3.1132 \(\int \frac{x^4}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=83 \[ \frac{x \sqrt [4]{a+b x^4}}{2 b}+\frac{\sqrt{a} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \left (a+b x^4\right )^{3/4}} \]

[Out]

(x*(a + b*x^4)^(1/4))/(2*b) + (Sqrt[a]*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCo
t[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*Sqrt[b]*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.102876, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \sqrt [4]{a+b x^4}}{2 b}+\frac{\sqrt{a} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^4)^(3/4),x]

[Out]

(x*(a + b*x^4)^(1/4))/(2*b) + (Sqrt[a]*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCo
t[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*Sqrt[b]*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 11.8845, size = 70, normalized size = 0.84 \[ \frac{\sqrt{a} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{2 \sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{x \sqrt [4]{a + b x^{4}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2,
2)/(2*sqrt(b)*(a + b*x**4)**(3/4)) + x*(a + b*x**4)**(1/4)/(2*b)

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Mathematica [C]  time = 0.042783, size = 62, normalized size = 0.75 \[ \frac{x \left (-a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{2 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^4+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^4 + a)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 2.44469, size = 37, normalized size = 0.45 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((3/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/4)*
gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^4 + a)^(3/4), x)