3.204 \(\int \frac{1}{\left (c+d x+e x^2\right ) \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=1493 \[ \text{result too large to display} \]

[Out]

(2*e*ArcTan[(Sqrt[-((16*a*e^4 + b*(d - Sqrt[d^2 - 4*c*e])^4)/(e^2*(d - Sqrt[d^2
- 4*c*e])^2))]*x)/(2*Sqrt[a + b*x^4])])/(Sqrt[d^2 - 4*c*e]*(d - Sqrt[d^2 - 4*c*e
])*Sqrt[-((16*a*e^4 + b*(d - Sqrt[d^2 - 4*c*e])^4)/(e^2*(d - Sqrt[d^2 - 4*c*e])^
2))]) - (2*e*ArcTan[(Sqrt[-((16*a*e^4 + b*(d + Sqrt[d^2 - 4*c*e])^4)/(e^2*(d + S
qrt[d^2 - 4*c*e])^2))]*x)/(2*Sqrt[a + b*x^4])])/(Sqrt[d^2 - 4*c*e]*(d + Sqrt[d^2
- 4*c*e])*Sqrt[-((16*a*e^4 + b*(d + Sqrt[d^2 - 4*c*e])^4)/(e^2*(d + Sqrt[d^2 -
4*c*e])^2))]) - (e^2*ArcTanh[(4*a*e^2 + b*(d - Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt
[2]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^
2 - 2*c*e)]*Sqrt[a + b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2
*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (e^2*ArcTan
h[(4*a*e^2 + b*(d + Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2*
e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*Sqrt[a + b*x^4]
)])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4
+ b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (b^(1/4)*e*(d - Sqrt[d^2 - 4*c*e])*(Sq
rt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Arc
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[d^2 - 4*c*e]*(4*e^2 + (Sqrt[b]*(d
- Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqrt[a + b*x^4]) - (b^(1/4)*e*(d + Sqrt[d^2 - 4
*c*e])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellip
ticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[d^2 - 4*c*e]*(4*e^2 + (S
qrt[b]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqrt[a + b*x^4]) + (e*(4*e^2 - (Sqrt[
b]*(d - Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/
(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[a]*(4*e^2 + (Sqrt[b]*(d - Sqrt[d^2 -
4*c*e])^2)/Sqrt[a])^2)/(16*Sqrt[b]*e^2*(d - Sqrt[d^2 - 4*c*e])^2), 2*ArcTan[(b^
(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[d^2 - 4*c*e]*(d - Sqrt[d^2 - 4*
c*e])*(4*e^2 + (Sqrt[b]*(d - Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqrt[a + b*x^4]) - (
e*(4*e^2 - (Sqrt[b]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*(Sqrt[a] + Sqrt[b]*x^2)*
Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[a]*(4*e^2 + (Sqrt[b
]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])^2)/(16*Sqrt[b]*e^2*(d + Sqrt[d^2 - 4*c*e])
^2), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[d^2 - 4*c*e]*(
d + Sqrt[d^2 - 4*c*e])*(4*e^2 + (Sqrt[b]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqr
t[a + b*x^4])

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Rubi [A]  time = 15.2873, antiderivative size = 1493, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((c + d*x + e*x^2)*Sqrt[a + b*x^4]),x]

[Out]

(2*e*ArcTan[(Sqrt[-((16*a*e^4 + b*(d - Sqrt[d^2 - 4*c*e])^4)/(e^2*(d - Sqrt[d^2
- 4*c*e])^2))]*x)/(2*Sqrt[a + b*x^4])])/(Sqrt[d^2 - 4*c*e]*(d - Sqrt[d^2 - 4*c*e
])*Sqrt[-((16*a*e^4 + b*(d - Sqrt[d^2 - 4*c*e])^4)/(e^2*(d - Sqrt[d^2 - 4*c*e])^
2))]) - (2*e*ArcTan[(Sqrt[-((16*a*e^4 + b*(d + Sqrt[d^2 - 4*c*e])^4)/(e^2*(d + S
qrt[d^2 - 4*c*e])^2))]*x)/(2*Sqrt[a + b*x^4])])/(Sqrt[d^2 - 4*c*e]*(d + Sqrt[d^2
- 4*c*e])*Sqrt[-((16*a*e^4 + b*(d + Sqrt[d^2 - 4*c*e])^4)/(e^2*(d + Sqrt[d^2 -
4*c*e])^2))]) - (e^2*ArcTanh[(4*a*e^2 + b*(d - Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt
[2]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^
2 - 2*c*e)]*Sqrt[a + b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2
*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (e^2*ArcTan
h[(4*a*e^2 + b*(d + Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2*
e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*Sqrt[a + b*x^4]
)])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4
+ b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (b^(1/4)*e*(d - Sqrt[d^2 - 4*c*e])*(Sq
rt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Arc
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[d^2 - 4*c*e]*(4*e^2 + (Sqrt[b]*(d
- Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqrt[a + b*x^4]) - (b^(1/4)*e*(d + Sqrt[d^2 - 4
*c*e])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellip
ticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[d^2 - 4*c*e]*(4*e^2 + (S
qrt[b]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqrt[a + b*x^4]) + (e*(4*e^2 - (Sqrt[
b]*(d - Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/
(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[a]*(4*e^2 + (Sqrt[b]*(d - Sqrt[d^2 -
4*c*e])^2)/Sqrt[a])^2)/(16*Sqrt[b]*e^2*(d - Sqrt[d^2 - 4*c*e])^2), 2*ArcTan[(b^
(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[d^2 - 4*c*e]*(d - Sqrt[d^2 - 4*
c*e])*(4*e^2 + (Sqrt[b]*(d - Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqrt[a + b*x^4]) - (
e*(4*e^2 - (Sqrt[b]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*(Sqrt[a] + Sqrt[b]*x^2)*
Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[a]*(4*e^2 + (Sqrt[b
]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])^2)/(16*Sqrt[b]*e^2*(d + Sqrt[d^2 - 4*c*e])
^2), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[d^2 - 4*c*e]*(
d + Sqrt[d^2 - 4*c*e])*(4*e^2 + (Sqrt[b]*(d + Sqrt[d^2 - 4*c*e])^2)/Sqrt[a])*Sqr
t[a + b*x^4])

