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

Optimal. Leaf size=552 \[ -\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^4+c d^4\right )^2}-\frac{e^3}{(d+e x) \left (a e^4+c d^4\right )}+\frac{4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.808323, antiderivative size = 552, normalized size of antiderivative = 1., 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 = {6725, 1876, 1248, 635, 205, 260, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^4+c d^4\right )^2}-\frac{e^3}{(d+e x) \left (a e^4+c d^4\right )}+\frac{4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \]

Antiderivative was successfully verified.

[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 6725

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]

Rule 1876

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 1248

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 635

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 205

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

Rule 260

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

Rule 1168

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 1162

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+c x^4\right )} \, dx &=\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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.706478, size = 524, normalized size = 0.95 \[ \frac{-\frac{\sqrt{2} \sqrt [4]{c} \left (a^{3/2} e^6-3 \sqrt{a} c d^4 e^2-3 a \sqrt{c} d^2 e^4+c^{3/2} d^6\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (a^{3/2} e^6-3 \sqrt{a} c d^4 e^2-3 a \sqrt{c} d^2 e^4+c^{3/2} d^6\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{2 \sqrt [4]{c} \left (\sqrt{a} e^2-\sqrt{c} d^2\right ) \left (-4 a^{3/4} \sqrt [4]{c} d e^3-4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt{2} \sqrt{a} \sqrt{c} d^2 e^2+\sqrt{2} a e^4+\sqrt{2} c d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt [4]{c} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (4 a^{3/4} \sqrt [4]{c} d e^3+4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt{2} \sqrt{a} \sqrt{c} d^2 e^2+\sqrt{2} a e^4+\sqrt{2} c d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-8 c d^3 e^3 \log \left (a+c x^4\right )-\frac{8 e^3 \left (a e^4+c d^4\right )}{d+e x}+32 c d^3 e^3 \log (d+e x)}{8 \left (a e^4+c d^4\right )^2} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.01, size = 866, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out