3.1024 \(\int x^5 \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{2 a^2 x^2}{15 b \sqrt [4]{a+b x^4}}+\frac{1}{9} x^6 \left (a+b x^4\right )^{3/4}+\frac{a x^2 \left (a+b x^4\right )^{3/4}}{15 b} \]

[Out]

(-2*a^2*x^2)/(15*b*(a + b*x^4)^(1/4)) + (a*x^2*(a + b*x^4)^(3/4))/(15*b) + (x^6*
(a + b*x^4)^(3/4))/9 + (2*a^(5/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]]/2, 2])/(15*b^(3/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.171353, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{2 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{2 a^2 x^2}{15 b \sqrt [4]{a+b x^4}}+\frac{1}{9} x^6 \left (a+b x^4\right )^{3/4}+\frac{a x^2 \left (a+b x^4\right )^{3/4}}{15 b} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^4)^(3/4),x]

[Out]

(-2*a^2*x^2)/(15*b*(a + b*x^4)^(1/4)) + (a*x^2*(a + b*x^4)^(3/4))/(15*b) + (x^6*
(a + b*x^4)^(3/4))/9 + (2*a^(5/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]]/2, 2])/(15*b^(3/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{15 b} + \frac{a x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{15 b} + \frac{x^{6} \left (a + b x^{4}\right )^{\frac{3}{4}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*Integral((a + b*x**2)**(-1/4), (x, x**2))/(15*b) + a*x**2*(a + b*x**4)**(3
/4)/(15*b) + x**6*(a + b*x**4)**(3/4)/9

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Mathematica [C]  time = 0.0612009, size = 80, normalized size = 0.64 \[ \frac{x^2 \left (-3 a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+3 a^2+8 a b x^4+5 b^2 x^8\right )}{45 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(3*a^2 + 8*a*b*x^4 + 5*b^2*x^8 - 3*a^2*(1 + (b*x^4)/a)^(1/4)*Hypergeometric
2F1[1/4, 1/2, 3/2, -((b*x^4)/a)]))/(45*b*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.038, 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. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)*x^5, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(3/4)*x^5, x)

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Sympy [A]  time = 5.80497, size = 29, normalized size = 0.23 \[ \frac{a^{\frac{3}{4}} x^{6}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*x**6*hyper((-3/4, 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. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)*x^5, x)