\(\int \frac {1}{(d+e x)^2 (a+c x^4)} \, dx\) [399]

Optimal result

Integrand size = 17, antiderivative size = 552 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6857, 1890, 1262, 649, 211, 266, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=-\frac {\sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e^2 \left (3 c d^4-a e^4\right )+\sqrt {c} d^2 \left (c d^4-3 a e^4\right )\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e^2 \left (3 c d^4-a e^4\right )+\sqrt {c} d^2 \left (c d^4-3 a e^4\right )\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt {c} d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (c d^4-a e^4\right )}{\sqrt {a} \left (a e^4+c d^4\right )^2}-\frac {e^3}{(d+e x) \left (a e^4+c d^4\right )}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \]

[In]

Int[1/((d + e*x)^2*(a + c*x^4)),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\frac {-\frac {8 e^3 \left (c d^4+a e^4\right )}{d+e x}+\frac {2 \sqrt [4]{c} \left (-\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (\sqrt {2} c d^4-4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt {2} \sqrt {a} \sqrt {c} d^2 e^2-4 a^{3/4} \sqrt [4]{c} d e^3+\sqrt {2} a e^4\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {2 \sqrt [4]{c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {2} c d^4+4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt {2} \sqrt {a} \sqrt {c} d^2 e^2+4 a^{3/4} \sqrt [4]{c} d e^3+\sqrt {2} a e^4\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+32 c d^3 e^3 \log (d+e x)-\frac {\sqrt {2} \sqrt [4]{c} \left (c^{3/2} d^6-3 \sqrt {a} c d^4 e^2-3 a \sqrt {c} d^2 e^4+a^{3/2} e^6\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (c^{3/2} d^6-3 \sqrt {a} c d^4 e^2-3 a \sqrt {c} d^2 e^4+a^{3/2} e^6\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}-8 c d^3 e^3 \log \left (a+c x^4\right )}{8 \left (c d^4+a e^4\right )^2} \]

[In]

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

[Out]

((-8*e^3*(c*d^4 + a*e^4))/(d + e*x) + (2*c^(1/4)*(-(Sqrt[c]*d^2) + Sqrt[a]*e^2)*(Sqrt[2]*c*d^4 - 4*a^(1/4)*c^(
3/4)*d^3*e + 4*Sqrt[2]*Sqrt[a]*Sqrt[c]*d^2*e^2 - 4*a^(3/4)*c^(1/4)*d*e^3 + Sqrt[2]*a*e^4)*ArcTan[1 - (Sqrt[2]*
c^(1/4)*x)/a^(1/4)])/a^(3/4) + (2*c^(1/4)*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[2]*c*d^4 + 4*a^(1/4)*c^(3/4)*d^3*e
+ 4*Sqrt[2]*Sqrt[a]*Sqrt[c]*d^2*e^2 + 4*a^(3/4)*c^(1/4)*d*e^3 + Sqrt[2]*a*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)
/a^(1/4)])/a^(3/4) + 32*c*d^3*e^3*Log[d + e*x] - (Sqrt[2]*c^(1/4)*(c^(3/2)*d^6 - 3*Sqrt[a]*c*d^4*e^2 - 3*a*Sqr
t[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) + (Sqrt[2]*c^(1/4)
*(c^(3/2)*d^6 - 3*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)
*x + Sqrt[c]*x^2])/a^(3/4) - 8*c*d^3*e^3*Log[a + c*x^4])/(8*(c*d^4 + a*e^4)^2)

