3.1225 \(\int \frac{x^{10}}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=134 \[ \frac{7 a^{5/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{5/2} \sqrt [4]{a-b x^4}}-\frac{7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac{7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac{x^7 \left (a-b x^4\right )^{3/4}}{10 b} \]

[Out]

(-7*a^2*(a - b*x^4)^(3/4))/(40*b^3*x) - (7*a*x^3*(a - b*x^4)^(3/4))/(60*b^2) - (
x^7*(a - b*x^4)^(3/4))/(10*b) + (7*a^(5/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[Arc
Csc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*b^(5/2)*(a - b*x^4)^(1/4))

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Rubi [A]  time = 0.190772, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{7 a^{5/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{5/2} \sqrt [4]{a-b x^4}}-\frac{7 a^2 \left (a-b x^4\right )^{3/4}}{40 b^3 x}-\frac{7 a x^3 \left (a-b x^4\right )^{3/4}}{60 b^2}-\frac{x^7 \left (a-b x^4\right )^{3/4}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^10/(a - b*x^4)^(1/4),x]

[Out]

(-7*a^2*(a - b*x^4)^(3/4))/(40*b^3*x) - (7*a*x^3*(a - b*x^4)^(3/4))/(60*b^2) - (
x^7*(a - b*x^4)^(3/4))/(10*b) + (7*a^(5/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[Arc
Csc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*b^(5/2)*(a - b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 23.8705, size = 116, normalized size = 0.87 \[ \frac{7 a^{\frac{5}{2}} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{40 b^{\frac{5}{2}} \sqrt [4]{a - b x^{4}}} - \frac{7 a^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{40 b^{3} x} - \frac{7 a x^{3} \left (a - b x^{4}\right )^{\frac{3}{4}}}{60 b^{2}} - \frac{x^{7} \left (a - b x^{4}\right )^{\frac{3}{4}}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(-b*x**4+a)**(1/4),x)

[Out]

7*a**(5/2)*x*(-a/(b*x**4) + 1)**(1/4)*elliptic_e(asin(sqrt(a)/(sqrt(b)*x**2))/2,
2)/(40*b**(5/2)*(a - b*x**4)**(1/4)) - 7*a**2*(a - b*x**4)**(3/4)/(40*b**3*x) -
7*a*x**3*(a - b*x**4)**(3/4)/(60*b**2) - x**7*(a - b*x**4)**(3/4)/(10*b)

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Mathematica [C]  time = 0.0723251, size = 80, normalized size = 0.6 \[ \frac{x^3 \left (7 a^2 \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-7 a^2+a b x^4+6 b^2 x^8\right )}{60 b^2 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a - b*x^4)^(1/4),x]

[Out]

(x^3*(-7*a^2 + a*b*x^4 + 6*b^2*x^8 + 7*a^2*(1 - (b*x^4)/a)^(1/4)*Hypergeometric2
F1[1/4, 3/4, 7/4, (b*x^4)/a]))/(60*b^2*(a - b*x^4)^(1/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{{x}^{10}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(-b*x^4+a)^(1/4),x)

[Out]

int(x^10/(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(-b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^10/(-b*x^4 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(-b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^10/(-b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 5.30036, size = 39, normalized size = 0.29 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(-b*x**4+a)**(1/4),x)

[Out]

x**11*gamma(11/4)*hyper((1/4, 11/4), (15/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**
(1/4)*gamma(15/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(-b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^10/(-b*x^4 + a)^(1/4), x)