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

Optimal. Leaf size=73 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.120486, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.5972, size = 63, normalized size = 0.86 \[ \frac{\sqrt [4]{a} \sqrt{1 - \frac{c x^{4}}{a}} \Pi \left (- \frac{\sqrt{a} e}{\sqrt{c} d}; \operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{c} d \sqrt{- a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*sqrt(1 - c*x**4/a)*elliptic_pi(-sqrt(a)*e/(sqrt(c)*d), asin(c**(1/4)*x/
a**(1/4)), -1)/(c**(1/4)*d*sqrt(-a + c*x**4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0704225, size = 92, normalized size = 1.26 \[ -\frac{i \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{d \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 99, normalized size = 1.4 \[{\frac{1}{d}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{e}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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