3.797 \(\int \frac{\left (a+c x^4\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=126 \[ \frac{2 c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} \sqrt{a+c x^4}}-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7}-\frac{2 c \sqrt{a+c x^4}}{7 x^3} \]

[Out]

(-2*c*Sqrt[a + c*x^4])/(7*x^3) - (a + c*x^4)^(3/2)/(7*x^7) + (2*c^(7/4)*(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])/(7*a^(1/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.0967737, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} \sqrt{a+c x^4}}-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7}-\frac{2 c \sqrt{a+c x^4}}{7 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x^8,x]

[Out]

(-2*c*Sqrt[a + c*x^4])/(7*x^3) - (a + c*x^4)^(3/2)/(7*x^7) + (2*c^(7/4)*(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])/(7*a^(1/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 10.2677, size = 114, normalized size = 0.9 \[ - \frac{2 c \sqrt{a + c x^{4}}}{7 x^{3}} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{7 x^{7}} + \frac{2 c^{\frac{7}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{7 \sqrt [4]{a} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(3/2)/x**8,x)

[Out]

-2*c*sqrt(a + c*x**4)/(7*x**3) - (a + c*x**4)**(3/2)/(7*x**7) + 2*c**(7/4)*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)/(7*a**(1/4)*sqrt(a + c*x**4))

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Mathematica [C]  time = 0.246471, size = 106, normalized size = 0.84 \[ \frac{-\frac{a^2+4 a c x^4+3 c^2 x^8}{x^7}-\frac{4 i c^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{7 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(3/2)/x^8,x]

[Out]

(-((a^2 + 4*a*c*x^4 + 3*c^2*x^8)/x^7) - ((4*I)*c^2*Sqrt[1 + (c*x^4)/a]*EllipticF
[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(7*Sqrt
[a + c*x^4])

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Maple [C]  time = 0.02, size = 105, normalized size = 0.8 \[ -{\frac{a}{7\,{x}^{7}}\sqrt{c{x}^{4}+a}}-{\frac{3\,c}{7\,{x}^{3}}\sqrt{c{x}^{4}+a}}+{\frac{4\,{c}^{2}}{7}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x^8,x)

[Out]

-1/7*a*(c*x^4+a)^(1/2)/x^7-3/7*c*(c*x^4+a)^(1/2)/x^3+4/7*c^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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^8,x, algorithm="maxima")

[Out]

integrate((c*x^4 + a)^(3/2)/x^8, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^8,x, algorithm="fricas")

[Out]

integral((c*x^4 + a)^(3/2)/x^8, x)

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Sympy [A]  time = 5.50176, size = 46, normalized size = 0.37 \[ \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(3/2)/x**8,x)

[Out]

a**(3/2)*gamma(-7/4)*hyper((-7/4, -3/2), (-3/4,), c*x**4*exp_polar(I*pi)/a)/(4*x
**7*gamma(-3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^8,x, algorithm="giac")

[Out]

integrate((c*x^4 + a)^(3/2)/x^8, x)