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

Optimal. Leaf size=751 \[ -\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}+\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}+\frac{e}{a^2 d^2 x}-\frac{1}{3 a^2 d x^3}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2} \]

[Out]

-1/(3*a^2*d*x^3) + e/(a^2*d^2*x) - (c^2*x*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (e^(11/2)*ArcTan[
(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2) + (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a
]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d - Sqr
t[a]*e)*(c*d^2 + 2*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) + (c
^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*
(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)
*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)
^2)

________________________________________________________________________________________

Rubi [A]  time = 0.686109, antiderivative size = 751, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {1336, 205, 1179, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}+\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}+\frac{e}{a^2 d^2 x}-\frac{1}{3 a^2 d x^3}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*a^2*d*x^3) + e/(a^2*d^2*x) - (c^2*x*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (e^(11/2)*ArcTan[
(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2) + (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a
]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d - Sqr
t[a]*e)*(c*d^2 + 2*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) + (c
^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*
(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)
*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)
^2)

Rule 1336

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q,
0] || IntegersQ[m, q])

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 1179

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

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}{x^4 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\int \left (\frac{1}{a^2 d x^4}-\frac{e}{a^2 d^2 x^2}+\frac{e^6}{d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac{c^2 \left (d-e x^2\right )}{a \left (c d^2+a e^2\right ) \left (a+c x^4\right )^2}-\frac{c^2 \left (c d^2+2 a e^2\right ) \left (d-e x^2\right )}{a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac{1}{3 a^2 d x^3}+\frac{e}{a^2 d^2 x}+\frac{e^6 \int \frac{1}{d+e x^2} \, dx}{d^2 \left (c d^2+a e^2\right )^2}-\frac{c^2 \int \frac{d-e x^2}{\left (a+c x^4\right )^2} \, dx}{a \left (c d^2+a e^2\right )}-\frac{\left (c^2 \left (c d^2+2 a e^2\right )\right ) \int \frac{d-e x^2}{a+c x^4} \, dx}{a^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{1}{3 a^2 d x^3}+\frac{e}{a^2 d^2 x}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac{c^2 \int \frac{-3 d+e x^2}{a+c x^4} \, dx}{4 a^2 \left (c d^2+a e^2\right )}-\frac{\left (c \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 a^2 \left (c d^2+a e^2\right )^2}-\frac{\left (c \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 a^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{1}{3 a^2 d x^3}+\frac{e}{a^2 d^2 x}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}-\frac{\left (c \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a^2 \left (c d^2+a e^2\right )}-\frac{\left (c \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a^2 \left (c d^2+a e^2\right )}-\frac{\left (c \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 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 a^2 \left (c d^2+a e^2\right )^2}-\frac{\left (c \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 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 a^2 \left (c d^2+a e^2\right )^2}+\frac{\left (c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}+\frac{\left (c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}\\ &=-\frac{1}{3 a^2 d x^3}+\frac{e}{a^2 d^2 x}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}-\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}-\frac{\left (c \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\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^2 \left (c d^2+a e^2\right )}-\frac{\left (c \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\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^2 \left (c d^2+a e^2\right )}+\frac{\left (c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\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^{9/4} \left (c d^2+a e^2\right )}+\frac{\left (c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\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^{9/4} \left (c d^2+a e^2\right )}-\frac{\left (c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\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} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac{\left (c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\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} a^{9/4} \left (c d^2+a e^2\right )^2}\\ &=-\frac{1}{3 a^2 d x^3}+\frac{e}{a^2 d^2 x}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac{c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}+\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}-\frac{c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}-\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}-\frac{\left (c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\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 )}{8 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}+\frac{\left (c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\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 )}{8 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}\\ &=-\frac{1}{3 a^2 d x^3}+\frac{e}{a^2 d^2 x}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac{c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}+\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac{c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}-\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac{c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}+\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}-\frac{c^{5/4} \left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{9/4} \left (c d^2+a e^2\right )}-\frac{c^{5/4} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (c d^2+2 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^{9/4} \left (c d^2+a e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.39944, size = 513, normalized size = 0.68 \[ \frac{1}{96} \left (-\frac{24 c^2 x \left (d-e x^2\right )}{a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{3 \sqrt{2} c^{5/4} \left (9 a^{3/2} e^3+5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{3 \sqrt{2} c^{5/4} \left (9 a^{3/2} e^3+5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{6 \sqrt{2} c^{5/4} \left (-9 a^{3/2} e^3-5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{6 \sqrt{2} c^{5/4} \left (9 a^{3/2} e^3+5 \sqrt{a} c d^2 e-11 a \sqrt{c} d e^2-7 c^{3/2} d^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{96 e}{a^2 d^2 x}-\frac{32}{a^2 d x^3}+\frac{96 e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32/(a^2*d*x^3) + (96*e)/(a^2*d^2*x) - (24*c^2*x*(d - e*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (96*e^(11/2
)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2) + (6*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 - 5*Sqrt[a]*c*d
^2*e + 11*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2 + a*e^2)^
2) + (6*Sqrt[2]*c^(5/4)*(-7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*ArcTan[1 + (
Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2 + a*e^2)^2) + (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^
2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c
*d^2 + a*e^2)^2) - (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c*d^2 + a*e^2)^2))/96

