3.1118 \(\int \frac{x^5}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=82 \[ \frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{2 a^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}} \]

[Out]

(x^2*(a + b*x^4)^(1/4))/(3*b) - (2*a^(3/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTa
n[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*b^(3/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.113308, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{2 a^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^4)^(3/4),x]

[Out]

(x^2*(a + b*x^4)^(1/4))/(3*b) - (2*a^(3/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTa
n[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*b^(3/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 11.1281, size = 70, normalized size = 0.85 \[ - \frac{2 a^{\frac{3}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 b^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{x^{2} \sqrt [4]{a + b x^{4}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**4+a)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.0455685, size = 64, normalized size = 0.78 \[ \frac{x^2 \left (-a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+a+b x^4\right )}{3 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^4)^(3/4),x]

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{5} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)^(3/4),x)

[Out]

int(x^5/(b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^5/(b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^5/(b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 2.67918, size = 27, normalized size = 0.33 \[ \frac{x^{6}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**4+a)**(3/4),x)

[Out]

x**6*hyper((3/4, 3/2), (5/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^5/(b*x^4 + a)^(3/4), x)