\(\int \frac {1}{(d+e x) (a+c x^4)^3} \, dx\) [412]

Optimal result

Integrand size = 17, antiderivative size = 1352 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\frac {c x \left (7 d^3-6 d^2 e x+5 d e^2 x^2\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{8 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}+\frac {e^4 \left (a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^9 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d^2 e^5 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^2}-\frac {3 \sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d e^4 \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d e^4 \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d e^4 \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d e^4 \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}-\frac {e^{11} \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^3} \]

[Out]

1/32*c*x*(5*d*e^2*x^2-6*d^2*e*x+7*d^3)/a^2/(a*e^4+c*d^4)/(c*x^4+a)+1/8*(a*e^3+c*x*(d*e^2*x^2-d^2*e*x+d^3))/a/(
a*e^4+c*d^4)/(c*x^4+a)^2+1/4*e^4*(a*e^3+c*x*(d*e^2*x^2-d^2*e*x+d^3))/a/(a*e^4+c*d^4)^2/(c*x^4+a)+e^11*ln(e*x+d
)/(a*e^4+c*d^4)^3-1/4*e^11*ln(c*x^4+a)/(a*e^4+c*d^4)^3-1/4*d^2*e^5*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(3/2)
/(a*e^4+c*d^4)^2-3/16*d^2*e*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(5/2)/(a*e^4+c*d^4)-1/2*d^2*e^9*arctan(x^2*c
^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^3/a^(1/2)-1/8*c^(1/4)*d*e^8*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^
(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/8*c^(1/4)*d*e^8*ln(a^(1/4)*c^(1/4)*x*2^(1/
2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(1/4)*d*e^8*arctan(-1
+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(1/4)*d*e^8*arctan
(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)-1/32*c^(1/4)*d*e^4*ln(
-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^2*2^(1/2)+1
/32*c^(1/4)*d*e^4*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^
4+c*d^4)^2*2^(1/2)+1/16*c^(1/4)*d*e^4*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)
/(a*e^4+c*d^4)^2*2^(1/2)+1/16*c^(1/4)*d*e^4*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+3*d^2*c^(1/2))/a^
(7/4)/(a*e^4+c*d^4)^2*2^(1/2)-1/256*c^(1/4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-5*e^2*a^(1/
2)+21*d^2*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)*2^(1/2)+1/256*c^(1/4)*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(
1/2))*(-5*e^2*a^(1/2)+21*d^2*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)*2^(1/2)+1/128*c^(1/4)*d*arctan(-1+c^(1/4)*x*2^(1/
2)/a^(1/4))*(5*e^2*a^(1/2)+21*d^2*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)*2^(1/2)+1/128*c^(1/4)*d*arctan(1+c^(1/4)*x*2
^(1/2)/a^(1/4))*(5*e^2*a^(1/2)+21*d^2*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)*2^(1/2)

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 1352, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {6874, 1868, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\frac {\log (d+e x) e^{11}}{\left (c d^4+a e^4\right )^3}-\frac {\log \left (c x^4+a\right ) e^{11}}{4 \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^9}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^8}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^8}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^8}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^8}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{4 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac {\left (a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )\right ) e^4}{4 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {3 \sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{16 a^{5/2} \left (c d^4+a e^4\right )}+\frac {a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{8 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )^2}-\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {c x \left (7 d^3-6 e x d^2+5 e^2 x^2 d\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (c x^4+a\right )} \]

[In]

Int[1/((d + e*x)*(a + c*x^4)^3),x]

[Out]

