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

Optimal. Leaf size=416 \[ -\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}-\frac{\sqrt{c} d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )}+\frac{e^3 \log (d+e x)}{a e^4+c d^4} \]

[Out]

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

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Rubi [A]  time = 0.428482, antiderivative size = 416, 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} \[ -\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}-\frac{\sqrt{c} d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )}+\frac{e^3 \log (d+e x)}{a e^4+c d^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a + c*x^4)),x]

[Out]

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

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

Mathematica [A]  time = 0.153133, size = 404, normalized size = 0.97 \[ \frac{-2 a^{3/4} e^3 \log \left (a+c x^4\right )+8 a^{3/4} e^3 \log (d+e x)-\sqrt{2} c^{3/4} d^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} c^{3/4} d^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-2 \sqrt [4]{c} d \left (-2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{c} d \left (2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{2} \sqrt{a} \sqrt [4]{c} d e^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\sqrt{2} \sqrt{a} \sqrt [4]{c} d e^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{8 a^{3/4} \left (a e^4+c d^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^(1/4)*d*(Sqrt[2]*Sqrt[c]*d^2 - 2*a^(1/4)*c^(1/4)*d*e + Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)] + 2*c^(1/4)*d*(Sqrt[2]*Sqrt[c]*d^2 + 2*a^(1/4)*c^(1/4)*d*e + Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt
[2]*c^(1/4)*x)/a^(1/4)] + 8*a^(3/4)*e^3*Log[d + e*x] - Sqrt[2]*c^(3/4)*d^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/
4)*x + Sqrt[c]*x^2] + Sqrt[2]*Sqrt[a]*c^(1/4)*d*e^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + S
qrt[2]*c^(3/4)*d^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Sqrt[2]*Sqrt[a]*c^(1/4)*d*e^2*Log[
Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - 2*a^(3/4)*e^3*Log[a + c*x^4])/(8*a^(3/4)*(c*d^4 + a*e^4))

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Maple [A]  time = 0.007, size = 433, normalized size = 1. \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) }{a{e}^{4}+c{d}^{4}}}+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 8\,a{e}^{4}+8\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,a{e}^{4}+4\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,a{e}^{4}+4\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{ce{d}^{2}}{2\,a{e}^{4}+2\,c{d}^{4}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{d{e}^{2}\sqrt{2}}{8\,a{e}^{4}+8\,c{d}^{4}}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{d{e}^{2}\sqrt{2}}{4\,a{e}^{4}+4\,c{d}^{4}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{d{e}^{2}\sqrt{2}}{4\,a{e}^{4}+4\,c{d}^{4}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{{e}^{3}\ln \left ( c{x}^{4}+a \right ) }{4\,a{e}^{4}+4\,c{d}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.17897, size = 517, normalized size = 1.24 \begin{align*} \frac{\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} - 2 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e + \sqrt{2} \sqrt{a c} a c^{2} e^{2}\right )}} + \frac{\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} + 2 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e + \sqrt{2} \sqrt{a c} a c^{2} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{4} + \sqrt{2} a^{2} c^{2} e^{4}\right )}} - \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{4} + \sqrt{2} a^{2} c^{2} e^{4}\right )}} - \frac{e^{3} \log \left ({\left | c x^{4} + a \right |}\right )}{4 \,{\left (c d^{4} + a e^{4}\right )}} + \frac{e^{4} \log \left ({\left | x e + d \right |}\right )}{c d^{4} e + a e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*c^3)^(1/4)*c^2*d*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^3*d^2 - 2*(a*
c^3)^(1/4)*a*c^2*d*e + sqrt(2)*sqrt(a*c)*a*c^2*e^2) + 1/2*(a*c^3)^(1/4)*c^2*d*arctan(1/2*sqrt(2)*(2*x - sqrt(2
)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^3*d^2 + 2*(a*c^3)^(1/4)*a*c^2*d*e + sqrt(2)*sqrt(a*c)*a*c^2*e^2) + 1/
4*((a*c^3)^(1/4)*c^2*d^3 - (a*c^3)^(3/4)*d*e^2)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^3*d^
4 + sqrt(2)*a^2*c^2*e^4) - 1/4*((a*c^3)^(1/4)*c^2*d^3 - (a*c^3)^(3/4)*d*e^2)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) +
sqrt(a/c))/(sqrt(2)*a*c^3*d^4 + sqrt(2)*a^2*c^2*e^4) - 1/4*e^3*log(abs(c*x^4 + a))/(c*d^4 + a*e^4) + e^4*log(
abs(x*e + d))/(c*d^4*e + a*e^5)