3.823 \(\int \frac{1}{x^8 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=132 \[ \frac{5 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 a^{9/4} \sqrt{a+b x^4}}+\frac{5 b \sqrt{a+b x^4}}{21 a^2 x^3}-\frac{\sqrt{a+b x^4}}{7 a x^7} \]

[Out]

-Sqrt[a + b*x^4]/(7*a*x^7) + (5*b*Sqrt[a + b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*(Sq
rt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Arc
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(42*a^(9/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.106155, antiderivative size = 132, 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 \[ \frac{5 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 a^{9/4} \sqrt{a+b x^4}}+\frac{5 b \sqrt{a+b x^4}}{21 a^2 x^3}-\frac{\sqrt{a+b x^4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*Sqrt[a + b*x^4]),x]

[Out]

-Sqrt[a + b*x^4]/(7*a*x^7) + (5*b*Sqrt[a + b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*(Sq
rt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Arc
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(42*a^(9/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 10.9396, size = 119, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{4}}}{7 a x^{7}} + \frac{5 b \sqrt{a + b x^{4}}}{21 a^{2} x^{3}} + \frac{5 b^{\frac{7}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{42 a^{\frac{9}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(b*x**4+a)**(1/2),x)

[Out]

-sqrt(a + b*x**4)/(7*a*x**7) + 5*b*sqrt(a + b*x**4)/(21*a**2*x**3) + 5*b**(7/4)*
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)/(42*a**(9/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.244999, size = 106, normalized size = 0.8 \[ \frac{-\frac{3 a^2}{x^7}-\frac{5 i b^2 \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}}}+\frac{2 a b}{x^3}+5 b^2 x}{21 a^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*Sqrt[a + b*x^4]),x]

[Out]

((-3*a^2)/x^7 + (2*a*b)/x^3 + 5*b^2*x - ((5*I)*b^2*Sqrt[1 + (b*x^4)/a]*EllipticF
[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[b])/Sqrt[a]])/(21*a^2
*Sqrt[a + b*x^4])

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Maple [C]  time = 0.021, size = 113, normalized size = 0.9 \[ -{\frac{1}{7\,a{x}^{7}}\sqrt{b{x}^{4}+a}}+{\frac{5\,b}{21\,{x}^{3}{a}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{5\,{b}^{2}}{21\,{a}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(b*x^4+a)^(1/2),x)

[Out]

-1/7*(b*x^4+a)^(1/2)/a/x^7+5/21*b*(b*x^4+a)^(1/2)/x^3/a^2+5/21/a^2*b^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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*x^8),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{4} + a} x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*x^8),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x^4 + a)*x^8), x)

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Sympy [A]  time = 4.44487, size = 44, normalized size = 0.33 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(b*x**4+a)**(1/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*x**
7*gamma(-3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*x^8),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^8), x)