(c*x*(7*d^3 - 6*d^2*e*x + 5*d*e^2*x^2))/(32*a^2*(c*d^4 + a*e^4)*(a + c*x^4)) + (a*e^3 + c*x*(d^3 - d^2*e*x + d
*e^2*x^2))/(8*a*(c*d^4 + a*e^4)*(a + c*x^4)^2) + (e^4*(a*e^3 + c*x*(d^3 - d^2*e*x + d*e^2*x^2)))/(4*a*(c*d^4 +
a*e^4)^2*(a + c*x^4)) - (Sqrt[c]*d^2*e^9*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^3) - (Sqrt
[c]*d^2*e^5*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^2) - (3*Sqrt[c]*d^2*e*ArcTan[(Sqrt[c]*x^
2)/Sqrt[a]])/(16*a^(5/2)*(c*d^4 + a*e^4)) - (c^(1/4)*d*e^8*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) - (c^(1/4)*d*e^4*(3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[
1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^2) - (c^(1/4)*d*(21*Sqrt[c]*d^2 + 5*Sqrt[
a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)) + (c^(1/4)*d*e^8*(Sqrt[
c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(1/4
)*d*e^4*(3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e
^4)^2) + (c^(1/4)*d*(21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(1
1/4)*(c*d^4 + a*e^4)) + (e^11*Log[d + e*x])/(c*d^4 + a*e^4)^3 - (c^(1/4)*d*e^8*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) - (c^(1/4)*d*e^4*(3
*Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4
+ a*e^4)^2) - (c^(1/4)*d*(21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x
^2])/(128*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)) + (c^(1/4)*d*e^8*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(1/4)*d*e^4*(3*Sqrt[c]*d^2 - Sqr
t[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^2) + (c^
(1/4)*d*(21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*
a^(11/4)*(c*d^4 + a*e^4)) - (e^11*Log[a + c*x^4])/(4*(c*d^4 + a*e^4)^3)

