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

Optimal. Leaf size=204 \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} \Pi \left (\frac{\left (b-\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d \sqrt{-a+b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.578772, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} \Pi \left (\frac{\left (b-\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d \sqrt{-a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 153.612, size = 411, normalized size = 2.01 \[ - \frac{\sqrt{2} \sqrt{c} \sqrt{b + \sqrt{4 a c + b^{2}}} \left (\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | - \frac{2 \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{\sqrt{\frac{\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1}{\frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1}} \sqrt{- a + b x^{2} + c x^{4}} \left (b e - 2 c d + e \sqrt{4 a c + b^{2}}\right )} + \frac{\sqrt{2} e \left (b + \sqrt{4 a c + b^{2}}\right )^{\frac{3}{2}} \left (\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1\right ) \Pi \left (1 - \frac{e \left (b + \sqrt{4 a c + b^{2}}\right )}{2 c d}; \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | - \frac{2 \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{2 \sqrt{c} d \sqrt{\frac{\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1}{\frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1}} \sqrt{- a + b x^{2} + c x^{4}} \left (b e - 2 c d + e \sqrt{4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.226377, size = 216, normalized size = 1.06 \[ -\frac{i \sqrt{\frac{\sqrt{4 a c+b^2}+b+2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \Pi \left (\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} d \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{-a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.036, size = 198, normalized size = 1. \[{\frac{1}{d}\sqrt{1-{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{4\,ac+{b}^{2}}}}\sqrt{1-{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{4\,ac+{b}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}x,2\,{\frac{ae}{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) d}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{b}{2\,a}}-{\frac{1}{2\,a}\sqrt{4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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