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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(1/(e*x**2+d*x+c)/(b*x**4+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 8.50006, size = 653, normalized size = 0.44 \[ -\frac{\sqrt [4]{-1} \sqrt{2} \sqrt{-\frac{i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} x}} \left (\sqrt{b} x^2+i \sqrt{a}\right ) \left (\sqrt [4]{-1} \sqrt [4]{a} \left (-\left (\sqrt{b} \left (d \sqrt{d^2-4 c e}-2 c e+d^2\right )-2 i \sqrt{a} e^2\right ) \Pi \left (\frac{2 (-1)^{3/4} \sqrt [4]{a} e-i \sqrt [4]{b} \left (\sqrt{d^2-4 c e}-d\right )}{2 \sqrt [4]{-1} \sqrt [4]{a} e+\sqrt [4]{b} \left (\sqrt{d^2-4 c e}-d\right )};\left .\sin ^{-1}\left (\sqrt{-\frac{i \left (\sqrt [4]{b} x+\sqrt [4]{-1} \sqrt [4]{a}\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} x}}\right )\right |-1\right )-\left (\sqrt{b} \left (d \sqrt{d^2-4 c e}+2 c e-d^2\right )+2 i \sqrt{a} e^2\right ) \Pi \left (-\frac{i \left (2 \sqrt [4]{-1} \sqrt [4]{a} e+\sqrt [4]{b} \left (d+\sqrt{d^2-4 c e}\right )\right )}{\sqrt [4]{b} \left (d+\sqrt{d^2-4 c e}\right )-2 \sqrt [4]{-1} \sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt{-\frac{i \left (\sqrt [4]{b} x+\sqrt [4]{-1} \sqrt [4]{a}\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} x}}\right )\right |-1\right )\right )+\sqrt [4]{b} \sqrt{d^2-4 c e} \left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{b} d-i \sqrt{a} e-\sqrt{b} c\right ) F\left (\left .\sin ^{-1}\left (\sqrt{-\frac{i \left (\sqrt [4]{b} x+\sqrt [4]{-1} \sqrt [4]{a}\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} x}}\right )\right |-1\right )\right )}{\sqrt [4]{a} \sqrt{\frac{\sqrt{b} x^2+i \sqrt{a}}{\left (\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} x\right )^2}} \sqrt{a+b x^4} \sqrt{d^2-4 c e} \left (-i \sqrt{a} \sqrt{b} \left (d^2-2 c e\right )-a e^2+b c^2\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((c + d*x + e*x^2)*Sqrt[a + b*x^4]),x]

[Out]