Rule 210

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

Rule 211

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

Rule 266

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

Rule 281

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

Rule 631

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

Rule 642

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

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1868

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

Rule 1869

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

Rule 1890

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{12}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (c d^4+a e^4\right ) \left (a+c x^4\right )^3}-\frac {c e^4 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}-\frac {c e^8 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}\right ) \, dx \\ & = \frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e^8\right ) \int \frac {-d^3+d^2 e x-d e^2 x^2+e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e^4\right ) \int \frac {-d^3+d^2 e x-d e^2 x^2+e^3 x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^2}+\frac {c \int \frac {d^3-d^2 e x+d e^2 x^2-e^3 x^3}{\left (a+c x^4\right )^3} \, dx}{c d^4+a e^4} \\ & = \frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{8 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}+\frac {e^4 \left (a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e^8\right ) \int \left (\frac {-d^3-d e^2 x^2}{a+c x^4}+\frac {x \left (d^2 e+e^3 x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \frac {3 d^3-2 d^2 e x+d e^2 x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^2}-\frac {c \int \frac {-7 d^3+6 d^2 e x-5 d e^2 x^2}{\left (a+c x^4\right )^2} \, dx}{8 a \left (c d^4+a e^4\right )} \\ & = \frac {c x \left (7 d^3-6 d^2 e x+5 d e^2 x^2\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{8 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}+\frac {e^4 \left (a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e^8\right ) \int \frac {-d^3-d e^2 x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e^8\right ) \int \frac {x \left (d^2 e+e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \left (-\frac {2 d^2 e x}{a+c x^4}+\frac {3 d^3+d e^2 x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^2}+\frac {c \int \frac {21 d^3-12 d^2 e x+5 d e^2 x^2}{a+c x^4} \, dx}{32 a^2 \left (c d^4+a e^4\right )} \\ & = \frac {c x \left (7 d^3-6 d^2 e x+5 d e^2 x^2\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{8 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}+\frac {e^4 \left (a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e^8\right ) \text {Subst}\left (\int \frac {d^2 e+e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (d e^8 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (d e^8 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \frac {3 d^3+d e^2 x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^2}-\frac {\left (c d^2 e^5\right ) \int \frac {x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )^2}+\frac {c \int \left (-\frac {12 d^2 e x}{a+c x^4}+\frac {21 d^3+5 d e^2 x^2}{a+c x^4}\right ) \, dx}{32 a^2 \left (c d^4+a e^4\right )} \\ & = \frac {c x \left (7 d^3-6 d^2 e x+5 d e^2 x^2\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{8 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}+\frac {e^4 \left (a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c d^2 e^9\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}-\frac {\left (c e^{11}\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (d e^8 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}+\frac {\left (d e^8 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}-\frac {\left (\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\left (\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\left (c d^2 e^5\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^2}+\frac {c \int \frac {21 d^3+5 d e^2 x^2}{a+c x^4} \, dx}{32 a^2 \left (c d^4+a e^4\right )}-\frac {\left (3 c d^2 e\right ) \int \frac {x}{a+c x^4} \, dx}{8 a^2 \left (c d^4+a e^4\right )} \\ & = \frac {c x \left (7 d^3-6 d^2 e x+5 d e^2 x^2\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (a+c x^4\right )}+\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{8 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )^2}+\frac {e^4 \left (a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^9 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac {e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {e^{11} \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^3}+\frac {\left (\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\left (\sqrt [4]{c} d e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\left (d e^4 \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^2}+\frac {\left (d e^4 \left (\frac {3 \sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} d e^4 \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {\left (3 c d^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2 \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}-5 e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{64 a^2 \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {21 \sqrt {c} d^2}{\sqrt {a}}+5 e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{64 a^2 \left (c d^4+a e^4\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 835, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\frac {\frac {32 \left (c d^4+a e^4\right )^2 \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )}{a \left (a+c x^4\right )^2}+\frac {8 \left (c d^4+a e^4\right ) \left (8 a^2 e^7+c^2 d^5 x \left (7 d^2-6 d e x+5 e^2 x^2\right )+a c d e^4 x \left (15 d^2-14 d e x+13 e^2 x^2\right )\right )}{a^2 \left (a+c x^4\right )}-\frac {2 \sqrt [4]{c} d \left (21 \sqrt {2} c^{5/2} d^{10}-24 \sqrt [4]{a} c^{9/4} d^9 e+5 \sqrt {2} \sqrt {a} c^2 d^8 e^2+66 \sqrt {2} a c^{3/2} d^6 e^4-80 a^{5/4} c^{5/4} d^5 e^5+18 \sqrt {2} a^{3/2} c d^4 e^6+77 \sqrt {2} a^2 \sqrt {c} d^2 e^8-120 a^{9/4} \sqrt [4]{c} d e^9+45 \sqrt {2} a^{5/2} e^{10}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {2 \sqrt [4]{c} d \left (21 \sqrt {2} c^{5/2} d^{10}+24 \sqrt [4]{a} c^{9/4} d^9 e+5 \sqrt {2} \sqrt {a} c^2 d^8 e^2+66 \sqrt {2} a c^{3/2} d^6 e^4+80 a^{5/4} c^{5/4} d^5 e^5+18 \sqrt {2} a^{3/2} c d^4 e^6+77 \sqrt {2} a^2 \sqrt {c} d^2 e^8+120 a^{9/4} \sqrt [4]{c} d e^9+45 \sqrt {2} a^{5/2} e^{10}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+256 e^{11} \log (d+e x)+\frac {\sqrt {2} \sqrt [4]{c} \left (-21 c^{5/2} d^{11}+5 \sqrt {a} c^2 d^9 e^2-66 a c^{3/2} d^7 e^4+18 a^{3/2} c d^5 e^6-77 a^2 \sqrt {c} d^3 e^8+45 a^{5/2} d e^{10}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (21 c^{5/2} d^{11}-5 \sqrt {a} c^2 d^9 e^2+66 a c^{3/2} d^7 e^4-18 a^{3/2} c d^5 e^6+77 a^2 \sqrt {c} d^3 e^8-45 a^{5/2} d e^{10}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4}}-64 e^{11} \log \left (a+c x^4\right )}{256 \left (c d^4+a e^4\right )^3} \]

[In]

Integrate[1/((d + e*x)*(a + c*x^4)^3),x]

[Out]

