3.1117 \(\int \frac{x^9}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{4 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a x^2 \sqrt [4]{a+b x^4}}{7 b^2}+\frac{x^6 \sqrt [4]{a+b x^4}}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.151385, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{4 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a x^2 \sqrt [4]{a+b x^4}}{7 b^2}+\frac{x^6 \sqrt [4]{a+b x^4}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.8906, size = 92, normalized size = 0.88 \[ \frac{4 a^{\frac{5}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{7 b^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{2 a x^{2} \sqrt [4]{a + b x^{4}}}{7 b^{2}} + \frac{x^{6} \sqrt [4]{a + b x^{4}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.0585937, size = 79, normalized size = 0.76 \[ \frac{x^2 \left (2 a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 a^2-a b x^4+b^2 x^8\right )}{7 b^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^4)^(3/4),x]

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{9} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)^(3/4),x)

[Out]

int(x^9/(b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^9/(b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^9/(b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 4.27567, size = 27, normalized size = 0.26 \[ \frac{x^{10}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)**(3/4),x)

[Out]

x**10*hyper((3/4, 5/2), (7/2,), b*x**4*exp_polar(I*pi)/a)/(10*a**(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^9/(b*x^4 + a)^(3/4), x)