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.64

method	result	size
default	\(-\frac {c \left (\frac {\left (3 a \,d^{2} e^{4}-c \,d^{6}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-2 a d \,e^{5}+2 c \,d^{5} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (a \,e^{6}-3 c \,d^{4} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+d^{3} e^{3} \ln \left (c \,x^{4}+a \right )\right )}{\left (e^{4} a +d^{4} c \right )^{2}}-\frac {e^{3}}{\left (e^{4} a +d^{4} c \right ) \left (e x +d \right )}+\frac {4 c \,d^{3} e^{3} \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{2}}\)	\(354\)
risch	\(-\frac {e^{3}}{\left (e^{4} a +d^{4} c \right ) \left (e x +d \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{5} e^{8}+2 d^{4} c \,a^{4} e^{4}+c^{2} d^{8} a^{3}\right ) \textit {\_Z}^{4}+16 a^{3} c \,d^{3} e^{3} \textit {\_Z}^{3}+20 a^{2} c \,d^{2} e^{2} \textit {\_Z}^{2}+8 a c d e \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{5} e^{14}+7 a^{4} c \,d^{4} e^{10}-a^{3} c^{2} d^{8} e^{6}-3 a^{2} c^{3} d^{12} e^{2}\right ) \textit {\_R}^{4}+\left (44 a^{3} c \,d^{3} e^{9}+40 a^{2} c^{2} d^{7} e^{5}-4 a \,c^{3} d^{11} e \right ) \textit {\_R}^{3}+\left (79 a^{2} c \,d^{2} e^{8}+62 a \,c^{2} d^{6} e^{4}-c^{3} d^{10}\right ) \textit {\_R}^{2}+\left (32 a c d \,e^{7}+16 c^{2} d^{5} e^{3}\right ) \textit {\_R} +4 c \,e^{6}\right ) x +\left (6 a^{5} d \,e^{13}+10 a^{4} c \,d^{5} e^{9}+2 a^{3} c^{2} d^{9} e^{5}-2 a^{2} c^{3} d^{13} e \right ) \textit {\_R}^{4}+\left (a^{4} e^{12}+33 a^{3} c \,d^{4} e^{8}+31 a^{2} c^{2} d^{8} e^{4}-a \,c^{3} d^{12}\right ) \textit {\_R}^{3}+\left (64 a^{2} c \,d^{3} e^{7}+16 a \,c^{2} d^{7} e^{3}\right ) \textit {\_R}^{2}+32 a c \,d^{2} e^{6} \textit {\_R} +4 c d \,e^{5}\right )\right )}{4}+\frac {4 c \,d^{3} e^{3} \ln \left (e x +d \right )}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}\)	\(460\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\frac {4 \, c d^{3} e^{3} \log \left (e x + d\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} - \frac {e^{3}}{c d^{5} + a d e^{4} + {\left (c d^{4} e + a e^{5}\right )} x} - \frac {c {\left (\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {5}{4}} d^{3} e^{3} - c^{2} d^{6} + 3 \, \sqrt {a} c^{\frac {3}{2}} d^{4} e^{2} + 3 \, a c d^{2} e^{4} - a^{\frac {3}{2}} \sqrt {c} 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 {3}{4}} c^{\frac {5}{4}} d^{3} e^{3} + c^{2} d^{6} - 3 \, \sqrt {a} c^{\frac {3}{2}} d^{4} e^{2} - 3 \, a c d^{2} e^{4} + a^{\frac {3}{2}} \sqrt {c} 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 (\sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{6} + 3 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{4} e^{2} - 3 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{2} e^{4} - \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} e^{6} + 4 \, \sqrt {a} c^{2} d^{5} e - 4 \, a^{\frac {3}{2}} c d 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 (\sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{6} + 3 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{4} e^{2} - 3 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{2} e^{4} - \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} e^{6} - 4 \, \sqrt {a} c^{2} d^{5} e + 4 \, a^{\frac {3}{2}} c d 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 )}}{8 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )}} \]

[In]

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

[Out]

4*c*d^3*e^3*log(e*x + d)/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8) - e^3/(c*d^5 + a*d*e^4 + (c*d^4*e + a*e^5)*x) - 1
/8*c*(sqrt(2)*(4*sqrt(2)*a^(3/4)*c^(5/4)*d^3*e^3 - c^2*d^6 + 3*sqrt(a)*c^(3/2)*d^4*e^2 + 3*a*c*d^2*e^4 - a^(3/
2)*sqrt(c)*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^(3/4)*c^(5/4)*d^3*e^3 + c^2*d^6 - 3*sqrt(a)*c^(3/2)*d^4*e^2 - 3*a*c*d^2*e^4 + a^(3/2)*sqrt(c)*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*(sqrt(2)*a^(1/4)*c^(9/4)*d^6 + 3*sqrt(2)*a
^(3/4)*c^(7/4)*d^4*e^2 - 3*sqrt(2)*a^(5/4)*c^(5/4)*d^2*e^4 - sqrt(2)*a^(7/4)*c^(3/4)*e^6 + 4*sqrt(a)*c^2*d^5*e
- 4*a^(3/2)*c*d*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)) - 2*(sqrt(2)*a^(1/4)*c^(9/4)*d^6 + 3*sqrt(2)*a^(3/4)*c^(7/4)*d^4*e^2 - 3*sqr
t(2)*a^(5/4)*c^(5/4)*d^2*e^4 - sqrt(2)*a^(7/4)*c^(3/4)*e^6 - 4*sqrt(a)*c^2*d^5*e + 4*a^(3/2)*c*d*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)))/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)

