3.830 \(\int \frac{x^4}{\left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=100 \[ \frac{8 a^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{7 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{4 a x \sqrt [4]{a+b x^2}}{7 b^2}+\frac{2 x^3 \sqrt [4]{a+b x^2}}{7 b} \]

[Out]

(-4*a*x*(a + b*x^2)^(1/4))/(7*b^2) + (2*x^3*(a + b*x^2)^(1/4))/(7*b) + (8*a^(5/2
)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(7*b^(5/2)*
(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.0997195, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{8 a^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{7 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{4 a x \sqrt [4]{a+b x^2}}{7 b^2}+\frac{2 x^3 \sqrt [4]{a+b x^2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(3/4),x]

[Out]

(-4*a*x*(a + b*x^2)^(1/4))/(7*b^2) + (2*x^3*(a + b*x^2)^(1/4))/(7*b) + (8*a^(5/2
)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(7*b^(5/2)*
(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 12.0548, size = 90, normalized size = 0.9 \[ \frac{8 a^{\frac{5}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{7 b^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{4 a x \sqrt [4]{a + b x^{2}}}{7 b^{2}} + \frac{2 x^{3} \sqrt [4]{a + b x^{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.062158, size = 78, normalized size = 0.78 \[ \frac{2 \left (2 a^2 x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )-2 a^2 x-a b x^3+b^2 x^5\right )}{7 b^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^2)^(3/4),x]

[Out]

(2*(-2*a^2*x - a*b*x^3 + b^2*x^5 + 2*a^2*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2
F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(7*b^2*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(3/4),x)

[Out]

int(x^4/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^4/(b*x^2 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^4/(b*x^2 + a)^(3/4), x)

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Sympy [A]  time = 2.6327, size = 27, normalized size = 0.27 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^2 + a)^(3/4), x)