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*(-a*e^2+6*c*d^2)*x/c^2+4/3*d*e^3*x^3/c+1/5*e^4*x^5/c-1/8*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2)
)*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4-4*d*e*(-a*e^2+c*d^2)*a^(1/2)*c^(1/2))/a^(3/4)/c^(9/4)*2^(1/2)+1/8*ln(a^(1/4)*
c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4-4*d*e*(-a*e^2+c*d^2)*a^(1/2)*c^(1/2))/a^
(3/4)/c^(9/4)*2^(1/2)+1/4*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4+4*d*e*(-a*e^2+c*
d^2)*a^(1/2)*c^(1/2))/a^(3/4)/c^(9/4)*2^(1/2)+1/4*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^4-6*a*c*d^2*e^2+a
^2*e^4+4*d*e*(-a*e^2+c*d^2)*a^(1/2)*c^(1/2))/a^(3/4)/c^(9/4)*2^(1/2)

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Rubi [A]  time = 0.45, antiderivative size = 437, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.34, size = 444, normalized size = 1.02 \[ \frac {160 a^{3/4} c^{5/4} d e^3 x^3+24 a^{3/4} c^{5/4} e^4 x^5-120 a^{3/4} \sqrt [4]{c} e^2 x \left (a e^2-6 c d^2\right )-15 \sqrt {2} \left (4 a^{3/2} \sqrt {c} d e^3+a^2 e^4-4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+15 \sqrt {2} \left (4 a^{3/2} \sqrt {c} d e^3+a^2 e^4-4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )-30 \sqrt {2} \left (-4 a^{3/2} \sqrt {c} d e^3+a^2 e^4+4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+30 \sqrt {2} \left (-4 a^{3/2} \sqrt {c} d e^3+a^2 e^4+4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{120 a^{3/4} c^{9/4}} \]

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))

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fricas [B]  time = 4.26, size = 2878, normalized size = 6.59 \[ \text {result too large to display} \]

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

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giac [A]  time = 0.19, size = 498, normalized size = 1.14 \[ \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{4}} + \frac {3 \, c^{4} x^{5} e^{4} + 20 \, c^{4} d x^{3} e^{3} + 90 \, c^{4} d^{2} x e^{2} - 15 \, a c^{3} x e^{4}}{15 \, c^{5}} \]

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

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maple [B]  time = 0.01, size = 741, normalized size = 1.70 \[ \frac {e^{4} x^{5}}{5 c}+\frac {4 d \,e^{3} x^{3}}{3 c}-\frac {\sqrt {2}\, a d \,e^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{\left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, a d \,e^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{\left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, a d \,e^{3} \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{2 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {a \,e^{4} x}{c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, a \,e^{4} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, a \,e^{4} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, a \,e^{4} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{4} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{4} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{4} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 a}+\frac {\sqrt {2}\, d^{3} e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{\left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, d^{3} e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{\left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, d^{3} e \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{2 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {6 d^{2} e^{2} x}{c}-\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{2 c}-\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{2 c}-\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} e^{2} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{4 c} \]

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+1/8/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+1/4/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^4-1/2/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

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maxima [A]  time = 2.45, size = 432, normalized size = 0.99 \[ \frac {3 \, c e^{4} x^{5} + 20 \, c d e^{3} x^{3} + 15 \, {\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x}{15 \, c^{2}} + \frac {\frac {2 \, \sqrt {2} {\left (c^{\frac {5}{2}} d^{4} + 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} - 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (c^{\frac {5}{2}} d^{4} + 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} - 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (c^{\frac {5}{2}} d^{4} - 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} + 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (c^{\frac {5}{2}} d^{4} - 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} + 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c^{2}} \]

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]

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

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mupad [B]  time = 5.08, size = 4022, normalized size = 9.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^4/(a + c*x^4),x)

