3.1086 \(\int \frac{x^{13}}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=152 \[ \frac{8 a^{7/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a+b x^4}}-\frac{8 a^3 x^2}{39 b^3 \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2 \left (a+b x^4\right )^{3/4}}{39 b^3}-\frac{10 a x^6 \left (a+b x^4\right )^{3/4}}{117 b^2}+\frac{x^{10} \left (a+b x^4\right )^{3/4}}{13 b} \]

[Out]

(-8*a^3*x^2)/(39*b^3*(a + b*x^4)^(1/4)) + (4*a^2*x^2*(a + b*x^4)^(3/4))/(39*b^3)
- (10*a*x^6*(a + b*x^4)^(3/4))/(117*b^2) + (x^10*(a + b*x^4)^(3/4))/(13*b) + (8
*a^(7/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3
9*b^(7/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.223283, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{8 a^{7/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a+b x^4}}-\frac{8 a^3 x^2}{39 b^3 \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2 \left (a+b x^4\right )^{3/4}}{39 b^3}-\frac{10 a x^6 \left (a+b x^4\right )^{3/4}}{117 b^2}+\frac{x^{10} \left (a+b x^4\right )^{3/4}}{13 b} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(a + b*x^4)^(1/4),x]

[Out]

(-8*a^3*x^2)/(39*b^3*(a + b*x^4)^(1/4)) + (4*a^2*x^2*(a + b*x^4)^(3/4))/(39*b^3)
- (10*a*x^6*(a + b*x^4)^(3/4))/(117*b^2) + (x^10*(a + b*x^4)^(3/4))/(13*b) + (8
*a^(7/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3
9*b^(7/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{4} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{39 b^{3}} - \frac{8 a^{3} x^{2}}{39 b^{3} \sqrt [4]{a + b x^{4}}} + \frac{4 a^{2} x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{39 b^{3}} - \frac{10 a x^{6} \left (a + b x^{4}\right )^{\frac{3}{4}}}{117 b^{2}} + \frac{x^{10} \left (a + b x^{4}\right )^{\frac{3}{4}}}{13 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(b*x**4+a)**(1/4),x)

[Out]

4*a**4*Integral((a + b*x**2)**(-5/4), (x, x**2))/(39*b**3) - 8*a**3*x**2/(39*b**
3*(a + b*x**4)**(1/4)) + 4*a**2*x**2*(a + b*x**4)**(3/4)/(39*b**3) - 10*a*x**6*(
a + b*x**4)**(3/4)/(117*b**2) + x**10*(a + b*x**4)**(3/4)/(13*b)

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Mathematica [C]  time = 0.076458, size = 91, normalized size = 0.6 \[ \frac{x^2 \left (-12 a^3 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+12 a^3+2 a^2 b x^4-a b^2 x^8+9 b^3 x^{12}\right )}{117 b^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/(a + b*x^4)^(1/4),x]

[Out]

(x^2*(12*a^3 + 2*a^2*b*x^4 - a*b^2*x^8 + 9*b^3*x^12 - 12*a^3*(1 + (b*x^4)/a)^(1/
4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)]))/(117*b^3*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{{x}^{13}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(b*x^4+a)^(1/4),x)

[Out]

int(x^13/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^13/(b*x^4 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^13/(b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 8.56957, size = 27, normalized size = 0.18 \[ \frac{x^{14}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{14 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(b*x**4+a)**(1/4),x)

[Out]

x**14*hyper((1/4, 7/2), (9/2,), b*x**4*exp_polar(I*pi)/a)/(14*a**(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^13/(b*x^4 + a)^(1/4), x)