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

Optimal. Leaf size=459 \[ \frac{\left (\frac{2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt{b^2-4 a c}}+e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )-\frac{2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^3 (4 c d-b e)}{3 c^2}+\frac{e^4 x^5}{5 c} \]

[Out]

(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e)*x^3)/
(3*c^2) + (e^4*x^5)/(5*c) + ((e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d
+ a*e)) + (2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*
a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(S
qrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*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)) - (2
*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2
*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]
)

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Rubi [A]  time = 4.61514, antiderivative size = 459, 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 \[ \frac{\left (\frac{2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt{b^2-4 a c}}+e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )-\frac{2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^3 (4 c d-b e)}{3 c^2}+\frac{e^4 x^5}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e)*x^3)/
(3*c^2) + (e^4*x^5)/(5*c) + ((e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d
+ a*e)) + (2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*
a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(S
qrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*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)) - (2
*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2
*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]
)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.67091, size = 570, normalized size = 1.24 \[ \frac{e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac{\left (4 c^3 d^2 e \left (d \sqrt{b^2-4 a c}-3 a e-b d\right )+2 c^2 e^2 \left (-3 b d \left (d \sqrt{b^2-4 a c}-2 a e\right )+a e \left (a e-2 d \sqrt{b^2-4 a c}\right )+3 b^2 d^2\right )+2 b c e^3 \left (2 b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}-2 a b e-2 b^2 d\right )+b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+2 c^4 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (-4 c^3 d^2 e \left (d \sqrt{b^2-4 a c}+3 a e+b d\right )+2 c^2 e^2 \left (3 b d \left (d \sqrt{b^2-4 a c}+2 a e\right )+a e \left (2 d \sqrt{b^2-4 a c}+a e\right )+3 b^2 d^2\right )-2 b c e^3 \left (2 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+2 b^2 d\right )+b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+2 c^4 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^3 x^3 (4 c d-b e)}{3 c^2}+\frac{e^4 x^5}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e)*x^3)/
(3*c^2) + (e^4*x^5)/(5*c) + ((2*c^4*d^4 + b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 + 4*c^
3*d^2*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*b*c*e^3*(-2*b^2*d + 2*b*Sqrt[
b^2 - 4*a*c]*d - 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(3*b^2*d^2 - 3*b*d
*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + a*e*(-2*Sqrt[b^2 - 4*a*c]*d + a*e)))*ArcTan[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c
]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((2*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 -
 4*c^3*d^2*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 3*a*e) - 2*b*c*e^3*(2*b^2*d + a*Sqrt[b
^2 - 4*a*c]*e + 2*b*(Sqrt[b^2 - 4*a*c]*d + a*e)) + 2*c^2*e^2*(3*b^2*d^2 + a*e*(2
*Sqrt[b^2 - 4*a*c]*d + a*e) + 3*b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)))*ArcTan[(Sqrt
[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*
Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Maple [B]  time = 0.061, size = 1888, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \, c^{2} e^{4} x^{5} + 5 \,{\left (4 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 15 \,{\left (6 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} +{\left (b^{2} - a c\right )} e^{4}\right )} x}{15 \, c^{3}} + \frac{\int \frac{c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2} + 4 \, a b c d e^{3} -{\left (a b^{2} - a^{2} c\right )} e^{4} +{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d e^{3} -{\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*c^2*e^4*x^5 + 5*(4*c^2*d*e^3 - b*c*e^4)*x^3 + 15*(6*c^2*d^2*e^2 - 4*b*c*
d*e^3 + (b^2 - a*c)*e^4)*x)/c^3 + integrate((c^3*d^4 - 6*a*c^2*d^2*e^2 + 4*a*b*c
*d*e^3 - (a*b^2 - a^2*c)*e^4 + (4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*(b^2*c - a*c^2
)*d*e^3 - (b^3 - 2*a*b*c)*e^4)*x^2)/(c*x^4 + b*x^2 + a), x)/c^3

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.23661, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done

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