3.829 \(\int \frac{x^6}{\left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{80 a^{7/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a+b x^2\right )^{3/4}}+\frac{40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac{20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac{2 x^5 \sqrt [4]{a+b x^2}}{11 b} \]

[Out]

(40*a^2*x*(a + b*x^2)^(1/4))/(77*b^3) - (20*a*x^3*(a + b*x^2)^(1/4))/(77*b^2) +
(2*x^5*(a + b*x^2)^(1/4))/(11*b) - (80*a^(7/2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[A
rcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.138766, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{80 a^{7/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a+b x^2\right )^{3/4}}+\frac{40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac{20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac{2 x^5 \sqrt [4]{a+b x^2}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^2)^(3/4),x]

[Out]

(40*a^2*x*(a + b*x^2)^(1/4))/(77*b^3) - (20*a*x^3*(a + b*x^2)^(1/4))/(77*b^2) +
(2*x^5*(a + b*x^2)^(1/4))/(11*b) - (80*a^(7/2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[A
rcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 16.3914, size = 114, normalized size = 0.92 \[ - \frac{80 a^{\frac{7}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{7}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{40 a^{2} x \sqrt [4]{a + b x^{2}}}{77 b^{3}} - \frac{20 a x^{3} \sqrt [4]{a + b x^{2}}}{77 b^{2}} + \frac{2 x^{5} \sqrt [4]{a + b x^{2}}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**2+a)**(3/4),x)

[Out]

-80*a**(7/2)*(1 + b*x**2/a)**(3/4)*elliptic_f(atan(sqrt(b)*x/sqrt(a))/2, 2)/(77*
b**(7/2)*(a + b*x**2)**(3/4)) + 40*a**2*x*(a + b*x**2)**(1/4)/(77*b**3) - 20*a*x
**3*(a + b*x**2)**(1/4)/(77*b**2) + 2*x**5*(a + b*x**2)**(1/4)/(11*b)

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Mathematica [C]  time = 0.0763399, size = 90, normalized size = 0.73 \[ \frac{2 \left (-20 a^3 x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+20 a^3 x+10 a^2 b x^3-3 a b^2 x^5+7 b^3 x^7\right )}{77 b^3 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^2)^(3/4),x]

[Out]

(2*(20*a^3*x + 10*a^2*b*x^3 - 3*a*b^2*x^5 + 7*b^3*x^7 - 20*a^3*x*(1 + (b*x^2)/a)
^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(77*b^3*(a + b*x^2)^(3/4
))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{6} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^2+a)^(3/4),x)

[Out]

int(x^6/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^6/(b*x^2 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^6/(b*x^2 + a)^(3/4), x)

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Sympy [A]  time = 3.11487, size = 27, normalized size = 0.22 \[ \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**2+a)**(3/4),x)

[Out]

x**7*hyper((3/4, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/(7*a**(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^2 + a)^(3/4), x)