((32*(c*d^4 + a*e^4)^2*(a*e^3 + c*d*x*(d^2 - d*e*x + e^2*x^2)))/(a*(a + c*x^4)^2) + (8*(c*d^4 + a*e^4)*(8*a^2*
e^7 + c^2*d^5*x*(7*d^2 - 6*d*e*x + 5*e^2*x^2) + a*c*d*e^4*x*(15*d^2 - 14*d*e*x + 13*e^2*x^2)))/(a^2*(a + c*x^4
)) - (2*c^(1/4)*d*(21*Sqrt[2]*c^(5/2)*d^10 - 24*a^(1/4)*c^(9/4)*d^9*e + 5*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 + 66*Sqr
t[2]*a*c^(3/2)*d^6*e^4 - 80*a^(5/4)*c^(5/4)*d^5*e^5 + 18*Sqrt[2]*a^(3/2)*c*d^4*e^6 + 77*Sqrt[2]*a^2*Sqrt[c]*d^
2*e^8 - 120*a^(9/4)*c^(1/4)*d*e^9 + 45*Sqrt[2]*a^(5/2)*e^10)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4)
+ (2*c^(1/4)*d*(21*Sqrt[2]*c^(5/2)*d^10 + 24*a^(1/4)*c^(9/4)*d^9*e + 5*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 + 66*Sqrt[
2]*a*c^(3/2)*d^6*e^4 + 80*a^(5/4)*c^(5/4)*d^5*e^5 + 18*Sqrt[2]*a^(3/2)*c*d^4*e^6 + 77*Sqrt[2]*a^2*Sqrt[c]*d^2*
e^8 + 120*a^(9/4)*c^(1/4)*d*e^9 + 45*Sqrt[2]*a^(5/2)*e^10)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) +
256*e^11*Log[d + e*x] + (Sqrt[2]*c^(1/4)*(-21*c^(5/2)*d^11 + 5*Sqrt[a]*c^2*d^9*e^2 - 66*a*c^(3/2)*d^7*e^4 + 1
8*a^(3/2)*c*d^5*e^6 - 77*a^2*Sqrt[c]*d^3*e^8 + 45*a^(5/2)*d*e^10)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/a^(11/4) + (Sqrt[2]*c^(1/4)*(21*c^(5/2)*d^11 - 5*Sqrt[a]*c^2*d^9*e^2 + 66*a*c^(3/2)*d^7*e^4 - 18*a
^(3/2)*c*d^5*e^6 + 77*a^2*Sqrt[c]*d^3*e^8 - 45*a^(5/2)*d*e^10)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[
c]*x^2])/a^(11/4) - 64*e^11*Log[a + c*x^4])/(256*(c*d^4 + a*e^4)^3)

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 677, normalized size of antiderivative = 0.50

method	result	size
default	\(\frac {c \left (\frac {\frac {c d \,e^{2} \left (13 a^{2} e^{8}+18 a c \,d^{4} e^{4}+5 c^{2} d^{8}\right ) x^{7}}{32 a^{2}}-\frac {c \,d^{2} e \left (7 a^{2} e^{8}+10 a c \,d^{4} e^{4}+3 c^{2} d^{8}\right ) x^{6}}{16 a^{2}}+\frac {d^{3} c \left (15 a^{2} e^{8}+22 a c \,d^{4} e^{4}+7 c^{2} d^{8}\right ) x^{5}}{32 a^{2}}+\left (\frac {1}{4} a \,e^{11}+\frac {1}{4} d^{4} e^{7} c \right ) x^{4}+\frac {d \,e^{2} \left (17 a^{2} e^{8}+26 a c \,d^{4} e^{4}+9 c^{2} d^{8}\right ) x^{3}}{32 a}-\frac {d^{2} e \left (9 a^{2} e^{8}+14 a c \,d^{4} e^{4}+5 c^{2} d^{8}\right ) x^{2}}{16 a}+\frac {d^{3} \left (19 a^{2} e^{8}+30 a c \,d^{4} e^{4}+11 c^{2} d^{8}\right ) x}{32 a}+\frac {e^{3} \left (3 a^{2} e^{8}+4 a c \,d^{4} e^{4}+c^{2} d^{8}\right )}{8 c}}{\left (c \,x^{4}+a \right )^{2}}+\frac {\frac {\left (77 a^{2} d^{3} e^{8}+66 a c \,d^{7} e^{4}+21 c^{2} d^{11}\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 (-60 a^{2} d^{2} e^{9}-40 a c \,d^{6} e^{5}-12 c^{2} d^{10} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (45 a^{2} d \,e^{10}+18 a c \,d^{5} e^{6}+5 c^{2} d^{9} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}-\frac {8 a^{2} e^{11} \ln \left (c \,x^{4}+a \right )}{c}}{32 a^{2}}\right )}{\left (e^{4} a +d^{4} c \right )^{3}}+\frac {e^{11} \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{3}}\)	\(677\)
risch	\(\text {Expression too large to display}\)	\(1312\)

