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

Optimal. Leaf size=130 \[ \frac{5 a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 b^{9/4} \sqrt{a+b x^4}}-\frac{5 a x \sqrt{a+b x^4}}{21 b^2}+\frac{x^5 \sqrt{a+b x^4}}{7 b} \]

[Out]

(-5*a*x*Sqrt[a + b*x^4])/(21*b^2) + (x^5*Sqrt[a + b*x^4])/(7*b) + (5*a^(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*b^(9/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.110391, antiderivative size = 130, 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 \[ \frac{5 a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 b^{9/4} \sqrt{a+b x^4}}-\frac{5 a x \sqrt{a+b x^4}}{21 b^2}+\frac{x^5 \sqrt{a+b x^4}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[a + b*x^4],x]

[Out]

(-5*a*x*Sqrt[a + b*x^4])/(21*b^2) + (x^5*Sqrt[a + b*x^4])/(7*b) + (5*a^(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*b^(9/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 11.0792, size = 117, normalized size = 0.9 \[ \frac{5 a^{\frac{7}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{42 b^{\frac{9}{4}} \sqrt{a + b x^{4}}} - \frac{5 a x \sqrt{a + b x^{4}}}{21 b^{2}} + \frac{x^{5} \sqrt{a + b x^{4}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.280929, size = 106, normalized size = 0.82 \[ \frac{-\frac{5 i a^2 \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}}}-5 a^2 x-2 a b x^5+3 b^2 x^9}{21 b^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*a^2*x - 2*a*b*x^5 + 3*b^2*x^9 - ((5*I)*a^2*Sqrt[1 + (b*x^4)/a]*EllipticF[I*A
rcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[b])/Sqrt[a]])/(21*b^2*Sqr
t[a + b*x^4])

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Maple [C]  time = 0.051, size = 111, normalized size = 0.9 \[{\frac{{x}^{5}}{7\,b}\sqrt{b{x}^{4}+a}}-{\frac{5\,ax}{21\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{5\,{a}^{2}}{21\,{b}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^4+a)^(1/2),x)

[Out]

1/7*x^5*(b*x^4+a)^(1/2)/b-5/21*a*x*(b*x^4+a)^(1/2)/b^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. \[ \int \frac{x^{8}}{\sqrt{b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^8/sqrt(b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^8/sqrt(b*x^4 + a), x)

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Sympy [A]  time = 3.18558, size = 37, normalized size = 0.28 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*
gamma(13/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^8/sqrt(b*x^4 + a), x)