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

Optimal. Leaf size=437 \[ -\frac{\left (a^2 e^4-6 a c d^2 e^2-4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{9/4}}+\frac{\left (a^2 e^4-6 a c d^2 e^2-4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{9/4}}-\frac{\left (a^2 e^4-6 a c d^2 e^2+4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{9/4}}+\frac{\left (a^2 e^4-6 a c d^2 e^2+4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{9/4}}+\frac{e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c} \]

[Out]

(e^2*(6*c*d^2 - a*e^2)*x)/c^2 + (4*d*e^3*x^3)/(3*c) + (e^4*x^5)/(5*c) - ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 +
4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - a*e^2))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(9/4)) +
((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 + 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - a*e^2))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^
(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(9/4)) - ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 - 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - 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^(9/4)) + ((c^2*d^4 - 6*a*c*d
^2*e^2 + a^2*e^4 - 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - 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^(9/4))

________________________________________________________________________________________

Rubi [A]  time = 0.453181, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {1171, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (a^2 e^4-6 a c d^2 e^2-4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{9/4}}+\frac{\left (a^2 e^4-6 a c d^2 e^2-4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{9/4}}-\frac{\left (a^2 e^4-6 a c d^2 e^2+4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{9/4}}+\frac{\left (a^2 e^4-6 a c d^2 e^2+4 \sqrt{a} \sqrt{c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{9/4}}+\frac{e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^4/(a + c*x^4),x]

[Out]

(e^2*(6*c*d^2 - a*e^2)*x)/c^2 + (4*d*e^3*x^3)/(3*c) + (e^4*x^5)/(5*c) - ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 +
4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - a*e^2))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(9/4)) +
((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 + 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - a*e^2))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^
(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(9/4)) - ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 - 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - 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^(9/4)) + ((c^2*d^4 - 6*a*c*d
^2*e^2 + a^2*e^4 - 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - 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^(9/4))

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

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{\left (d+e x^2\right )^4}{a+c x^4} \, dx &=\int \left (\frac{e^2 \left (6 c d^2-a e^2\right )}{c^2}+\frac{4 d e^3 x^2}{c}+\frac{e^4 x^4}{c}+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x^2}{c^2 \left (a+c x^4\right )}\right ) \, dx\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c}+\frac{\int \frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x^2}{a+c x^4} \, dx}{c^2}\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c}-\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 c^2}+\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 c^2}\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c}+\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c^2}+\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c^2}\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c}+\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{7/4}}\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c}-\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{\left (4 c d^3 e-4 a d e^3+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{\left (4 c d^3 e-4 a d e^3-\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4}{\sqrt{a} \sqrt{c}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{7/4}}\\ \end{align*}

Mathematica [A]  time = 0.341162, size = 444, normalized size = 1.02 \[ \frac{-15 \sqrt{2} \left (4 a^{3/2} \sqrt{c} d e^3+a^2 e^4-4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+15 \sqrt{2} \left (4 a^{3/2} \sqrt{c} d e^3+a^2 e^4-4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-30 \sqrt{2} \left (-4 a^{3/2} \sqrt{c} d e^3+a^2 e^4+4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+30 \sqrt{2} \left (-4 a^{3/2} \sqrt{c} d e^3+a^2 e^4+4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+160 a^{3/4} c^{5/4} d e^3 x^3+24 a^{3/4} c^{5/4} e^4 x^5-120 a^{3/4} \sqrt [4]{c} e^2 x \left (a e^2-6 c d^2\right )}{120 a^{3/4} c^{9/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-120*a^(3/4)*c^(1/4)*e^2*(-6*c*d^2 + a*e^2)*x + 160*a^(3/4)*c^(5/4)*d*e^3*x^3 + 24*a^(3/4)*c^(5/4)*e^4*x^5 -
30*Sqrt[2]*(c^2*d^4 + 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 - 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*ArcTan[1 -
(Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 30*Sqrt[2]*(c^2*d^4 + 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 - 4*a^(3/2)*Sqrt[
c]*d*e^3 + a^2*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 15*Sqrt[2]*(c^2*d^4 - 4*Sqrt[a]*c^(3/2)*d^3*e -
6*a*c*d^2*e^2 + 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 15
*Sqrt[2]*(c^2*d^4 - 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 + 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(120*a^(3/4)*c^(9/4))

