3.1087 \(\int \frac{x^9}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{4 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2}{15 b^2 \sqrt [4]{a+b x^4}}-\frac{2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac{x^6 \left (a+b x^4\right )^{3/4}}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.177607, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{4 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2}{15 b^2 \sqrt [4]{a+b x^4}}-\frac{2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac{x^6 \left (a+b x^4\right )^{3/4}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^4)^(1/4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.059169, size = 80, normalized size = 0.62 \[ \frac{x^2 \left (6 a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-6 a^2-a b x^4+5 b^2 x^8\right )}{45 b^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^4)^(1/4),x]

[Out]

(x^2*(-6*a^2 - a*b*x^4 + 5*b^2*x^8 + 6*a^2*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2
F1[1/4, 1/2, 3/2, -((b*x^4)/a)]))/(45*b^2*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{9}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)^(1/4),x)

[Out]

int(x^9/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^9/(b*x^4 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^9/(b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 4.41651, size = 27, normalized size = 0.21 \[ \frac{x^{10}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)**(1/4),x)

[Out]

x**10*hyper((1/4, 5/2), (7/2,), b*x**4*exp_polar(I*pi)/a)/(10*a**(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^9/(b*x^4 + a)^(1/4), x)