3.400 \(\int \frac{1}{(d+e x)^3 (a+c x^4)} \, dx\)

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

[Out]

-e^3/(2*(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 - 1
2*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]*Sq
rt[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]*Sq
rt[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)

________________________________________________________________________________________

Rubi [A]  time = 0.949867, antiderivative size = 680, 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, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 205, 260} \[ -\frac{c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt{a} \sqrt{c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac{c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt{a} \sqrt{c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac{\sqrt{c} e \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )^3}-\frac{c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt{a} \sqrt{c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac{c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt{a} \sqrt{c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac{c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (a e^4+c d^4\right )^3}-\frac{4 c d^3 e^3}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac{e^3}{2 (d+e x)^2 \left (a e^4+c d^4\right )}+\frac{2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (a e^4+c d^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + c*x^4)),x]

[Out]

-e^3/(2*(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 - 1
2*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]*Sq
rt[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]*Sq
rt[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 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 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]

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]

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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \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 )^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 ) \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 )^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*}

Mathematica [A]  time = 0.986304, size = 738, normalized size = 1.09 \[ \frac{-\sqrt{2} c^{3/4} d (d+e x)^2 \left (10 a^{3/2} \sqrt{c} d^2 e^6+3 a^2 e^8-6 \sqrt{a} c^{3/2} d^6 e^2-12 a c d^4 e^4+c^2 d^8\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} c^{3/4} d (d+e x)^2 \left (10 a^{3/2} \sqrt{c} d^2 e^6+3 a^2 e^8-6 \sqrt{a} c^{3/2} d^6 e^2-12 a c d^4 e^4+c^2 d^8\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-2 \sqrt{c} (d+e x)^2 \left (-10 \sqrt{2} a^{3/2} c^{3/4} d^3 e^6+24 a^{5/4} c d^4 e^5+3 \sqrt{2} a^2 \sqrt [4]{c} d e^8-2 a^{9/4} e^9+6 \sqrt{2} \sqrt{a} c^{7/4} d^7 e^2-12 \sqrt{2} a c^{5/4} d^5 e^4-6 \sqrt [4]{a} c^2 d^8 e+\sqrt{2} c^{9/4} d^9\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt{c} (d+e x)^2 \left (-10 \sqrt{2} a^{3/2} c^{3/4} d^3 e^6-24 a^{5/4} c d^4 e^5+3 \sqrt{2} a^2 \sqrt [4]{c} d e^8+2 a^{9/4} e^9+6 \sqrt{2} \sqrt{a} c^{7/4} d^7 e^2-12 \sqrt{2} a c^{5/4} d^5 e^4+6 \sqrt [4]{a} c^2 d^8 e+\sqrt{2} c^{9/4} d^9\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+4 a^{3/4} c d^2 e^3 (d+e x)^2 \left (3 a e^4-5 c d^4\right ) \log \left (a+c x^4\right )-32 a^{3/4} c d^3 e^3 (d+e x) \left (a e^4+c d^4\right )+16 a^{3/4} c d^2 e^3 (d+e x)^2 \left (5 c d^4-3 a e^4\right ) \log (d+e x)-4 a^{3/4} e^3 \left (a e^4+c d^4\right )^2}{8 a^{3/4} (d+e x)^2 \left (a e^4+c d^4\right )^3} \]

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 [B]  time = 0.013, size = 1201, normalized size = 1.8 \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)^3/(c*x^4+a),x)

