3.4 \(\int \frac{d+e x^4}{a-c x^8} \, dx\)

Optimal. Leaf size=329 \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}} \]

[Out]

((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[
1 - (Sqrt[2]*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 + (Sqrt[2]
*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTanh[(c^(1/8)*x)/a^(1/8)])/(4*
a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[
2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sq
rt[2]*a^(7/8)*c^(1/8))

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Rubi [A]  time = 0.209223, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {1417, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^4)/(a - c*x^8),x]

[Out]

((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[
1 - (Sqrt[2]*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 + (Sqrt[2]
*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTanh[(c^(1/8)*x)/a^(1/8)])/(4*
a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[
2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sq
rt[2]*a^(7/8)*c^(1/8))

Rule 1417

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-(a/c), 2]}, Dist[(d + e*q)/
2, Int[1/(a + c*q*x^n), x], x] + Dist[(d - e*q)/2, Int[1/(a - c*q*x^n), x], x]] /; FreeQ[{a, c, d, e, n}, x] &
& EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 208

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

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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 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])

Rubi steps

\begin{align*} \int \frac{d+e x^4}{a-c x^8} \, dx &=\frac{1}{2} \left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{a+\sqrt{a} \sqrt{c} x^4} \, dx+\frac{1}{2} \left (d+\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{a-\sqrt{a} \sqrt{c} x^4} \, dx\\ &=\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{\sqrt [4]{a}-\sqrt [4]{c} x^2}{a+\sqrt{a} \sqrt{c} x^4} \, dx}{4 \sqrt [4]{a}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{\sqrt [4]{a}+\sqrt [4]{c} x^2}{a+\sqrt{a} \sqrt{c} x^4} \, dx}{4 \sqrt [4]{a}}+\frac{\left (d+\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{\sqrt [4]{a}-\sqrt [4]{c} x^2} \, dx}{4 a^{3/4}}+\frac{\left (d+\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{\sqrt [4]{a}+\sqrt [4]{c} x^2} \, dx}{4 a^{3/4}}\\ &=\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 a^{3/4} \sqrt [4]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 a^{3/4} \sqrt [4]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{\frac{\sqrt{2} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{-\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt [8]{a} x}{\sqrt [8]{c}}-x^2} \, dx}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \int \frac{\frac{\sqrt{2} \sqrt [8]{a}}{\sqrt [8]{c}}-2 x}{-\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2} \sqrt [8]{a} x}{\sqrt [8]{c}}-x^2} \, dx}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}\\ &=\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [4]{a}-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [4]{a}+\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}\\ &=\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [4]{a}-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [4]{a}+\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}\\ \end{align*}

Mathematica [A]  time = 0.136066, size = 425, normalized size = 1.29 \[ \frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a c^{5/8}}-\frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a c^{5/8}}-\frac{\left (a^{5/8} e+\sqrt [8]{a} \sqrt{c} d\right ) \log \left (\sqrt [8]{a}-\sqrt [8]{c} x\right )}{8 a c^{5/8}}-\frac{\left (-a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \log \left (\sqrt [8]{a}+\sqrt [8]{c} x\right )}{8 a c^{5/8}}+\frac{\left (a^{5/8} e+\sqrt [8]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a c^{5/8}}-\frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [8]{c} x-\sqrt{2} \sqrt [8]{a}}{\sqrt{2} \sqrt [8]{a}}\right )}{4 \sqrt{2} a c^{5/8}}-\frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2} \sqrt [8]{a}}\right )}{4 \sqrt{2} a c^{5/8}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^4)/(a - c*x^8),x]

[Out]

((a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)
*e)*ArcTan[(-(Sqrt[2]*a^(1/8)) + 2*c^(1/8)*x)/(Sqrt[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*
d) + a^(5/8)*e)*ArcTan[(Sqrt[2]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - ((a^(1/8)*S
qrt[c]*d + a^(5/8)*e)*Log[a^(1/8) - c^(1/8)*x])/(8*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) - a^(5/8)*e)*Log[a^(1/8
) + c^(1/8)*x])/(8*a*c^(5/8)) + ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x +
c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8
)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8))

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Maple [C]  time = 0.015, size = 39, normalized size = 0.1 \begin{align*}{\frac{1}{8\,c}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}-a \right ) }{\frac{ \left ( -{{\it \_R}}^{4}e-d \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^4+d)/(-c*x^8+a),x)

[Out]

1/8/c*sum((-_R^4*e-d)/_R^7*ln(x-_R),_R=RootOf(_Z^8*c-a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{e x^{4} + d}{c x^{8} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(-c*x^8+a),x, algorithm="maxima")

[Out]

-integrate((e*x^4 + d)/(c*x^8 - a), x)

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Fricas [B]  time = 3.30455, size = 6946, normalized size = 21.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(-c*x^8+a),x, algorithm="fricas")

[Out]

