3.269 \(\int \frac{\left (d+e x^2\right )^2}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=238 \[ \frac{\left (\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}+e (2 c d-b e)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (e (2 c d-b e)-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x}{c} \]

[Out]

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

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Rubi [A]  time = 1.23784, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}+e (2 c d-b e)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (e (2 c d-b e)-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{2} \int \frac{1}{c}\, dx + \frac{\sqrt{2} \left (- b e \left (b e - 2 c d\right ) + 2 c \left (a e^{2} - c d^{2}\right ) - e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt{2} \left (- b e \left (b e - 2 c d\right ) + 2 c \left (a e^{2} - c d^{2}\right ) + e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)**2/(c*x**4+b*x**2+a),x)

[Out]

e**2*Integral(1/c, x) + sqrt(2)*(-b*e*(b*e - 2*c*d) + 2*c*(a*e**2 - c*d**2) - e*
sqrt(-4*a*c + b**2)*(b*e - 2*c*d))*atan(sqrt(2)*sqrt(c)*x/sqrt(b + sqrt(-4*a*c +
b**2)))/(2*c**(3/2)*sqrt(b + sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2)) - sqrt(2
)*(-b*e*(b*e - 2*c*d) + 2*c*(a*e**2 - c*d**2) + e*sqrt(-4*a*c + b**2)*(b*e - 2*c
*d))*atan(sqrt(2)*sqrt(c)*x/sqrt(b - sqrt(-4*a*c + b**2)))/(2*c**(3/2)*sqrt(b -
sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.632238, size = 269, normalized size = 1.13 \[ \frac{\frac{\sqrt{2} \left (-2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+2 \sqrt{c} e^2 x}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*e^2*x + (Sqrt[2]*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(
b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(2*c^2*d^2
+ b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcT
an[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b +
Sqrt[b^2 - 4*a*c]]))/(2*c^(3/2))