[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)-6*e^7*c*d^2/(a*e^4+c*d^4)^3*ln(e*x+d)*a+1
0*e^3*c^2*d^6/(a*e^4+c*d^4)^3*ln(e*x+d)+3/4*c/(a*e^4+c*d^4)^3*(a/c)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x+1)*d*e^8-3*c^2/(a*e^4+c*d^4)^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^5*e^4+1/4*c^3/(a*e^4+c
*d^4)^3*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^9+3/4*c/(a*e^4+c*d^4)^3*(a/c)^(1/4)*a*2^(1/2)*
arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d*e^8-3*c^2/(a*e^4+c*d^4)^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1
)*d^5*e^4+1/4*c^3/(a*e^4+c*d^4)^3*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^9+3/8*c/(a*e^4+c*d^4
)^3*(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*e^8-3/2*c^2/(a*e^4+c*d^4)^3*(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^5*e^4+1/8*c^3/(a*e^4+c*d^4)^3*(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^9-1/2*c/(a*e^4+c*d^4)^3/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(
1/2))*a^2*e^9+6*c^2/(a*e^4+c*d^4)^3/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*a*d^4*e^5-3/2*c^3/(a*e^4+c*d^4)^3/(a
*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^8*e-5/4*c/(a*e^4+c*d^4)^3/(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*d^3*e^6+3/4*c^2/(a*e^4+c*d^4)^3/(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^7*e^2-5/2*c/(a*e^4+c*d
^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*a*d^3*e^6+3/2*c^2/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^7*e^2-5/2*c/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x-1)*a*d^3*e^6+3/2*c^2/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^7*e^2+3/2*c/(a*e
^4+c*d^4)^3*ln(c*x^4+a)*a*d^2*e^7-5/2*c^2/(a*e^4+c*d^4)^3*ln(c*x^4+a)*d^6*e^3

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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)^3/(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)^3/(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)**3/(c*x**4+a),x)

[Out]

Timed out

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Giac [A]  time = 1.60219, size = 1266, normalized size = 1.86 \begin{align*} \text{result too large to display} \end{align*}

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]

1/2*((a*c^3)^(1/4)*a*c^4*d^3 + sqrt(2)*a^2*c^4*e^3 - 3*(a*c^3)^(3/4)*a*c^2*d*e^2)*arctan(1/2*sqrt(2)*(2*x + sq
rt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^5*d^6 - 6*(a*c^3)^(1/4)*a^2*c^4*d^5*e + 9*sqrt(2)*sqrt(a*c)*a^2
*c^4*d^4*e^2 + 9*sqrt(2)*a^3*c^4*d^2*e^4 - 16*(a*c^3)^(3/4)*a^2*c^2*d^3*e^3 - 6*(a*c^3)^(1/4)*a^3*c^3*d*e^5 +
sqrt(2)*sqrt(a*c)*a^3*c^3*e^6) + 1/2*((a*c^3)^(1/4)*a*c^4*d^3 - sqrt(2)*a^2*c^4*e^3 - 3*(a*c^3)^(3/4)*a*c^2*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^5*d^6 + 6*(a*c^3)^(1/4)*a^2*c^
4*d^5*e + 9*sqrt(2)*sqrt(a*c)*a^2*c^4*d^4*e^2 + 9*sqrt(2)*a^3*c^4*d^2*e^4 + 16*(a*c^3)^(3/4)*a^2*c^2*d^3*e^3 +
6*(a*c^3)^(1/4)*a^3*c^3*d*e^5 + sqrt(2)*sqrt(a*c)*a^3*c^3*e^6) + 1/4*((a*c^3)^(1/4)*c^3*d^9 - 6*(a*c^3)^(3/4)
*c*d^7*e^2 - 12*(a*c^3)^(1/4)*a*c^2*d^5*e^4 + 10*(a*c^3)^(3/4)*a*d^3*e^6 + 3*(a*c^3)^(1/4)*a^2*c*d*e^8)*log(x^
2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^4*d^12 + 3*sqrt(2)*a^2*c^3*d^8*e^4 + 3*sqrt(2)*a^3*c^2*d^4
*e^8 + sqrt(2)*a^4*c*e^12) - 1/4*((a*c^3)^(1/4)*c^3*d^9 - 6*(a*c^3)^(3/4)*c*d^7*e^2 - 12*(a*c^3)^(1/4)*a*c^2*d
^5*e^4 + 10*(a*c^3)^(3/4)*a*d^3*e^6 + 3*(a*c^3)^(1/4)*a^2*c*d*e^8)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c)
)/(sqrt(2)*a*c^4*d^12 + 3*sqrt(2)*a^2*c^3*d^8*e^4 + 3*sqrt(2)*a^3*c^2*d^4*e^8 + sqrt(2)*a^4*c*e^12) - 1/2*(5*c
^2*d^6*e^3 - 3*a*c*d^2*e^7)*log(abs(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)*log(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)