3.827 \(\int \frac{1}{x^2 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=232 \[ \frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )} \]

[Out]

-(Sqrt[a + b*x^4]/(a*x)) + (Sqrt[b]*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2
)) - (b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2
]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b*x^4]) + (b^
(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellipt
icF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.194308, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[a + b*x^4]),x]

[Out]

-(Sqrt[a + b*x^4]/(a*x)) + (Sqrt[b]*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2
)) - (b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2
]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b*x^4]) + (b^
(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellipt
icF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 22.8872, size = 202, normalized size = 0.87 \[ \frac{\sqrt{b} x \sqrt{a + b x^{4}}}{a \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{\sqrt{a + b x^{4}}}{a x} - \frac{\sqrt [4]{b} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{\sqrt [4]{b} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*x*sqrt(a + b*x**4)/(a*(sqrt(a) + sqrt(b)*x**2)) - sqrt(a + b*x**4)/(a*x)
- b**(1/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x*
*2)*elliptic_e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(a**(3/4)*sqrt(a + b*x**4)) + b
**(1/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*
elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(2*a**(3/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.470396, size = 121, normalized size = 0.52 \[ \frac{-\frac{a+b x^4}{a x}-i \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{\frac{b x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-((a + b*x^4)/(a*x)) - I*Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[1 + (b*x^4)/a]*(Ellipti
cE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[b])/Sqrt[a]]*x], -1]))/Sqrt[a + b*x^4]

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Maple [C]  time = 0.014, size = 115, normalized size = 0.5 \[ -{\frac{1}{ax}\sqrt{b{x}^{4}+a}}+{i\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^4+a)^(1/2),x)

[Out]

-(b*x^4+a)^(1/2)/a/x+I*b^(1/2)/a^(1/2)/(I/a^(1/2)*b^(1/2))^(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)*(EllipticF(x*(I
/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(b*x^4 + a)*x^2), x)

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Sympy [A]  time = 2.38786, size = 39, normalized size = 0.17 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*x*ga
mma(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^2), x)