3.995 \(\int x^5 \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}+\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b} \]

[Out]

(a*x^2*(a + b*x^4)^(1/4))/(21*b) + (x^6*(a + b*x^4)^(1/4))/7 - (2*a^(5/2)*(1 + (
b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*b^(3/2)*(a +
b*x^4)^(3/4))

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Rubi [A]  time = 0.147767, antiderivative size = 101, 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 \[ -\frac{2 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}+\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^4)^(1/4),x]

[Out]

(a*x^2*(a + b*x^4)^(1/4))/(21*b) + (x^6*(a + b*x^4)^(1/4))/7 - (2*a^(5/2)*(1 + (
b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*b^(3/2)*(a +
b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 14.7811, size = 87, normalized size = 0.86 \[ - \frac{2 a^{\frac{5}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{21 b^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{a x^{2} \sqrt [4]{a + b x^{4}}}{21 b} + \frac{x^{6} \sqrt [4]{a + b x^{4}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**4+a)**(1/4),x)

[Out]

-2*a**(5/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2, 2)/(2
1*b**(3/2)*(a + b*x**4)**(3/4)) + a*x**2*(a + b*x**4)**(1/4)/(21*b) + x**6*(a +
b*x**4)**(1/4)/7

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Mathematica [C]  time = 0.0576033, size = 78, normalized size = 0.77 \[ \frac{x^2 \left (-a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+a^2+4 a b x^4+3 b^2 x^8\right )}{21 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^4)^(1/4),x]

[Out]

(x^2*(a^2 + 4*a*b*x^4 + 3*b^2*x^8 - a^2*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[
1/2, 3/4, 3/2, -((b*x^4)/a)]))/(21*b*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{5}\sqrt [4]{b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^4+a)^(1/4),x)

[Out]

int(x^5*(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^5,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)*x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^5,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)*x^5, x)

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Sympy [A]  time = 3.18603, size = 29, normalized size = 0.29 \[ \frac{\sqrt [4]{a} x^{6}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**6*hyper((-1/4, 3/2), (5/2,), b*x**4*exp_polar(I*pi)/a)/6

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^5,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)*x^5, x)