________________________________________________________________________________________

Maple [B]  time = 0.048, size = 741, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 28.7822, size = 6198, normalized size = 14.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*c*e^4*x^5 + 80*c*d*e^3*x^3 + 15*c^2*sqrt(-(8*c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*
d*e^7 + a*c^4*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^
4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))*
log((c^8*d^16 - 24*a*c^7*d^14*e^2 - 36*a^2*c^6*d^12*e^4 + 88*a^3*c^5*d^10*e^6 + 198*a^4*c^4*d^8*e^8 + 88*a^5*c
^3*d^6*e^10 - 36*a^6*c^2*d^4*e^12 - 24*a^7*c*d^2*e^14 + a^8*e^16)*x + (a*c^8*d^12 - 34*a^2*c^7*d^10*e^2 + 239*
a^3*c^6*d^8*e^4 - 476*a^4*c^5*d^6*e^6 + 239*a^5*c^4*d^4*e^8 - 34*a^6*c^3*d^2*e^10 + a^7*c^2*e^12 + 4*(a^3*c^8*
d^3*e - a^4*c^7*d*e^3)*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 64
70*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))*
sqrt(-(8*c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 + a*c^4*sqrt(-(c^8*d^16 - 56*a*c^7*d^14
*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c
^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))) - 15*c^2*sqrt(-(8*c^3*d^7*e - 56*a*c^2*d^5*e
^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 + a*c^4*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*
a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a
^8*e^16)/(a^3*c^9)))/(a*c^4))*log((c^8*d^16 - 24*a*c^7*d^14*e^2 - 36*a^2*c^6*d^12*e^4 + 88*a^3*c^5*d^10*e^6 +
198*a^4*c^4*d^8*e^8 + 88*a^5*c^3*d^6*e^10 - 36*a^6*c^2*d^4*e^12 - 24*a^7*c*d^2*e^14 + a^8*e^16)*x - (a*c^8*d^1
2 - 34*a^2*c^7*d^10*e^2 + 239*a^3*c^6*d^8*e^4 - 476*a^4*c^5*d^6*e^6 + 239*a^5*c^4*d^4*e^8 - 34*a^6*c^3*d^2*e^1
0 + a^7*c^2*e^12 + 4*(a^3*c^8*d^3*e - a^4*c^7*d*e^3)*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^
4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2
*e^14 + a^8*e^16)/(a^3*c^9)))*sqrt(-(8*c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 + a*c^4*s
qrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 397
6*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))) + 15*c^2*sqrt(
-(8*c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 - a*c^4*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2
+ 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^
4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))*log((c^8*d^16 - 24*a*c^7*d^14*e^2 - 36*a^2*c^6*d^1
2*e^4 + 88*a^3*c^5*d^10*e^6 + 198*a^4*c^4*d^8*e^8 + 88*a^5*c^3*d^6*e^10 - 36*a^6*c^2*d^4*e^12 - 24*a^7*c*d^2*e
^14 + a^8*e^16)*x + (a*c^8*d^12 - 34*a^2*c^7*d^10*e^2 + 239*a^3*c^6*d^8*e^4 - 476*a^4*c^5*d^6*e^6 + 239*a^5*c^
4*d^4*e^8 - 34*a^6*c^3*d^2*e^10 + a^7*c^2*e^12 - 4*(a^3*c^8*d^3*e - a^4*c^7*d*e^3)*sqrt(-(c^8*d^16 - 56*a*c^7*
d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a
^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))*sqrt(-(8*c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d
^3*e^5 - 8*a^3*d*e^7 - a*c^4*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^
6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c
^9)))/(a*c^4))) - 15*c^2*sqrt(-(8*c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 - a*c^4*sqrt(-
(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5
*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))*log((c^8*d^16 - 24*a
*c^7*d^14*e^2 - 36*a^2*c^6*d^12*e^4 + 88*a^3*c^5*d^10*e^6 + 198*a^4*c^4*d^8*e^8 + 88*a^5*c^3*d^6*e^10 - 36*a^6
*c^2*d^4*e^12 - 24*a^7*c*d^2*e^14 + a^8*e^16)*x - (a*c^8*d^12 - 34*a^2*c^7*d^10*e^2 + 239*a^3*c^6*d^8*e^4 - 47
6*a^4*c^5*d^6*e^6 + 239*a^5*c^4*d^4*e^8 - 34*a^6*c^3*d^2*e^10 + a^7*c^2*e^12 - 4*(a^3*c^8*d^3*e - a^4*c^7*d*e^
3)*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 -
3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))*sqrt(-(8*c^3*d^7*e -
56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 - a*c^4*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d
^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7
*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))) + 60*(6*c*d^2*e^2 - a*e^4)*x)/c^2

