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

Optimal. Leaf size=399 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{a e}{d}-b-\frac{c d}{e}}}{\sqrt{-a+b x^2-c x^4}}\right )}{2 d \sqrt{-\frac{a e}{d}-b-\frac{c d}{e}}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt{-a+b x^2-c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \Pi \left (-\frac{\sqrt{a} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right )^2}{4 \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{-a+b x^2-c x^4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right )} \]

[Out]

ArcTan[(Sqrt[-b - (c*d)/e - (a*e)/d]*x)/Sqrt[-a + b*x^2 - c*x^4]]/(2*d*Sqrt[-b -
(c*d)/e - (a*e)/d]) + (c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a - b*x^2 + c*x^4)
/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 + b/(Sqr
t[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[-a + b*x^2 - c*x^4])
- (((Sqrt[c]*d)/Sqrt[a] + e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a - b*x^2 + c*x^4)/(S
qrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[a]*((Sqrt[c]*d)/Sqrt[a] - e)^2)/(4*Sq
rt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 + b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1
/4)*c^(1/4)*d*((Sqrt[c]*d)/Sqrt[a] - e)*Sqrt[-a + b*x^2 - c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.485705, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{a e}{d}-b-\frac{c d}{e}}}{\sqrt{-a+b x^2-c x^4}}\right )}{2 d \sqrt{-\frac{a e}{d}-b-\frac{c d}{e}}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt{-a+b x^2-c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{4 \sqrt [4]{c} d \sqrt{-a+b x^2-c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.7169, size = 338, normalized size = 0.85 \[ \frac{\operatorname{atan}{\left (\frac{x \sqrt{- \frac{a e}{d} - b - \frac{c d}{e}}}{\sqrt{- a + b x^{2} - c x^{4}}} \right )}}{2 d \sqrt{- \frac{a e}{d} - b - \frac{c d}{e}}} - \frac{\sqrt [4]{c} \sqrt{- \frac{- a + b x^{2} - c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} + \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} \left (\sqrt{a} e - \sqrt{c} d\right ) \sqrt{- a + b x^{2} - c x^{4}}} + \frac{\sqrt{- \frac{- a + b x^{2} - c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e + \sqrt{c} d\right ) \Pi \left (- \frac{\sqrt{a} \left (e - \frac{\sqrt{c} d}{\sqrt{a}}\right )^{2}}{4 \sqrt{c} d e}; 2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} + \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{a} e - \sqrt{c} d\right ) \sqrt{- a + b x^{2} - c x^{4}}} \]

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]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.21763, size = 207, normalized size = 0.52 \[ -\frac{i \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \Pi \left (-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|-\frac{b+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{2} d \sqrt{-\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{-a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[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), 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])

_______________________________________________________________________________________

Maple [A]  time = 0.035, size = 199, normalized size = 0.5 \[{\frac{1}{d}\sqrt{1-{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1-{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}x,2\,{\frac{ae}{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) d}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{b}{2\,a}}-{\frac{1}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}-a}}}} \]

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))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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]

Exception raised: TypeError