1/2*((a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) +
4*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))^(1/4)*arctan(((3*a^3*c^5*d^6*e + 19*a^4*c^4*d^4*e^3 + 9*a^5*c^3*d^2*e^5 + a
^6*c^2*e^7 - (a^6*c^6*d^3 + 3*a^7*c^5*d*e^2)*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*
d^2*e^6 + a^4*e^8)/(a^7*c^5)))*sqrt(((c^4*d^8 + 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e
^8)*x^2 - (2*a^6*c^4*d*e*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(
a^7*c^5)) - a^2*c^4*d^6 - 7*a^3*c^3*d^4*e^2 - 7*a^4*c^2*d^2*e^4 - a^5*c*e^6)*sqrt((a^3*c^2*sqrt((c^4*d^8 + 12*
a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e + 4*a*d*e^3)/(a^3*c^2)
))/(c^4*d^8 + 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8))*sqrt((a^3*c^2*sqrt((c^4*d^8 +
12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e + 4*a*d*e^3)/(a^3*
c^2)) + ((a^6*c^6*d^3 + 3*a^7*c^5*d*e^2)*x*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^
2*e^6 + a^4*e^8)/(a^7*c^5)) - (3*a^3*c^5*d^6*e + 19*a^4*c^4*d^4*e^3 + 9*a^5*c^3*d^2*e^5 + a^6*c^2*e^7)*x)*sqrt
((a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c
*d^3*e + 4*a*d*e^3)/(a^3*c^2)))*((a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2
*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))^(1/4)/(c^5*d^10 + 3*a*c^4*d^8*e^2 - 14*a^2*c^3*
d^6*e^4 + 14*a^3*c^2*d^4*e^6 - 3*a^4*c*d^2*e^8 - a^5*e^10)) - 1/2*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2
+ 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(1/4)*arctan
(-((3*a^3*c^5*d^6*e + 19*a^4*c^4*d^4*e^3 + 9*a^5*c^3*d^2*e^5 + a^6*c^2*e^7 + (a^6*c^6*d^3 + 3*a^7*c^5*d*e^2)*s
qrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)))*sqrt(((c^4*d^8
+ 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*x^2 + (2*a^6*c^4*d*e*sqrt((c^4*d^8 + 12*a*
c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + a^2*c^4*d^6 + 7*a^3*c^3*d^4*e^2 +
7*a^4*c^2*d^2*e^4 + a^5*c*e^6)*sqrt(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c
*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2)))/(c^4*d^8 + 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^
4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8))*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*
c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(3/4) - ((a^6*c^6*d^3 + 3*a^7*c^5*d*e^2)*x
*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + (3*a^3*c^5*d
^6*e + 19*a^4*c^4*d^4*e^3 + 9*a^5*c^3*d^2*e^5 + a^6*c^2*e^7)*x)*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 +
38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(3/4))/(c^5*d^
10 + 3*a*c^4*d^8*e^2 - 14*a^2*c^3*d^6*e^4 + 14*a^3*c^2*d^4*e^6 - 3*a^4*c*d^2*e^8 - a^5*e^10)) + 1/8*((a^3*c^2*
sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e + 4
*a*d*e^3)/(a^3*c^2))^(1/4)*log(-(c^3*d^6 + 5*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 - a^3*e^6)*x + (a^5*c^3*e*sqrt((c
^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - a*c^3*d^5 - 6*a^2*c^
2*d^3*e^2 - a^3*c*d*e^4)*((a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 +
a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))^(1/4)) - 1/8*((a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^
2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))^(1/4)*log(
-(c^3*d^6 + 5*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 - a^3*e^6)*x - (a^5*c^3*e*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*
a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - a*c^3*d^5 - 6*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*((a^3*
c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e
+ 4*a*d*e^3)/(a^3*c^2))^(1/4)) - 1/8*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^
3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(1/4)*log(-(c^3*d^6 + 5*a*c^2*d^4*e^2 -
5*a^2*c*d^2*e^4 - a^3*e^6)*x + (a^5*c^3*e*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2
*e^6 + a^4*e^8)/(a^7*c^5)) + a*c^3*d^5 + 6*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*
d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(1/4
)) + 1/8*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c
^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(1/4)*log(-(c^3*d^6 + 5*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 - a^3*e^6)*x
- (a^5*c^3*e*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) +
a*c^3*d^5 + 6*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*(-(a^3*c^2*sqrt((c^4*d^8 + 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4
+ 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))^(1/4))

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Sympy [A]  time = 12.4586, size = 202, normalized size = 0.61 \begin{align*} - \operatorname{RootSum}{\left (16777216 t^{8} a^{7} c^{5} + t^{4} \left (- 32768 a^{5} c^{3} d e^{3} - 32768 a^{4} c^{4} d^{3} e\right ) - a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} - c^{4} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{- 32768 t^{5} a^{5} c^{3} e + 40 t a^{3} c d e^{4} + 80 t a^{2} c^{2} d^{3} e^{2} + 8 t a c^{3} d^{5}}{a^{3} e^{6} + 5 a^{2} c d^{2} e^{4} - 5 a c^{2} d^{4} e^{2} - c^{3} d^{6}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**4+d)/(-c*x**8+a),x)

[Out]

-RootSum(16777216*_t**8*a**7*c**5 + _t**4*(-32768*a**5*c**3*d*e**3 - 32768*a**4*c**4*d**3*e) - a**4*e**8 + 4*a
**3*c*d**2*e**6 - 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 - c**4*d**8, Lambda(_t, _t*log(x + (-32768*_t**5*
a**5*c**3*e + 40*_t*a**3*c*d*e**4 + 80*_t*a**2*c**2*d**3*e**2 + 8*_t*a*c**3*d**5)/(a**3*e**6 + 5*a**2*c*d**2*e
**4 - 5*a*c**2*d**4*e**2 - c**3*d**6))))

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Giac [B]  time = 1.22088, size = 855, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(-c*x^8+a),x, algorithm="giac")

[Out]

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