3.1146 \(\int \frac{x^9}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{12 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a+b x^4}}-\frac{6 a x^2}{5 b^2 \sqrt [4]{a+b x^4}}+\frac{x^6}{5 b \sqrt [4]{a+b x^4}} \]

[Out]

(-6*a*x^2)/(5*b^2*(a + b*x^4)^(1/4)) + x^6/(5*b*(a + b*x^4)^(1/4)) + (12*a^(3/2)
*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*b^(5/2)
*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.153423, 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 \[ \frac{12 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a+b x^4}}-\frac{6 a x^2}{5 b^2 \sqrt [4]{a+b x^4}}+\frac{x^6}{5 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^4)^(5/4),x]

[Out]

(-6*a*x^2)/(5*b^2*(a + b*x^4)^(1/4)) + x^6/(5*b*(a + b*x^4)^(1/4)) + (12*a^(3/2)
*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*b^(5/2)
*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{6 a^{2} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{5 b^{2}} - \frac{6 a x^{2}}{5 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{6}}{5 b \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)**(5/4),x)

[Out]

6*a**2*Integral((a + b*x**2)**(-5/4), (x, x**2))/(5*b**2) - 6*a*x**2/(5*b**2*(a
+ b*x**4)**(1/4)) + x**6/(5*b*(a + b*x**4)**(1/4))

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Mathematica [C]  time = 0.0583521, size = 66, normalized size = 0.63 \[ \frac{x^2 \left (-6 a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+6 a+b x^4\right )}{5 b^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^4)^(5/4),x]

[Out]

(x^2*(6*a + b*x^4 - 6*a*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -
((b*x^4)/a)]))/(5*b^2*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{{x}^{9} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)^(5/4),x)

[Out]

int(x^9/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate(x^9/(b*x^4 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(x^9/(b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 4.30363, size = 27, normalized size = 0.26 \[ \frac{x^{10}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^9/(b*x^4 + a)^(5/4), x)