[In]

int(1/(e*x+d)/(c*x^4+a)^3,x,method=_RETURNVERBOSE)

[Out]

c/(a*e^4+c*d^4)^3*((1/32*c*d*e^2*(13*a^2*e^8+18*a*c*d^4*e^4+5*c^2*d^8)/a^2*x^7-1/16*c*d^2*e*(7*a^2*e^8+10*a*c*
d^4*e^4+3*c^2*d^8)/a^2*x^6+1/32*d^3*c*(15*a^2*e^8+22*a*c*d^4*e^4+7*c^2*d^8)/a^2*x^5+(1/4*a*e^11+1/4*d^4*e^7*c)
*x^4+1/32*d*e^2*(17*a^2*e^8+26*a*c*d^4*e^4+9*c^2*d^8)/a*x^3-1/16*d^2*e*(9*a^2*e^8+14*a*c*d^4*e^4+5*c^2*d^8)/a*
x^2+1/32*d^3*(19*a^2*e^8+30*a*c*d^4*e^4+11*c^2*d^8)/a*x+1/8*e^3*(3*a^2*e^8+4*a*c*d^4*e^4+c^2*d^8)/c)/(c*x^4+a)
^2+1/32/a^2*(1/8*(77*a^2*d^3*e^8+66*a*c*d^7*e^4+21*c^2*d^11)*(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*(-60*a^2*d^2*e^9-40*a*c*d^6*e^5-12*c^2*d^10*e)/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+1/8*(4
5*a^2*d*e^10+18*a*c*d^5*e^6+5*c^2*d^9*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))-8
*a^2*e^11/c*ln(c*x^4+a)))+e^11*ln(e*x+d)/(a*e^4+c*d^4)^3

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x^4+a)^3,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x**4+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 1015, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

e^11*log(e*x + d)/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) - 1/256*c*(sqrt(2)*(32*sqrt(2)*a^(
11/4)*c^(1/4)*e^11 - 21*c^3*d^11 + 5*sqrt(a)*c^(5/2)*d^9*e^2 - 66*a*c^2*d^7*e^4 + 18*a^(3/2)*c^(3/2)*d^5*e^6 -
77*a^2*c*d^3*e^8 + 45*a^(5/2)*sqrt(c)*d*e^10)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)
*c^(5/4)) + sqrt(2)*(32*sqrt(2)*a^(11/4)*c^(1/4)*e^11 + 21*c^3*d^11 - 5*sqrt(a)*c^(5/2)*d^9*e^2 + 66*a*c^2*d^7
*e^4 - 18*a^(3/2)*c^(3/2)*d^5*e^6 + 77*a^2*c*d^3*e^8 - 45*a^(5/2)*sqrt(c)*d*e^10)*log(sqrt(c)*x^2 - sqrt(2)*a^
(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) - 2*(21*sqrt(2)*a^(1/4)*c^(13/4)*d^11 + 5*sqrt(2)*a^(3/4)*c^(11/4
)*d^9*e^2 + 66*sqrt(2)*a^(5/4)*c^(9/4)*d^7*e^4 + 18*sqrt(2)*a^(7/4)*c^(7/4)*d^5*e^6 + 77*sqrt(2)*a^(9/4)*c^(5/
4)*d^3*e^8 + 45*sqrt(2)*a^(11/4)*c^(3/4)*d*e^10 + 24*sqrt(a)*c^3*d^10*e + 80*a^(3/2)*c^2*d^6*e^5 + 120*a^(5/2)
*c*d^2*e^9)*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(sq
rt(a)*sqrt(c))*c^(5/4)) - 2*(21*sqrt(2)*a^(1/4)*c^(13/4)*d^11 + 5*sqrt(2)*a^(3/4)*c^(11/4)*d^9*e^2 + 66*sqrt(2
)*a^(5/4)*c^(9/4)*d^7*e^4 + 18*sqrt(2)*a^(7/4)*c^(7/4)*d^5*e^6 + 77*sqrt(2)*a^(9/4)*c^(5/4)*d^3*e^8 + 45*sqrt(
2)*a^(11/4)*c^(3/4)*d*e^10 - 24*sqrt(a)*c^3*d^10*e - 80*a^(3/2)*c^2*d^6*e^5 - 120*a^(5/2)*c*d^2*e^9)*arctan(1/
2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4
)))/(a^2*c^3*d^12 + 3*a^3*c^2*d^8*e^4 + 3*a^4*c*d^4*e^8 + a^5*e^12) + 1/32*(8*a^2*c*e^7*x^4 + 4*a^2*c*d^4*e^3
+ 12*a^3*e^7 + (5*c^3*d^5*e^2 + 13*a*c^2*d*e^6)*x^7 - 2*(3*c^3*d^6*e + 7*a*c^2*d^2*e^5)*x^6 + (7*c^3*d^7 + 15*
a*c^2*d^3*e^4)*x^5 + (9*a*c^2*d^5*e^2 + 17*a^2*c*d*e^6)*x^3 - 2*(5*a*c^2*d^6*e + 9*a^2*c*d^2*e^5)*x^2 + (11*a*
c^2*d^7 + 19*a^2*c*d^3*e^4)*x)/(a^4*c^2*d^8 + 2*a^5*c*d^4*e^4 + a^6*e^8 + (a^2*c^4*d^8 + 2*a^3*c^3*d^4*e^4 + a
^4*c^2*e^8)*x^8 + 2*(a^3*c^3*d^8 + 2*a^4*c^2*d^4*e^4 + a^5*c*e^8)*x^4)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 1311, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)/(c*x^4+a)^3,x, algorithm="giac")