Giac [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\frac {4 \, c d^{3} e^{4} \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 {c d^{3} e^{3} \log \left ({\left | c x^{4} + a \right |}\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} 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 )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + \sqrt {2} a^{2} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} 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 )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + \sqrt {2} a^{2} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )}} + \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{6} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{4} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{4} e^{2} + \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a e^{6}\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^{8} + 2 \, a^{2} c^{3} d^{4} e^{4} + a^{3} c^{2} e^{8}\right )}} - \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{6} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{4} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{4} e^{2} + \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a e^{6}\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^{8} + 2 \, a^{2} c^{3} d^{4} e^{4} + a^{3} c^{2} e^{8}\right )}} - \frac {c d^{4} e^{3} + a e^{7}}{{\left (c d^{4} + a e^{4}\right )}^{2} {\left (e x + d\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.35 (sec) , antiderivative size = 2436, normalized size of antiderivative = 4.41 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)*(d + e*x)^2),x)

[Out]

symsum(log((c^5*d*e^6 + c^5*e^7*x + 16*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 10
24*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a^4*c^4*e^13 + 256*root(512*a^4*c*d^4
*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e
*z + c, z, k)^2*a^2*c^5*d^3*e^8 + 496*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 102
4*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a^2*c^6*d^8*e^5 + 528*root(512*a^4*c*d
^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d
*e*z + c, z, k)^3*a^3*c^5*d^4*e^9 - 128*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1
024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^2*c^7*d^13*e^2 + 128*root(512*a^4*
c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*
c*d*e*z + c, z, k)^4*a^3*c^6*d^9*e^6 + 640*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4
+ 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^4*c^5*d^5*e^10 + 32*root(512*a^
4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*
a*c*d*e*z + c, z, k)*a*c^5*d^2*e^7 - 16*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1
024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a*c^7*d^12*e + 16*root(512*a^4*c*d^4
*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e
*z + c, z, k)*c^6*d^5*e^4*x - 4*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*
c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^2*c^7*d^10*e*x + 64*root(512*a^4*c*d^4*e^4*z^4
+ 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c,
z, k)^2*a*c^6*d^7*e^4 + 384*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^
3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^5*c^4*d*e^14 + 320*root(512*a^4*c*d^4*e^4*z^4
+ 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z
, k)^4*a^5*c^4*e^15*x + 248*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^
3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^2*a*c^6*d^6*e^5*x - 64*root(512*a^4*c*d^4*e^4*z^4
+ 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z
, k)^3*a*c^7*d^11*e^2*x + 32*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d
^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)*a*c^5*d*e^8*x + 316*root(512*a^4*c*d^4*e^4*z^4 +
256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z,
k)^2*a^2*c^5*d^2*e^9*x + 640*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d
^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a^2*c^6*d^7*e^6*x + 704*root(512*a^4*c*d^4*e^4*
z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z +
c, z, k)^3*a^3*c^5*d^3*e^10*x - 192*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*
a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^2*c^7*d^12*e^3*x - 64*root(512*a^4*c*d
^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d
*e*z + c, z, k)^4*a^3*c^6*d^8*e^7*x + 448*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 +
1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^4*c^5*d^4*e^11*x)/(a^2*e^8 + c^2
*d^8 + 2*a*c*d^4*e^4))*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3
*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k), k, 1, 4) - e^3/(c*d^5 + a*d*e^4 + a*e^5*x + c*d^4*e*x)
+ (4*c*d^3*e^3*log(d + e*x))/(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)