[Out]

atan((((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c - (4*(4*a*c^6*d^
4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*
e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d
^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*((a^4*e^8*(-a^3*c^9)^
(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5
- 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2)
)/(16*a^3*c^9))^(1/2)*1i + ((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4
))/c + (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(
1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^
9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*
((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3
- 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4
*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2)*1i)/(((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6
+ 70*a^2*c^2*d^4*e^4))/c - (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*((a^4*e^8*(-a^3*c^9)^(1/2) +
c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a
*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a
^3*c^9))^(1/2))/c^3)*((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7
+ 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(
1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2) - ((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2
- 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c + (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*((a^4*e^8*
(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*
c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3
*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e
+ 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^
2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2) + (8*(a^5*d*e^11 - c^5*d^11*
e - 3*a*c^4*d^9*e^3 + 3*a^4*c*d^3*e^9 - 2*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7))/c^3))*((a^4*e^8*(-a^3*c^9)^(1/
2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 -
28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(
16*a^3*c^9))^(1/2)*2i - x*((a*e^4)/c^2 - (6*d^2*e^2)/c) + atan((((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 -
28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c - (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*(-(a^4*e^8*(
-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d^7*e - 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c
^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*
c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*(-(a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d^7*e
- 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^
2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2)*1i + ((4*x*(a^4*e^8 + c^4*d^
8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c + (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^
5*d^2*e^2)*(-(a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d^7*e - 8*a^5*c^5*d*e^7 - 56*a^3
*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70
*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*(-(a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^
(1/2) + 8*a^2*c^8*d^7*e - 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c
^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2)*1i)/(
((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c - (4*(4*a*c^6*d^4 + 4*
a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*(-(a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d^7*e - 8
*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^
6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*(-(a^4*e^8*(-a^3*c^9)^(1/2
) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d^7*e - 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 2
8*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(1
6*a^3*c^9))^(1/2) - ((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c +
(4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2)*(-(a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) +
8*a^2*c^8*d^7*e - 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/
2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*(-(a^4
*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d^7*e - 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56
*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*
(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2) + (8*(a^5*d*e^11 - c^5*d^11*e - 3*a*c^4*d^9*e^3 + 3*a^4*c*d^3*e^9 - 2*a^
2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7))/c^3))*(-(a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) + 8*a^2*c^8*d
^7*e - 8*a^5*c^5*d*e^7 - 56*a^3*c^7*d^5*e^3 + 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*
c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2)*2i + (e^4*x^5)/(5*c) + (
4*d*e^3*x^3)/(3*c)

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sympy [A]  time = 3.75, size = 500, normalized size = 1.14 \[ x \left (- \frac {a e^{4}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \operatorname {RootSum} {\left (256 t^{4} a^{3} c^{9} + t^{2} \left (- 256 a^{5} c^{5} d e^{7} + 1792 a^{4} c^{6} d^{3} e^{5} - 1792 a^{3} c^{7} d^{5} e^{3} + 256 a^{2} c^{8} d^{7} e\right ) + a^{8} e^{16} + 8 a^{7} c d^{2} e^{14} + 28 a^{6} c^{2} d^{4} e^{12} + 56 a^{5} c^{3} d^{6} e^{10} + 70 a^{4} c^{4} d^{8} e^{8} + 56 a^{3} c^{5} d^{10} e^{6} + 28 a^{2} c^{6} d^{12} e^{4} + 8 a c^{7} d^{14} e^{2} + c^{8} d^{16}, \left (t \mapsto t \log {\left (x + \frac {256 t^{3} a^{4} c^{7} d e^{3} - 256 t^{3} a^{3} c^{8} d^{3} e + 4 t a^{7} c^{2} e^{12} - 264 t a^{6} c^{3} d^{2} e^{10} + 1980 t a^{5} c^{4} d^{4} e^{8} - 3696 t a^{4} c^{5} d^{6} e^{6} + 1980 t a^{3} c^{6} d^{8} e^{4} - 264 t a^{2} c^{7} d^{10} e^{2} + 4 t a c^{8} d^{12}}{a^{8} e^{16} - 24 a^{7} c d^{2} e^{14} - 36 a^{6} c^{2} d^{4} e^{12} + 88 a^{5} c^{3} d^{6} e^{10} + 198 a^{4} c^{4} d^{8} e^{8} + 88 a^{3} c^{5} d^{10} e^{6} - 36 a^{2} c^{6} d^{12} e^{4} - 24 a c^{7} d^{14} e^{2} + c^{8} d^{16}} \right )} \right )\right )} + \frac {4 d e^{3} x^{3}}{3 c} + \frac {e^{4} x^{5}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-a*e**4/c**2 + 6*d**2*e**2/c) + 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 - 2
4*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)

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