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Maple [B]  time = 0.034, size = 695, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^2*x/c-1/2/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-
4*a*c+b^2)^(1/2))*c)^(1/2))*b*e^2+2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arcta
n(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d*e+1/(-4*a*c+b^2)^(1/2)*2^(1/2)
/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^
(1/2))*a*e^2-1/2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a
rctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*e^2+1/(-4*a*c+b^2)^(1/2)
*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1
/2))*c)^(1/2))*b*d*e-c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/
2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d^2+1/2/c*2^(1/2)/((-b+(
-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2
))*b*e^2-2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*
a*c+b^2)^(1/2))*c)^(1/2))*d*e+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/
2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*e^2-1/2/c/
(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)
/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*e^2+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
)*b*d*e-c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c
*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{e^{2} x}{c} - \frac{-\int \frac{c d^{2} - a e^{2} +{\left (2 \, c d e - b e^{2}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^2/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92252, size = 6332, normalized size = 26.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*e^2*x - sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2
- 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*b^2*c^3 - 4*a^2*c
^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5
- 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8
*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2
*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14
*a*b*c^3*d^5*e^3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^2*c^3)*d^4
*e^4 + 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2
*e^6 + 2*(a^2*b^3 + a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*e^8)*x + sqrt(1/2)*((b^2*
c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*
c^3)*d^3*e^3 - (a*b^4*c - 11*a^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c -
4*a^3*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*e^6 - ((a*b^3*c^4 - 4*a
^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4 - 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^
2)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 -
2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*
(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*
c^6 - 4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b
^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*b^2*c^3 - 4*a^2*c^4)*sqrt
((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b
^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^
3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4
*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))) + sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d
^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e
^4 + (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^
3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2
- 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c
+ a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))*log(2*(c^5
*d^8 - 2*b*c^4*d^7*e + 14*a*b*c^3*d^5*e^3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a
*b^2*c^2 + 2*a^2*c^3)*d^4*e^4 + 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a*b^4 + 9*a
^2*b^2*c + 4*a^3*c^2)*d^2*e^6 + 2*(a^2*b^3 + a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*
e^8)*x - sqrt(1/2)*((b^2*c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4*e^2
+ 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - (a*b^4*c - 11*a^2*b^2*c^2 + 28*a^3*c^3)*
d^2*e^4 - 4*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2
)*e^6 - ((a*b^3*c^4 - 4*a^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4 - 4*a^3*c^5)*d*e + (a^2*
b^3*c^3 - 4*a^3*b*c^4)*e^2)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3
- 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2
- 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c
+ a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6
*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 + (a*
b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a
^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3
*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c
^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))) - sqrt(1/2)*c*sqr
t(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^
3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d
^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d
^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7
+ (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3
- 4*a^2*c^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14*a*b*c^3*d^5*e^3 + (b^2*c^3 -
4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^2*c^3)*d^4*e^4 + 6*(a*b^3*c + 3*a^2*b*c^
2)*d^3*e^5 - (a*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2*e^6 + 2*(a^2*b^3 + a^3*b*c)*d
*e^7 - (a^3*b^2 - a^4*c)*e^8)*x + sqrt(1/2)*((b^2*c^4 - 4*a*c^5)*d^6 - 7*(a*b^2*
c^3 - 4*a^2*c^4)*d^4*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - (a*b^4*c - 11*a
^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^5 + (a^2*b^4
- 5*a^3*b^2*c + 4*a^4*c^2)*e^6 + ((a*b^3*c^4 - 4*a^2*b*c^5)*d^2 - 4*(a^2*b^2*c^4
- 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^2)*sqrt((c^6*d^8 - 12*a*c^5*d^
6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^
4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7
+ (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))*sqrt(-(b*c^
3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b
^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6*d^8 - 12*a*c^5*d^6*e^2 +
8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 +
4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*
b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2
*c^4))) + sqrt(1/2)*c*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(
a*b^2*c - 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*s
qrt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(
a*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2
*b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6
- 4*a^3*c^7)))/(a*b^2*c^3 - 4*a^2*c^4))*log(2*(c^5*d^8 - 2*b*c^4*d^7*e + 14*a*b*
c^3*d^5*e^3 + (b^2*c^3 - 4*a*c^4)*d^6*e^2 - 5*(3*a*b^2*c^2 + 2*a^2*c^3)*d^4*e^4
+ 6*(a*b^3*c + 3*a^2*b*c^2)*d^3*e^5 - (a*b^4 + 9*a^2*b^2*c + 4*a^3*c^2)*d^2*e^6
+ 2*(a^2*b^3 + a^3*b*c)*d*e^7 - (a^3*b^2 - a^4*c)*e^8)*x - sqrt(1/2)*((b^2*c^4 -
4*a*c^5)*d^6 - 7*(a*b^2*c^3 - 4*a^2*c^4)*d^4*e^2 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*
d^3*e^3 - (a*b^4*c - 11*a^2*b^2*c^2 + 28*a^3*c^3)*d^2*e^4 - 4*(a^2*b^3*c - 4*a^3
*b*c^2)*d*e^5 + (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*e^6 + ((a*b^3*c^4 - 4*a^2*b*
c^5)*d^2 - 4*(a^2*b^2*c^4 - 4*a^3*c^5)*d*e + (a^2*b^3*c^3 - 4*a^3*b*c^4)*e^2)*sq
rt((c^6*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a
*b^2*c^3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*
b^3*c - a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 -
4*a^3*c^7)))*sqrt(-(b*c^3*d^4 - 8*a*c^3*d^3*e + 6*a*b*c^2*d^2*e^2 - 4*(a*b^2*c
- 2*a^2*c^2)*d*e^3 + (a*b^3 - 3*a^2*b*c)*e^4 - (a*b^2*c^3 - 4*a^2*c^4)*sqrt((c^6
*d^8 - 12*a*c^5*d^6*e^2 + 8*a*b*c^4*d^5*e^3 - 48*a^2*b*c^3*d^3*e^5 - 2*(a*b^2*c^
3 - 19*a^2*c^4)*d^4*e^4 + 4*(7*a^2*b^2*c^2 - 3*a^3*c^3)*d^2*e^6 - 8*(a^2*b^3*c -
a^3*b*c^2)*d*e^7 + (a^2*b^4 - 2*a^3*b^2*c + a^4*c^2)*e^8)/(a^2*b^2*c^6 - 4*a^3*
c^7)))/(a*b^2*c^3 - 4*a^2*c^4))))/c