[Out]

e^12*log(abs(e*x + 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) - 1/4*e^11*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) - 1/64*(75*sqrt(2)*a*c^2*d^2*e^3 - 51*sqrt(2)*sq
rt(a*c)*c^2*d^4*e - 21*(a*c^3)^(1/4)*c^2*d^5 - 45*(a*c^3)^(1/4)*a*c*d*e^4 - 122*(a*c^3)^(3/4)*d^3*e^2)*arctan(
1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^6 + 9*sqrt(2)*a^4*c^2*d^2*e^4 + 9*sqrt
(2)*sqrt(a*c)*a^3*c^2*d^4*e^2 + sqrt(2)*sqrt(a*c)*a^4*c*e^6 - 6*(a*c^3)^(1/4)*a^3*c^2*d^5*e - 6*(a*c^3)^(1/4)*
a^4*c*d*e^5 - 16*(a*c^3)^(3/4)*a^3*d^3*e^3) + 1/64*(75*sqrt(2)*a*c^2*d^2*e^3 + 51*sqrt(2)*sqrt(a*c)*c^2*d^4*e
+ 21*(a*c^3)^(1/4)*c^2*d^5 + 45*(a*c^3)^(1/4)*a*c*d*e^4 + 122*(a*c^3)^(3/4)*d^3*e^2)*arctan(1/2*sqrt(2)*(2*x -
sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^6 + 9*sqrt(2)*a^4*c^2*d^2*e^4 + 9*sqrt(2)*sqrt(a*c)*a^3*
c^2*d^4*e^2 + sqrt(2)*sqrt(a*c)*a^4*c*e^6 + 6*(a*c^3)^(1/4)*a^3*c^2*d^5*e + 6*(a*c^3)^(1/4)*a^4*c*d*e^5 + 16*(
a*c^3)^(3/4)*a^3*d^3*e^3) + 1/256*(21*sqrt(2)*(a*c^3)^(1/4)*c^4*d^11 + 66*sqrt(2)*(a*c^3)^(1/4)*a*c^3*d^7*e^4
+ 77*sqrt(2)*(a*c^3)^(1/4)*a^2*c^2*d^3*e^8 - 5*sqrt(2)*(a*c^3)^(3/4)*c^2*d^9*e^2 - 18*sqrt(2)*(a*c^3)^(3/4)*a*
c*d^5*e^6 - 45*sqrt(2)*(a*c^3)^(3/4)*a^2*d*e^10)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^5*d^12 +
3*a^4*c^4*d^8*e^4 + 3*a^5*c^3*d^4*e^8 + a^6*c^2*e^12) - 1/256*(21*sqrt(2)*(a*c^3)^(1/4)*c^4*d^11 + 66*sqrt(2)*
(a*c^3)^(1/4)*a*c^3*d^7*e^4 + 77*sqrt(2)*(a*c^3)^(1/4)*a^2*c^2*d^3*e^8 - 5*sqrt(2)*(a*c^3)^(3/4)*c^2*d^9*e^2 -
18*sqrt(2)*(a*c^3)^(3/4)*a*c*d^5*e^6 - 45*sqrt(2)*(a*c^3)^(3/4)*a^2*d*e^10)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) +
sqrt(a/c))/(a^3*c^5*d^12 + 3*a^4*c^4*d^8*e^4 + 3*a^5*c^3*d^4*e^8 + a^6*c^2*e^12) + 1/32*(4*a^2*c^2*d^8*e^3 +
16*a^3*c*d^4*e^7 + 12*a^4*e^11 + (5*c^4*d^9*e^2 + 18*a*c^3*d^5*e^6 + 13*a^2*c^2*d*e^10)*x^7 - 2*(3*c^4*d^10*e
+ 10*a*c^3*d^6*e^5 + 7*a^2*c^2*d^2*e^9)*x^6 + (7*c^4*d^11 + 22*a*c^3*d^7*e^4 + 15*a^2*c^2*d^3*e^8)*x^5 + 8*(a^
2*c^2*d^4*e^7 + a^3*c*e^11)*x^4 + (9*a*c^3*d^9*e^2 + 26*a^2*c^2*d^5*e^6 + 17*a^3*c*d*e^10)*x^3 - 2*(5*a*c^3*d^
10*e + 14*a^2*c^2*d^6*e^5 + 9*a^3*c*d^2*e^9)*x^2 + (11*a*c^3*d^11 + 30*a^2*c^2*d^7*e^4 + 19*a^3*c*d^3*e^8)*x)/
((c*d^4 + a*e^4)^3*(c*x^4 + a)^2*a^2)

