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

Optimal. Leaf size=125 \[ \frac{4 a^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^2}+\frac{1}{11} x^{10} \sqrt [4]{a+b x^4}+\frac{a x^6 \sqrt [4]{a+b x^4}}{77 b} \]

[Out]

(-2*a^2*x^2*(a + b*x^4)^(1/4))/(77*b^2) + (a*x^6*(a + b*x^4)^(1/4))/(77*b) + (x^
10*(a + b*x^4)^(1/4))/11 + (4*a^(7/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sq
rt[b]*x^2)/Sqrt[a]]/2, 2])/(77*b^(5/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.200105, antiderivative size = 125, 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 \[ \frac{4 a^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^2}+\frac{1}{11} x^{10} \sqrt [4]{a+b x^4}+\frac{a x^6 \sqrt [4]{a+b x^4}}{77 b} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a + b*x^4)^(1/4),x]

[Out]

(-2*a^2*x^2*(a + b*x^4)^(1/4))/(77*b^2) + (a*x^6*(a + b*x^4)^(1/4))/(77*b) + (x^
10*(a + b*x^4)^(1/4))/11 + (4*a^(7/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sq
rt[b]*x^2)/Sqrt[a]]/2, 2])/(77*b^(5/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 19.3451, size = 110, normalized size = 0.88 \[ \frac{4 a^{\frac{7}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{2 a^{2} x^{2} \sqrt [4]{a + b x^{4}}}{77 b^{2}} + \frac{a x^{6} \sqrt [4]{a + b x^{4}}}{77 b} + \frac{x^{10} \sqrt [4]{a + b x^{4}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(7/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2, 2)/(77
*b**(5/2)*(a + b*x**4)**(3/4)) - 2*a**2*x**2*(a + b*x**4)**(1/4)/(77*b**2) + a*x
**6*(a + b*x**4)**(1/4)/(77*b) + x**10*(a + b*x**4)**(1/4)/11

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Mathematica [C]  time = 0.0680639, size = 91, normalized size = 0.73 \[ \frac{x^2 \left (2 a^3 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 a^3-a^2 b x^4+8 a b^2 x^8+7 b^3 x^{12}\right )}{77 b^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(-2*a^3 - a^2*b*x^4 + 8*a*b^2*x^8 + 7*b^3*x^12 + 2*a^3*(1 + (b*x^4)/a)^(3/4
)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)]))/(77*b^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{x}^{9}\sqrt [4]{b{x}^{4}+a}\, dx \]

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. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)*x^9, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(1/4)*x^9, x)

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Sympy [A]  time = 5.82783, size = 29, normalized size = 0.23 \[ \frac{\sqrt [4]{a} x^{10}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)*x^9, x)