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

Optimal. Leaf size=601 \[ -\frac{108 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{1729 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{162 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{1729 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{324 a^3 x}{1729 b^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac{108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac{3}{19} x^5 \left (a+b x^2\right )^{2/3}+\frac{12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b} \]

[Out]

(-108*a^2*x*(a + b*x^2)^(2/3))/(1729*b^2) + (12*a*x^3*(a + b*x^2)^(2/3))/(247*b)
+ (3*x^5*(a + b*x^2)^(2/3))/19 - (324*a^3*x)/(1729*b^2*((1 - Sqrt[3])*a^(1/3) -
(a + b*x^2)^(1/3))) + (162*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10/3)*(a^(1/3) - (a + b
*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1
- Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/
3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqr
t[3]])/(1729*b^3*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])
*a^(1/3) - (a + b*x^2)^(1/3))^2)]) - (108*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) - (a
+ b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/
((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a
^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4
*Sqrt[3]])/(1729*b^3*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt
[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

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Rubi [A]  time = 1.03129, antiderivative size = 601, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{108 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{1729 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{162 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{1729 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{324 a^3 x}{1729 b^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac{108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac{3}{19} x^5 \left (a+b x^2\right )^{2/3}+\frac{12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^2)^(2/3),x]

[Out]

(-108*a^2*x*(a + b*x^2)^(2/3))/(1729*b^2) + (12*a*x^3*(a + b*x^2)^(2/3))/(247*b)
+ (3*x^5*(a + b*x^2)^(2/3))/19 - (324*a^3*x)/(1729*b^2*((1 - Sqrt[3])*a^(1/3) -
(a + b*x^2)^(1/3))) + (162*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10/3)*(a^(1/3) - (a + b
*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1
- Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/
3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqr
t[3]])/(1729*b^3*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])
*a^(1/3) - (a + b*x^2)^(1/3))^2)]) - (108*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) - (a
+ b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/
((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a
^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4
*Sqrt[3]])/(1729*b^3*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt
[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 46.8648, size = 502, normalized size = 0.84 \[ \frac{162 \sqrt [4]{3} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1729 b^{3} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} - \frac{108 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1729 b^{3} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} + \frac{324 a^{3} x}{1729 b^{2} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )} - \frac{108 a^{2} x \left (a + b x^{2}\right )^{\frac{2}{3}}}{1729 b^{2}} + \frac{12 a x^{3} \left (a + b x^{2}\right )^{\frac{2}{3}}}{247 b} + \frac{3 x^{5} \left (a + b x^{2}\right )^{\frac{2}{3}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

162*3**(1/4)*a**(10/3)*sqrt((a**(2/3) + a**(1/3)*(a + b*x**2)**(1/3) + (a + b*x*
*2)**(2/3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)*sqrt(sqrt(3) + 2
)*(a**(1/3) - (a + b*x**2)**(1/3))*elliptic_e(asin((a**(1/3)*(1 + sqrt(3)) - (a
+ b*x**2)**(1/3))/(-a**(1/3)*(-1 + sqrt(3)) - (a + b*x**2)**(1/3))), -7 + 4*sqrt
(3))/(1729*b**3*x*sqrt(-a**(1/3)*(a**(1/3) - (a + b*x**2)**(1/3))/(a**(1/3)*(-1
+ sqrt(3)) + (a + b*x**2)**(1/3))**2)) - 108*sqrt(2)*3**(3/4)*a**(10/3)*sqrt((a*
*(2/3) + a**(1/3)*(a + b*x**2)**(1/3) + (a + b*x**2)**(2/3))/(a**(1/3)*(-1 + sqr
t(3)) + (a + b*x**2)**(1/3))**2)*(a**(1/3) - (a + b*x**2)**(1/3))*elliptic_f(asi
n((a**(1/3)*(1 + sqrt(3)) - (a + b*x**2)**(1/3))/(-a**(1/3)*(-1 + sqrt(3)) - (a
+ b*x**2)**(1/3))), -7 + 4*sqrt(3))/(1729*b**3*x*sqrt(-a**(1/3)*(a**(1/3) - (a +
b*x**2)**(1/3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)) + 324*a**3
*x/(1729*b**2*(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))) - 108*a**2*x*(a +
b*x**2)**(2/3)/(1729*b**2) + 12*a*x**3*(a + b*x**2)**(2/3)/(247*b) + 3*x**5*(a
+ b*x**2)**(2/3)/19

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Mathematica [C]  time = 0.0583495, size = 90, normalized size = 0.15 \[ \frac{3 \left (36 a^3 x \sqrt [3]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-36 a^3 x-8 a^2 b x^3+119 a b^2 x^5+91 b^3 x^7\right )}{1729 b^2 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(-36*a^3*x - 8*a^2*b*x^3 + 119*a*b^2*x^5 + 91*b^3*x^7 + 36*a^3*x*(1 + (b*x^2)
/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(1729*b^2*(a + b*x^2)
^(1/3))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(2/3)*x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(2/3)*x^4, x)

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Sympy [A]  time = 3.28974, size = 29, normalized size = 0.05 \[ \frac{a^{\frac{2}{3}} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(2/3)*x^4, x)