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Maple [A]  time = 0.02, size = 932, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*x^3*e^3+1/4*c^3/(a*e^2+c*d^2)^2/a^2/(c*x^4+a)*x^3*e*d^2-1/4*c^2/(a*e^2+c*d
^2)^2/a/(c*x^4+a)*x*e^2*d-1/4*c^3/(a*e^2+c*d^2)^2/a^2/(c*x^4+a)*x*d^3-11/16*c^2/(a*e^2+c*d^2)^2/a^2*(1/c*a)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d*e^2-7/16*c^3/(a*e^2+c*d^2)^2/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/c*a)^(1/4)*x-1)*d^3-11/32*c^2/(a*e^2+c*d^2)^2/a^2*(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(
1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d*e^2-7/32*c^3/(a*e^2+c*d^2)^2/a^3*(1/c*a)^(1
/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^3-11
/16*c^2/(a*e^2+c*d^2)^2/a^2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d*e^2-7/16*c^3/(a*e^2+c*d^
2)^2/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^3+9/16*c/(a*e^2+c*d^2)^2/a/(1/c*a)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*e^3+5/16*c^2/(a*e^2+c*d^2)^2/a^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(1/c*a)^(1/4)*x-1)*d^2*e+9/32*c/(a*e^2+c*d^2)^2/a/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a
)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*e^3+5/32*c^2/(a*e^2+c*d^2)^2/a^2/(1/c*a)^(1/4)*2^(1/2)*l
n((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^2*e+9/16*c/(a*e^2
+c*d^2)^2/a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*e^3+5/16*c^2/(a*e^2+c*d^2)^2/a^2/(1/c*a)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^2*e-1/3/a^2/d/x^3+e/a^2/d^2/x+1/d^2*e^6/(a*e^2+c*d^2)^2/(d*e)
^(1/2)*arctan(e*x/(d*e)^(1/2))

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

[Out]

Timed out

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Giac [A]  time = 1.11836, size = 848, normalized size = 1.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 - 5*(a*c^3)^(3/4)*c*d^2*e - 9*(a*c^3)^(3/4)*a*e^3
)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2
+ sqrt(2)*a^5*c*e^4) - 1/8*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 - 5*(a*c^3)^(3/4)*c*d^2*e
- 9*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^4 + 2*
sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) - 1/16*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 +
5*(a*c^3)^(3/4)*c*d^2*e + 9*(a*c^3)^(3/4)*a*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^3*c^3
*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) + 1/16*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2
*d*e^2 + 5*(a*c^3)^(3/4)*c*d^2*e + 9*(a*c^3)^(3/4)*a*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2
)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) + arctan(x*e^(1/2)/sqrt(d))*e^(11/2)/((c^2*d^6
+ 2*a*c*d^4*e^2 + a^2*d^2*e^4)*sqrt(d)) + 1/4*(c^2*x^3*e - c^2*d*x)/((a^2*c*d^2 + a^3*e^2)*(c*x^4 + a)) + 1/3*
(3*x^2*e - d)/(a^2*d^2*x^3)