3.780 \(\int \frac{\sqrt{a+c x^4}}{x^6} \, dx\)

Optimal. Leaf size=258 \[ \frac{c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+c x^4}}-\frac{2 c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+c x^4}}+\frac{2 c^{3/2} x \sqrt{a+c x^4}}{5 a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{2 c \sqrt{a+c x^4}}{5 a x}-\frac{\sqrt{a+c x^4}}{5 x^5} \]

[Out]

-Sqrt[a + c*x^4]/(5*x^5) - (2*c*Sqrt[a + c*x^4])/(5*a*x) + (2*c^(3/2)*x*Sqrt[a +
c*x^4])/(5*a*(Sqrt[a] + Sqrt[c]*x^2)) - (2*c^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt
[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(5*a^(3/4)*Sqrt[a + c*x^4]) + (c^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a +
c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])
/(5*a^(3/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.239579, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+c x^4}}-\frac{2 c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+c x^4}}+\frac{2 c^{3/2} x \sqrt{a+c x^4}}{5 a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{2 c \sqrt{a+c x^4}}{5 a x}-\frac{\sqrt{a+c x^4}}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^6,x]

[Out]

-Sqrt[a + c*x^4]/(5*x^5) - (2*c*Sqrt[a + c*x^4])/(5*a*x) + (2*c^(3/2)*x*Sqrt[a +
c*x^4])/(5*a*(Sqrt[a] + Sqrt[c]*x^2)) - (2*c^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt
[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(5*a^(3/4)*Sqrt[a + c*x^4]) + (c^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a +
c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])
/(5*a^(3/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 28.8004, size = 230, normalized size = 0.89 \[ - \frac{\sqrt{a + c x^{4}}}{5 x^{5}} + \frac{2 c^{\frac{3}{2}} x \sqrt{a + c x^{4}}}{5 a \left (\sqrt{a} + \sqrt{c} x^{2}\right )} - \frac{2 c \sqrt{a + c x^{4}}}{5 a x} - \frac{2 c^{\frac{5}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{c^{\frac{5}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(1/2)/x**6,x)

[Out]

-sqrt(a + c*x**4)/(5*x**5) + 2*c**(3/2)*x*sqrt(a + c*x**4)/(5*a*(sqrt(a) + sqrt(
c)*x**2)) - 2*c*sqrt(a + c*x**4)/(5*a*x) - 2*c**(5/4)*sqrt((a + c*x**4)/(sqrt(a)
+ sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1
/4)), 1/2)/(5*a**(3/4)*sqrt(a + c*x**4)) + c**(5/4)*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)/(5*a**(3/4)*sqrt(a + c*x**4))

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Mathematica [C]  time = 0.47627, size = 133, normalized size = 0.52 \[ \frac{-\frac{\left (a+c x^4\right ) \left (a+2 c x^4\right )}{a x^5}-2 i c \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{\frac{c x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{5 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^6,x]

[Out]

(-(((a + c*x^4)*(a + 2*c*x^4))/(a*x^5)) - (2*I)*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*Sqrt
[1 + (c*x^4)/a]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Ellipti
cF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(5*Sqrt[a + c*x^4])

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Maple [C]  time = 0.019, size = 130, normalized size = 0.5 \[ -{\frac{1}{5\,{x}^{5}}\sqrt{c{x}^{4}+a}}-{\frac{2\,c}{5\,ax}\sqrt{c{x}^{4}+a}}+{{\frac{2\,i}{5}}{c}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^6,x)

[Out]

-1/5*(c*x^4+a)^(1/2)/x^5-2/5*c*(c*x^4+a)^(1/2)/a/x+2/5*I*c^(3/2)/a^(1/2)/(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)-EllipticE(x*(I/a^(1
/2)*c^(1/2))^(1/2),I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + a}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + a)/x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + a}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^6,x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + a)/x^6, x)

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Sympy [A]  time = 2.97086, size = 46, normalized size = 0.18 \[ \frac{\sqrt{a} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(1/2)/x**6,x)

[Out]

sqrt(a)*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), c*x**4*exp_polar(I*pi)/a)/(4*x*
*5*gamma(-1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + a}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^6,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + a)/x^6, x)