Mupad [B] (verification not implemented)

Time = 10.55 (sec) , antiderivative size = 2720, normalized size of antiderivative = 2.01 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)^3*(d + e*x)),x)

[Out]

symsum(log((194481*c^7*d^13*e^6 + 871362*a*c^6*d^9*e^10 + 425984*a^3*c^4*d*e^18 + 1148881*a^2*c^5*d^5*e^14)/(1
048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8)) + root(805306
368*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c^3*d^12*z^4 + 268435456*a^14*e^12*z^
4 + 268435456*a^11*e^11*z^3 + 43057152*a^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 100663296*a^8*e^10*z
^2 + 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z + 676881*a*c*d^4*e^4 + 194481*c^2*
d^8 + 1048576*a^2*e^8, z, k)*(root(805306368*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a
^11*c^3*d^12*z^4 + 268435456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 43057152*a^7*c*d^4*e^6*z^2 + 11599872*a
^6*c^2*d^8*e^2*z^2 + 100663296*a^8*e^10*z^2 + 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5
*e^9*z + 676881*a*c*d^4*e^4 + 194481*c^2*d^8 + 1048576*a^2*e^8, z, k)*(root(805306368*a^12*c^2*d^8*e^4*z^4 + 8
05306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c^3*d^12*z^4 + 268435456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3
+ 43057152*a^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 100663296*a^8*e^10*z^2 + 9652224*a^4*c*d^4*e^5*z
+ 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z + 676881*a*c*d^4*e^4 + 194481*c^2*d^8 + 1048576*a^2*e^8, z, k)
*(root(805306368*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c^3*d^12*z^4 + 268435456
*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 43057152*a^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 1006632
96*a^8*e^10*z^2 + 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z + 676881*a*c*d^4*e^4
+ 194481*c^2*d^8 + 1048576*a^2*e^8, z, k)*((402653184*a^15*c^4*d*e^22 - 134217728*a^10*c^9*d^21*e^2 - 13421772
8*a^11*c^8*d^17*e^6 + 805306368*a^12*c^7*d^13*e^10 + 1879048192*a^13*c^6*d^9*e^14 + 1476395008*a^14*c^5*d^5*e^
18)/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8)) + (x*(3
35544320*a^15*c^4*e^23 - 201326592*a^10*c^9*d^20*e^3 - 469762048*a^11*c^8*d^16*e^7 + 134217728*a^12*c^7*d^12*e
^11 + 1207959552*a^13*c^6*d^8*e^15 + 1140850688*a^14*c^5*d^4*e^19))/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^1
1*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8))) + (211288064*a^12*c^4*d*e^21 - 11010048*a^7*c^9*d^21
*e + 20447232*a^8*c^8*d^17*e^5 + 204472320*a^9*c^7*d^13*e^9 + 514850816*a^10*c^6*d^9*e^13 + 553123840*a^11*c^5
*d^5*e^17)/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8))
+ (x*(251658240*a^12*c^4*e^22 - 28311552*a^7*c^9*d^20*e^2 - 67108864*a^8*c^8*d^16*e^6 + 18874368*a^9*c^7*d^12*
e^10 + 377487360*a^10*c^6*d^8*e^14 + 571473920*a^11*c^5*d^4*e^18))/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11
*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8))) + (36962304*a^9*c^4*d*e^20 + 11010048*a^5*c^8*d^17*e^
4 + 57999360*a^6*c^7*d^13*e^8 + 138805248*a^7*c^6*d^9*e^12 + 141361152*a^8*c^5*d^5*e^16)/(1048576*(a^12*e^16 +
a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8)) + (x*(62914560*a^9*c^4*e^21 - 18
06336*a^4*c^9*d^20*e + 2670592*a^5*c^8*d^16*e^5 + 43032576*a^6*c^7*d^12*e^9 + 143179776*a^7*c^6*d^8*e^13 + 171
732992*a^8*c^5*d^4*e^17))/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10
*c^2*d^8*e^8))) + (4030464*a^6*c^4*d*e^19 + 576576*a^2*c^8*d^17*e^3 + 5061824*a^3*c^7*d^13*e^7 + 15959232*a^4*
c^6*d^9*e^11 + 17863744*a^5*c^5*d^5*e^15)/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d
^12*e^4 + 6*a^10*c^2*d^8*e^8)) + (x*(5242880*a^6*c^4*e^20 + 755136*a^2*c^8*d^16*e^4 + 6023488*a^3*c^7*d^12*e^8
+ 19579200*a^4*c^6*d^8*e^12 + 22240704*a^5*c^5*d^4*e^16))/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e
^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8))) + (x*(194481*c^7*d^12*e^7 + 871362*a*c^6*d^8*e^11 + 970321*a^
2*c^5*d^4*e^15))/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*
e^8)))*root(805306368*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c^3*d^12*z^4 + 2684
35456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 43057152*a^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 10
0663296*a^8*e^10*z^2 + 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z + 676881*a*c*d^4
*e^4 + 194481*c^2*d^8 + 1048576*a^2*e^8, z, k), k, 1, 4) + ((3*a*e^7 + c*d^4*e^3)/(8*(a^2*e^8 + c^2*d^8 + 2*a*
c*d^4*e^4)) + (x^5*(7*c^3*d^7 + 15*a*c^2*d^3*e^4))/(32*a^2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (x^2*(5*c^2*
d^6*e + 9*a*c*d^2*e^5))/(16*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (c*e^7*x^4)/(4*(a^2*e^8 + c^2*d^8 + 2*a*c
*d^4*e^4)) + (x*(11*c^2*d^7 + 19*a*c*d^3*e^4))/(32*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (x^6*(3*c^3*d^6*e
+ 7*a*c^2*d^2*e^5))/(16*a^2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (e^2*x^3*(9*c^2*d^5 + 17*a*c*d*e^4))/(32*a*
(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (e^2*x^7*(5*c^3*d^5 + 13*a*c^2*d*e^4))/(32*a^2*(a^2*e^8 + c^2*d^8 + 2*a
*c*d^4*e^4)))/(a^2 + c^2*x^8 + 2*a*c*x^4) + (e^11*log(d + e*x))/(a^3*e^12 + c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2
*c*d^4*e^8)