3.776 \(\int \frac{\sqrt{a+c x^4}}{x^4} \, dx\)

Optimal. Leaf size=107 \[ \frac{c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+c x^4}}-\frac{\sqrt{a+c x^4}}{3 x^3} \]

[Out]

-Sqrt[a + c*x^4]/(3*x^3) + (c^(3/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(3*a^(1/4
)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.0684904, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+c x^4}}-\frac{\sqrt{a+c x^4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^4,x]

[Out]

-Sqrt[a + c*x^4]/(3*x^3) + (c^(3/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(3*a^(1/4
)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 7.02261, size = 94, normalized size = 0.88 \[ - \frac{\sqrt{a + c x^{4}}}{3 x^{3}} + \frac{c^{\frac{3}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + c*x**4)/(3*x**3) + c**(3/4)*sqrt((a + 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)/(3*a*
*(1/4)*sqrt(a + c*x**4))

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Mathematica [C]  time = 0.209058, size = 92, normalized size = 0.86 \[ \frac{-\frac{a+c x^4}{x^3}-\frac{2 i c \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{3 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^4,x]

[Out]

(-((a + c*x^4)/x^3) - ((2*I)*c*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*S
qrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(3*Sqrt[a + c*x^4])

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Maple [C]  time = 0.015, size = 87, normalized size = 0.8 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{c{x}^{4}+a}}+{\frac{2\,c}{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^4,x)

[Out]

-1/3*(c*x^4+a)^(1/2)/x^3+2/3*c/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^
2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*
c^(1/2))^(1/2),I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + a)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^4 + a)/x^4, x)

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Sympy [A]  time = 2.48414, size = 42, normalized size = 0.39 \[ \frac{\sqrt{a} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + a)/x^4, x)