5 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} d e^{6} - 4 \, \sqrt {a} c^{2} d^{6} e - 12 \, a^{\frac {3}{2}} c d^{2} e^{5}\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 )}}{32 \, {\left (a c^{2} d^{8} + 2 \, a^{2} c d^{4} e^{4} + a^{3} e^{8}\right )}} + \frac {c d e^{2} x^{3} - c d^{2} e x^{2} + c d^{3} x + a e^{3}}{4 \, {\left (a^{2} c d^{4} + a^{3} e^{4} + {\left (a c^{2} d^{4} + a^{2} c e^{4}\right )} x^{4}\right )}} \]

[In]

integrate(1/(e*x+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

e^7*log(e*x + d)/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8) - 1/32*c*(sqrt(2)*(4*sqrt(2)*a^(7/4)*c^(1/4)*e^7 - 3*c^2*
d^7 + sqrt(a)*c^(3/2)*d^5*e^2 - 7*a*c*d^3*e^4 + 5*a^(3/2)*sqrt(c)*d*e^6)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(
1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(4*sqrt(2)*a^(7/4)*c^(1/4)*e^7 + 3*c^2*d^7 - sqrt(a)*c^(3/2)*d^5
*e^2 + 7*a*c*d^3*e^4 - 5*a^(3/2)*sqrt(c)*d*e^6)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4
)*c^(5/4)) - 2*(3*sqrt(2)*a^(1/4)*c^(9/4)*d^7 + sqrt(2)*a^(3/4)*c^(7/4)*d^5*e^2 + 7*sqrt(2)*a^(5/4)*c^(5/4)*d^
3*e^4 + 5*sqrt(2)*a^(7/4)*c^(3/4)*d*e^6 + 4*sqrt(a)*c^2*d^6*e + 12*a^(3/2)*c*d^2*e^5)*arctan(1/2*sqrt(2)*(2*sq
rt(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*(3*sqrt(
2)*a^(1/4)*c^(9/4)*d^7 + sqrt(2)*a^(3/4)*c^(7/4)*d^5*e^2 + 7*sqrt(2)*a^(5/4)*c^(5/4)*d^3*e^4 + 5*sqrt(2)*a^(7/
4)*c^(3/4)*d*e^6 - 4*sqrt(a)*c^2*d^6*e - 12*a^(3/2)*c*d^2*e^5)*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)))/(a*c^2*d^8 + 2*a^2*c*d^4*e^4 + a^3
*e^8) + 1/4*(c*d*e^2*x^3 - c*d^2*e*x^2 + c*d^3*x + a*e^3)/(a^2*c*d^4 + a^3*e^4 + (a*c^2*d^4 + a^2*c*e^4)*x^4)

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 795, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {e^{8} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{8} e + 2 \, a c d^{4} e^{5} + a^{2} e^{9}} - \frac {e^{7} \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )}} + \frac {{\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \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 )}{8 \, {\left (\sqrt {2} a^{2} c^{3} d^{4} + \sqrt {2} a^{3} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a^{2} c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{3} e - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d e^{3}\right )}} + \frac {{\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \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 )}{8 \, {\left (\sqrt {2} a^{2} c^{3} d^{4} + \sqrt {2} a^{3} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a^{2} c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{3} e + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d e^{3}\right )}} + \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{7} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e^{4} - \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{5} e^{2} - 5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{6}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{8} + 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} e^{8}\right )}} - \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{7} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e^{4} - \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{5} e^{2} - 5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{6}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{8} + 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} e^{8}\right )}} + \frac {a c d^{4} e^{3} + a^{2} e^{7} + {\left (c^{2} d^{5} e^{2} + a c d e^{6}\right )} x^{3} - {\left (c^{2} d^{6} e + a c d^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{7} + a c d^{3} e^{4}\right )} x}{4 \, {\left (c d^{4} + a e^{4}\right )}^{2} {\left (c x^{4} + a\right )} a} \]

[In]

