3.1240 \(\int \frac{x^{13}}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=133 \[ \frac{40 a^{7/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}-\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b} \]

[Out]

(-20*a^2*x^2*(a - b*x^4)^(1/4))/(77*b^3) - (10*a*x^6*(a - b*x^4)^(1/4))/(77*b^2)
- (x^10*(a - b*x^4)^(1/4))/(11*b) + (40*a^(7/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF
[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.211754, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{40 a^{7/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}-\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(a - b*x^4)^(3/4),x]

[Out]

(-20*a^2*x^2*(a - b*x^4)^(1/4))/(77*b^3) - (10*a*x^6*(a - b*x^4)^(1/4))/(77*b^2)
- (x^10*(a - b*x^4)^(1/4))/(11*b) + (40*a^(7/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF
[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 23.7624, size = 116, normalized size = 0.87 \[ \frac{40 a^{\frac{7}{2}} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{7}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{20 a^{2} x^{2} \sqrt [4]{a - b x^{4}}}{77 b^{3}} - \frac{10 a x^{6} \sqrt [4]{a - b x^{4}}}{77 b^{2}} - \frac{x^{10} \sqrt [4]{a - b x^{4}}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(-b*x**4+a)**(3/4),x)

[Out]

40*a**(7/2)*(1 - b*x**4/a)**(3/4)*elliptic_f(asin(sqrt(b)*x**2/sqrt(a))/2, 2)/(7
7*b**(7/2)*(a - b*x**4)**(3/4)) - 20*a**2*x**2*(a - b*x**4)**(1/4)/(77*b**3) - 1
0*a*x**6*(a - b*x**4)**(1/4)/(77*b**2) - x**10*(a - b*x**4)**(1/4)/(11*b)

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Mathematica [C]  time = 0.0867813, size = 92, normalized size = 0.69 \[ \frac{x^2 \left (20 a^3 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-20 a^3+10 a^2 b x^4+3 a b^2 x^8+7 b^3 x^{12}\right )}{77 b^3 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/(a - b*x^4)^(3/4),x]

[Out]

(x^2*(-20*a^3 + 10*a^2*b*x^4 + 3*a*b^2*x^8 + 7*b^3*x^12 + 20*a^3*(1 - (b*x^4)/a)
^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^4)/a]))/(77*b^3*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{{x}^{13} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(-b*x^4+a)^(3/4),x)

[Out]

int(x^13/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(-b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^13/(-b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(-b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^13/(-b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 8.52403, size = 29, normalized size = 0.22 \[ \frac{x^{14}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{14 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(-b*x**4+a)**(3/4),x)

[Out]

x**14*hyper((3/4, 7/2), (9/2,), b*x**4*exp_polar(2*I*pi)/a)/(14*a**(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(-b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^13/(-b*x^4 + a)^(3/4), x)