________________________________________________________________________________________

Sympy [A]  time = 4.09658, size = 500, normalized size = 1.14 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} c^{9} + t^{2} \left (- 256 a^{5} c^{5} d e^{7} + 1792 a^{4} c^{6} d^{3} e^{5} - 1792 a^{3} c^{7} d^{5} e^{3} + 256 a^{2} c^{8} d^{7} e\right ) + a^{8} e^{16} + 8 a^{7} c d^{2} e^{14} + 28 a^{6} c^{2} d^{4} e^{12} + 56 a^{5} c^{3} d^{6} e^{10} + 70 a^{4} c^{4} d^{8} e^{8} + 56 a^{3} c^{5} d^{10} e^{6} + 28 a^{2} c^{6} d^{12} e^{4} + 8 a c^{7} d^{14} e^{2} + c^{8} d^{16}, \left ( t \mapsto t \log{\left (x + \frac{256 t^{3} a^{4} c^{7} d e^{3} - 256 t^{3} a^{3} c^{8} d^{3} e + 4 t a^{7} c^{2} e^{12} - 264 t a^{6} c^{3} d^{2} e^{10} + 1980 t a^{5} c^{4} d^{4} e^{8} - 3696 t a^{4} c^{5} d^{6} e^{6} + 1980 t a^{3} c^{6} d^{8} e^{4} - 264 t a^{2} c^{7} d^{10} e^{2} + 4 t a c^{8} d^{12}}{a^{8} e^{16} - 24 a^{7} c d^{2} e^{14} - 36 a^{6} c^{2} d^{4} e^{12} + 88 a^{5} c^{3} d^{6} e^{10} + 198 a^{4} c^{4} d^{8} e^{8} + 88 a^{3} c^{5} d^{10} e^{6} - 36 a^{2} c^{6} d^{12} e^{4} - 24 a c^{7} d^{14} e^{2} + c^{8} d^{16}} \right )} \right )\right )} + \frac{4 d e^{3} x^{3}}{3 c} + \frac{e^{4} x^{5}}{5 c} - \frac{x \left (a e^{4} - 6 c d^{2} e^{2}\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**3*c**9 + _t**2*(-256*a**5*c**5*d*e**7 + 1792*a**4*c**6*d**3*e**5 - 1792*a**3*c**7*d**5*e*
*3 + 256*a**2*c**8*d**7*e) + a**8*e**16 + 8*a**7*c*d**2*e**14 + 28*a**6*c**2*d**4*e**12 + 56*a**5*c**3*d**6*e*
*10 + 70*a**4*c**4*d**8*e**8 + 56*a**3*c**5*d**10*e**6 + 28*a**2*c**6*d**12*e**4 + 8*a*c**7*d**14*e**2 + c**8*
d**16, Lambda(_t, _t*log(x + (256*_t**3*a**4*c**7*d*e**3 - 256*_t**3*a**3*c**8*d**3*e + 4*_t*a**7*c**2*e**12 -
264*_t*a**6*c**3*d**2*e**10 + 1980*_t*a**5*c**4*d**4*e**8 - 3696*_t*a**4*c**5*d**6*e**6 + 1980*_t*a**3*c**6*d
**8*e**4 - 264*_t*a**2*c**7*d**10*e**2 + 4*_t*a*c**8*d**12)/(a**8*e**16 - 24*a**7*c*d**2*e**14 - 36*a**6*c**2*
d**4*e**12 + 88*a**5*c**3*d**6*e**10 + 198*a**4*c**4*d**8*e**8 + 88*a**3*c**5*d**10*e**6 - 36*a**2*c**6*d**12*
e**4 - 24*a*c**7*d**14*e**2 + c**8*d**16)))) + 4*d*e**3*x**3/(3*c) + e**4*x**5/(5*c) - x*(a*e**4 - 6*c*d**2*e*
*2)/c**2

________________________________________________________________________________________

Giac [A]  time = 1.16816, size = 672, normalized size = 1.54 \begin{align*} \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac{1}{4}} a^{2} c e^{4} - 4 \, \left (a c^{3}\right )^{\frac{3}{4}} a d e^{3}\right )} \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 )}{4 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac{1}{4}} a^{2} c e^{4} - 4 \, \left (a c^{3}\right )^{\frac{3}{4}} a d e^{3}\right )} \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 )}{4 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac{1}{4}} a^{2} c e^{4} + 4 \, \left (a c^{3}\right )^{\frac{3}{4}} a d e^{3}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{4}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac{1}{4}} a^{2} c e^{4} + 4 \, \left (a c^{3}\right )^{\frac{3}{4}} a d e^{3}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{4}} + \frac{3 \, c^{4} x^{5} e^{4} + 20 \, c^{4} d x^{3} e^{3} + 90 \, c^{4} d^{2} x e^{2} - 15 \, a c^{3} x e^{4}}{15 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*((a*c^3)^(1/4)*c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + 4*(a*c^3)^(3/4)*c*d^3*e + (a*c^3)^(1/4)*a
^2*c*e^4 - 4*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) + 1/4*
sqrt(2)*((a*c^3)^(1/4)*c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + 4*(a*c^3)^(3/4)*c*d^3*e + (a*c^3)^(1/4)*a^2*c
*e^4 - 4*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) + 1/8*sqrt
(2)*((a*c^3)^(1/4)*c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 4*(a*c^3)^(3/4)*c*d^3*e + (a*c^3)^(1/4)*a^2*c*e^4
+ 4*(a*c^3)^(3/4)*a*d*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4) - 1/8*sqrt(2)*((a*c^3)^(1/4)*
c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 4*(a*c^3)^(3/4)*c*d^3*e + (a*c^3)^(1/4)*a^2*c*e^4 + 4*(a*c^3)^(3/4)*
a*d*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4) + 1/15*(3*c^4*x^5*e^4 + 20*c^4*d*x^3*e^3 + 90*c^
4*d^2*x*e^2 - 15*a*c^3*x*e^4)/c^5