-(((-1)^(1/4)*Sqrt[2]*Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*x))/((-1)^(1/4)*a
^(1/4) - b^(1/4)*x)]*(I*Sqrt[a] + Sqrt[b]*x^2)*(b^(1/4)*(-(Sqrt[b]*c) + (-1)^(1/
4)*a^(1/4)*b^(1/4)*d - I*Sqrt[a]*e)*Sqrt[d^2 - 4*c*e]*EllipticF[ArcSin[Sqrt[((-I
)*((-1)^(1/4)*a^(1/4) + b^(1/4)*x))/((-1)^(1/4)*a^(1/4) - b^(1/4)*x)]], -1] + (-
1)^(1/4)*a^(1/4)*(-(((-2*I)*Sqrt[a]*e^2 + Sqrt[b]*(d^2 - 2*c*e + d*Sqrt[d^2 - 4*
c*e]))*EllipticPi[(2*(-1)^(3/4)*a^(1/4)*e - I*b^(1/4)*(-d + Sqrt[d^2 - 4*c*e]))/
(2*(-1)^(1/4)*a^(1/4)*e + b^(1/4)*(-d + Sqrt[d^2 - 4*c*e])), ArcSin[Sqrt[((-I)*(
(-1)^(1/4)*a^(1/4) + b^(1/4)*x))/((-1)^(1/4)*a^(1/4) - b^(1/4)*x)]], -1]) - ((2*
I)*Sqrt[a]*e^2 + Sqrt[b]*(-d^2 + 2*c*e + d*Sqrt[d^2 - 4*c*e]))*EllipticPi[((-I)*
(2*(-1)^(1/4)*a^(1/4)*e + b^(1/4)*(d + Sqrt[d^2 - 4*c*e])))/(-2*(-1)^(1/4)*a^(1/
4)*e + b^(1/4)*(d + Sqrt[d^2 - 4*c*e])), ArcSin[Sqrt[((-I)*((-1)^(1/4)*a^(1/4) +
b^(1/4)*x))/((-1)^(1/4)*a^(1/4) - b^(1/4)*x)]], -1])))/(a^(1/4)*Sqrt[d^2 - 4*c*
e]*(b*c^2 - a*e^2 - I*Sqrt[a]*Sqrt[b]*(d^2 - 2*c*e))*Sqrt[(I*Sqrt[a] + Sqrt[b]*x
^2)/((-1)^(1/4)*a^(1/4) - b^(1/4)*x)^2]*Sqrt[a + b*x^4]))

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Maple [C]  time = 0.102, size = 1153, normalized size = 0.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x^2+d*x+c)/(b*x^4+a)^(1/2),x)

[Out]

-1/2/(-4*c*e+d^2)^(1/2)/(1/2*b/e^4*d^4-1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e^3*
d^2*c+b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)*arctanh(1/2/(1/2*b/e^4*d^4
-1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e^3*d^2*c+b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e
^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d^2-1/2/(1/2*b/e^4*d^4-1/2*b/e^4*(-4*c
*e+d^2)^(1/2)*d^3-2*b/e^3*d^2*c+b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)/
(b*x^4+a)^(1/2)*b*x^2/e^2*d*(-4*c*e+d^2)^(1/2)-1/(1/2*b/e^4*d^4-1/2*b/e^4*(-4*c*
e+d^2)^(1/2)*d^3-2*b/e^3*d^2*c+b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)/(
b*x^4+a)^(1/2)*b*x^2/e*c+1/(1/2*b/e^4*d^4-1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e
^3*d^2*c+b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*a)-2/(-
4*c*e+d^2)^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*e/(-d+(-4*c*e+d^2)^(1/2))*(1-I/a^(1/2
)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(
x*(I/a^(1/2)*b^(1/2))^(1/2),-4*I*a^(1/2)/b^(1/2)*e^2/(-d+(-4*c*e+d^2)^(1/2))^2,(
-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))+1/2/(-4*c*e+d^2)^(1/2)/(1/2
*b/e^4*d^4+1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e^3*d^2*c-b/e^3*d*(-4*c*e+d^2)^(
1/2)*c+b/e^2*c^2+a)^(1/2)*arctanh(1/2/(1/2*b/e^4*d^4+1/2*b/e^4*(-4*c*e+d^2)^(1/2
)*d^3-2*b/e^3*d^2*c-b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1
/2)*b*x^2/e^2*d^2+1/2/(1/2*b/e^4*d^4+1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e^3*d^
2*c-b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d*
(-4*c*e+d^2)^(1/2)-1/(1/2*b/e^4*d^4+1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e^3*d^2
*c-b/e^3*d*(-4*c*e+d^2)^(1/2)*c+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e*c+1/(
1/2*b/e^4*d^4+1/2*b/e^4*(-4*c*e+d^2)^(1/2)*d^3-2*b/e^3*d^2*c-b/e^3*d*(-4*c*e+d^2
)^(1/2)*c+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*a)-2/(-4*c*e+d^2)^(1/2)/(I/a^(1/2)*
b^(1/2))^(1/2)/(d+(-4*c*e+d^2)^(1/2))*e*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(
1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*b^(1/2))^(1/2),-
4*I*a^(1/2)/b^(1/2)/(d+(-4*c*e+d^2)^(1/2))^2*e^2,(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a
^(1/2)*b^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a}{\left (e x^{2} + d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*(e*x^2 + d*x + c)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a)*(e*x^2 + d*x + c)), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x**2+d*x+c)/(b*x**4+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a}{\left (e x^{2} + d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*(e*x^2 + d*x + c)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^4 + a)*(e*x^2 + d*x + c)), x)