integrate(1/(e*x+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

e^8*log(abs(e*x + d))/(c^2*d^8*e + 2*a*c*d^4*e^5 + a^2*e^9) - 1/4*e^7*log(abs(c*x^4 + a))/(c^2*d^8 + 2*a*c*d^4
*e^4 + a^2*e^8) + 1/8*(4*sqrt(2)*sqrt(a*c)*c^2*d^2*e + 3*(a*c^3)^(1/4)*c^2*d^3 + 5*(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))/(sqrt(2)*a^2*c^3*d^4 + sqrt(2)*a^3*c^2*e^4 + 4*sqrt(2)*s
qrt(a*c)*a^2*c^2*d^2*e^2 - 4*(a*c^3)^(1/4)*a^2*c^2*d^3*e - 4*(a*c^3)^(3/4)*a^2*d*e^3) + 1/8*(4*sqrt(2)*sqrt(a*
c)*c^2*d^2*e + 3*(a*c^3)^(1/4)*c^2*d^3 + 5*(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))/(sqrt(2)*a^2*c^3*d^4 + sqrt(2)*a^3*c^2*e^4 + 4*sqrt(2)*sqrt(a*c)*a^2*c^2*d^2*e^2 + 4*(a*c^3)^(1/
4)*a^2*c^2*d^3*e + 4*(a*c^3)^(3/4)*a^2*d*e^3) + 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^3*d^7 + 7*sqrt(2)*(a*c^3)^(1/4
)*a*c^2*d^3*e^4 - sqrt(2)*(a*c^3)^(3/4)*c*d^5*e^2 - 5*sqrt(2)*(a*c^3)^(3/4)*a*d*e^6)*log(x^2 + sqrt(2)*x*(a/c)
^(1/4) + sqrt(a/c))/(a^2*c^4*d^8 + 2*a^3*c^3*d^4*e^4 + a^4*c^2*e^8) - 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^3*d^7 +
7*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^3*e^4 - sqrt(2)*(a*c^3)^(3/4)*c*d^5*e^2 - 5*sqrt(2)*(a*c^3)^(3/4)*a*d*e^6)*log
(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^8 + 2*a^3*c^3*d^4*e^4 + a^4*c^2*e^8) + 1/4*(a*c*d^4*e^3 +
a^2*e^7 + (c^2*d^5*e^2 + a*c*d*e^6)*x^3 - (c^2*d^6*e + a*c*d^2*e^5)*x^2 + (c^2*d^7 + a*c*d^3*e^4)*x)/((c*d^4
+ a*e^4)^2*(c*x^4 + a)*a)

Mupad [B] (verification not implemented)

Time = 9.66 (sec) , antiderivative size = 1591, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)^2*(d + e*x)),x)

[Out]

e^3/(4*(a^2*e^4 + c^2*d^4*x^4 + a*c*d^4 + a*c*e^4*x^4)) + symsum(log((81*c^5*d^5*e^6 + 64*a*c^4*d*e^10)/(256*(
a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)) + root(131072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*
e^8*z^4 + 65536*a^7*e^7*z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3*e^5*z
+ 81*c*d^4 + 256*a*e^4, z, k)*(root(131072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*e^8*z^4 + 65
536*a^7*e^7*z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3*e^5*z + 81*c*d^4
+ 256*a*e^4, z, k)*(root(131072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*e^8*z^4 + 65536*a^7*e^7*
z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3*e^5*z + 81*c*d^4 + 256*a*e^4,
z, k)*(root(131072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*e^8*z^4 + 65536*a^7*e^7*z^3 + 5120*a
^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3*e^5*z + 81*c*d^4 + 256*a*e^4, z, k)*((983
04*a^9*c^4*d*e^14 - 32768*a^6*c^7*d^13*e^2 + 32768*a^7*c^6*d^9*e^6 + 163840*a^8*c^5*d^5*e^10)/(256*(a^6*e^8 +
a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)) + (x*(81920*a^9*c^4*e^15 - 49152*a^6*c^7*d^12*e^3 - 16384*a^7*c^6*d^8*e^7 + 11
4688*a^8*c^5*d^4*e^11))/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4))) + (52224*a^7*c^4*d*e^13 - 3072*a^4*c^
7*d^13*e + 13312*a^5*c^6*d^9*e^5 + 68608*a^6*c^5*d^5*e^9)/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)) + (x
*(61440*a^7*c^4*e^14 - 8192*a^4*c^7*d^12*e^2 - 4096*a^5*c^6*d^8*e^6 + 65536*a^6*c^5*d^4*e^10))/(256*(a^6*e^8 +
a^4*c^2*d^8 + 2*a^5*c*d^4*e^4))) + (8704*a^5*c^4*d*e^12 + 3584*a^3*c^6*d^9*e^4 + 15360*a^4*c^5*d^5*e^8)/(256*
(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)) + (x*(15360*a^5*c^4*e^13 - 576*a^2*c^7*d^12*e + 1920*a^3*c^6*d^8*e^
5 + 18880*a^4*c^5*d^4*e^9))/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4))) + (192*a*c^6*d^9*e^3 + 704*a^3*c^
4*d*e^11 + 1536*a^2*c^5*d^5*e^7)/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)) + (x*(1280*a^3*c^4*e^12 + 256
*a*c^6*d^8*e^4 + 2240*a^2*c^5*d^4*e^8))/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4))) + (81*c^5*d^4*e^7*x)/
(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)))*root(131072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536
*a^9*e^8*z^4 + 65536*a^7*e^7*z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3*
e^5*z + 81*c*d^4 + 256*a*e^4, z, k), k, 1, 4) + (e^7*log(d + e*x))/(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4) + (c*d^
3*x)/(4*(a^3*e^4 + a^2*c*d^4 + a*c^2*d^4*x^4 + a^2*c*e^4*x^4)) - (c*d^2*e*x^2)/(4*(a^3*e^4 + a^2*c*d^4 + a*c^2
*d^4*x^4 + a^2*c*e^4*x^4)) + (c*d*e^2*x^3)/(4*(a^3*e^4 + a^2*c*d^4 + a*c^2*d^4*x^4 + a^2*c*e^4*x^4))