3.815 \(\int \frac{x^6}{\sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{16 a^{7/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a+b x^2}}-\frac{16 a^3 x}{39 b^3 \sqrt [4]{a+b x^2}}+\frac{8 a^2 x \left (a+b x^2\right )^{3/4}}{39 b^3}-\frac{20 a x^3 \left (a+b x^2\right )^{3/4}}{117 b^2}+\frac{2 x^5 \left (a+b x^2\right )^{3/4}}{13 b} \]

[Out]

(-16*a^3*x)/(39*b^3*(a + b*x^2)^(1/4)) + (8*a^2*x*(a + b*x^2)^(3/4))/(39*b^3) -
(20*a*x^3*(a + b*x^2)^(3/4))/(117*b^2) + (2*x^5*(a + b*x^2)^(3/4))/(13*b) + (16*
a^(7/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(39*b
^(7/2)*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.165702, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{16 a^{7/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a+b x^2}}-\frac{16 a^3 x}{39 b^3 \sqrt [4]{a+b x^2}}+\frac{8 a^2 x \left (a+b x^2\right )^{3/4}}{39 b^3}-\frac{20 a x^3 \left (a+b x^2\right )^{3/4}}{117 b^2}+\frac{2 x^5 \left (a+b x^2\right )^{3/4}}{13 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^2)^(1/4),x]

[Out]

(-16*a^3*x)/(39*b^3*(a + b*x^2)^(1/4)) + (8*a^2*x*(a + b*x^2)^(3/4))/(39*b^3) -
(20*a*x^3*(a + b*x^2)^(3/4))/(117*b^2) + (2*x^5*(a + b*x^2)^(3/4))/(13*b) + (16*
a^(7/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(39*b
^(7/2)*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{8 a^{3} \int \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{39 b^{3}} + \frac{8 a^{2} x \left (a + b x^{2}\right )^{\frac{3}{4}}}{39 b^{3}} - \frac{20 a x^{3} \left (a + b x^{2}\right )^{\frac{3}{4}}}{117 b^{2}} + \frac{2 x^{5} \left (a + b x^{2}\right )^{\frac{3}{4}}}{13 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**3*Integral((a + b*x**2)**(-1/4), x)/(39*b**3) + 8*a**2*x*(a + b*x**2)**(3/
4)/(39*b**3) - 20*a*x**3*(a + b*x**2)**(3/4)/(117*b**2) + 2*x**5*(a + b*x**2)**(
3/4)/(13*b)

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Mathematica [C]  time = 0.0757217, size = 90, normalized size = 0.62 \[ \frac{2 \left (-12 a^3 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+12 a^3 x+2 a^2 b x^3-a b^2 x^5+9 b^3 x^7\right )}{117 b^3 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^2)^(1/4),x]

[Out]

(2*(12*a^3*x + 2*a^2*b*x^3 - a*b^2*x^5 + 9*b^3*x^7 - 12*a^3*x*(1 + (b*x^2)/a)^(1
/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(117*b^3*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{6}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^2+a)^(1/4),x)

[Out]

int(x^6/(b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^6/(b*x^2 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^6/(b*x^2 + a)^(1/4), x)

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Sympy [A]  time = 3.09115, size = 27, normalized size = 0.18 \[ \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**7*hyper((1/4, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/(7*a**(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^2 + a)^(1/4), x)