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Sympy [A]  time = 123.774, size = 920, normalized size = 3.87 \[ \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c^{5} - 128 a^{2} b^{2} c^{4} + 16 a b^{4} c^{3}\right ) + t^{2} \left (48 a^{3} b c^{2} e^{4} - 128 a^{3} c^{3} d e^{3} - 28 a^{2} b^{3} c e^{4} + 96 a^{2} b^{2} c^{2} d e^{3} - 96 a^{2} b c^{3} d^{2} e^{2} + 128 a^{2} c^{4} d^{3} e + 4 a b^{5} e^{4} - 16 a b^{4} c d e^{3} + 24 a b^{3} c^{2} d^{2} e^{2} - 32 a b^{2} c^{3} d^{3} e - 16 a b c^{4} d^{4} + 4 b^{3} c^{3} d^{4}\right ) + a^{4} e^{8} - 4 a^{3} b d e^{7} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} b^{2} d^{2} e^{6} - 12 a^{2} b c d^{3} e^{5} + 6 a^{2} c^{2} d^{4} e^{4} - 4 a b^{3} d^{3} e^{5} + 12 a b^{2} c d^{4} e^{4} - 12 a b c^{2} d^{5} e^{3} + 4 a c^{3} d^{6} e^{2} + b^{4} d^{4} e^{4} - 4 b^{3} c d^{5} e^{3} + 6 b^{2} c^{2} d^{6} e^{2} - 4 b c^{3} d^{7} e + c^{4} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{32 t^{3} a^{3} b c^{4} e^{2} - 128 t^{3} a^{3} c^{5} d e - 8 t^{3} a^{2} b^{3} c^{3} e^{2} + 32 t^{3} a^{2} b^{2} c^{4} d e + 32 t^{3} a^{2} b c^{5} d^{2} - 8 t^{3} a b^{3} c^{4} d^{2} - 4 t a^{4} c^{2} e^{6} + 8 t a^{3} b^{2} c e^{6} - 36 t a^{3} b c^{2} d e^{5} + 60 t a^{3} c^{3} d^{2} e^{4} - 2 t a^{2} b^{4} e^{6} + 12 t a^{2} b^{3} c d e^{5} - 30 t a^{2} b^{2} c^{2} d^{2} e^{4} + 40 t a^{2} b c^{3} d^{3} e^{3} - 60 t a^{2} c^{4} d^{4} e^{2} + 12 t a b c^{4} d^{5} e + 4 t a c^{5} d^{6} - 2 t b^{2} c^{4} d^{6}}{a^{4} c e^{8} - a^{3} b^{2} e^{8} + 2 a^{3} b c d e^{7} - 4 a^{3} c^{2} d^{2} e^{6} + 2 a^{2} b^{3} d e^{7} - 9 a^{2} b^{2} c d^{2} e^{6} + 18 a^{2} b c^{2} d^{3} e^{5} - 10 a^{2} c^{3} d^{4} e^{4} - a b^{4} d^{2} e^{6} + 6 a b^{3} c d^{3} e^{5} - 15 a b^{2} c^{2} d^{4} e^{4} + 14 a b c^{3} d^{5} e^{3} - 4 a c^{4} d^{6} e^{2} + b^{2} c^{3} d^{6} e^{2} - 2 b c^{4} d^{7} e + c^{5} d^{8}} \right )} \right )\right )} + \frac{e^{2} x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)**2/(c*x**4+b*x**2+a),x)

[Out]

RootSum(_t**4*(256*a**3*c**5 - 128*a**2*b**2*c**4 + 16*a*b**4*c**3) + _t**2*(48*
a**3*b*c**2*e**4 - 128*a**3*c**3*d*e**3 - 28*a**2*b**3*c*e**4 + 96*a**2*b**2*c**
2*d*e**3 - 96*a**2*b*c**3*d**2*e**2 + 128*a**2*c**4*d**3*e + 4*a*b**5*e**4 - 16*
a*b**4*c*d*e**3 + 24*a*b**3*c**2*d**2*e**2 - 32*a*b**2*c**3*d**3*e - 16*a*b*c**4
*d**4 + 4*b**3*c**3*d**4) + a**4*e**8 - 4*a**3*b*d*e**7 + 4*a**3*c*d**2*e**6 + 6
*a**2*b**2*d**2*e**6 - 12*a**2*b*c*d**3*e**5 + 6*a**2*c**2*d**4*e**4 - 4*a*b**3*
d**3*e**5 + 12*a*b**2*c*d**4*e**4 - 12*a*b*c**2*d**5*e**3 + 4*a*c**3*d**6*e**2 +
b**4*d**4*e**4 - 4*b**3*c*d**5*e**3 + 6*b**2*c**2*d**6*e**2 - 4*b*c**3*d**7*e +
c**4*d**8, Lambda(_t, _t*log(x + (32*_t**3*a**3*b*c**4*e**2 - 128*_t**3*a**3*c*
*5*d*e - 8*_t**3*a**2*b**3*c**3*e**2 + 32*_t**3*a**2*b**2*c**4*d*e + 32*_t**3*a*
*2*b*c**5*d**2 - 8*_t**3*a*b**3*c**4*d**2 - 4*_t*a**4*c**2*e**6 + 8*_t*a**3*b**2
*c*e**6 - 36*_t*a**3*b*c**2*d*e**5 + 60*_t*a**3*c**3*d**2*e**4 - 2*_t*a**2*b**4*
e**6 + 12*_t*a**2*b**3*c*d*e**5 - 30*_t*a**2*b**2*c**2*d**2*e**4 + 40*_t*a**2*b*
c**3*d**3*e**3 - 60*_t*a**2*c**4*d**4*e**2 + 12*_t*a*b*c**4*d**5*e + 4*_t*a*c**5
*d**6 - 2*_t*b**2*c**4*d**6)/(a**4*c*e**8 - a**3*b**2*e**8 + 2*a**3*b*c*d*e**7 -
4*a**3*c**2*d**2*e**6 + 2*a**2*b**3*d*e**7 - 9*a**2*b**2*c*d**2*e**6 + 18*a**2*
b*c**2*d**3*e**5 - 10*a**2*c**3*d**4*e**4 - a*b**4*d**2*e**6 + 6*a*b**3*c*d**3*e
**5 - 15*a*b**2*c**2*d**4*e**4 + 14*a*b*c**3*d**5*e**3 - 4*a*c**4*d**6*e**2 + b*
*2*c**3*d**6*e**2 - 2*b*c**4*d**7*e + c**5*d**8)))) + e**2*x/c

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GIAC/XCAS [A]  time = 1.24386, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done