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

Optimal. Leaf size=577 \[ \frac{27\ 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{2} b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{28 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{27 x \left (a+b x^2\right )^{2/3}}{14 b^2}+\frac{81 a x}{14 b^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac{3 x^3}{2 b \sqrt [3]{a+b x^2}} \]

[Out]

(-3*x^3)/(2*b*(a + b*x^2)^(1/3)) + (27*x*(a + b*x^2)^(2/3))/(14*b^2) + (81*a*x)/
(14*b^2*((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))) - (81*3^(1/4)*Sqrt[2 + Sqrt
[3]]*a^(4/3)*(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]*Ellipti
cE[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]])/(28*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)]) + (27*3^(3/4)*a^(
4/3)*(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[ArcSi
n[((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]])/(7*Sqrt[2]*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 = 0.872946, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{27\ 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{2} b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{28 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{27 x \left (a+b x^2\right )^{2/3}}{14 b^2}+\frac{81 a x}{14 b^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac{3 x^3}{2 b \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(4/3),x]

[Out]

(-3*x^3)/(2*b*(a + b*x^2)^(1/3)) + (27*x*(a + b*x^2)^(2/3))/(14*b^2) + (81*a*x)/
(14*b^2*((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))) - (81*3^(1/4)*Sqrt[2 + Sqrt
[3]]*a^(4/3)*(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]*Ellipti
cE[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]])/(28*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)]) + (27*3^(3/4)*a^(
4/3)*(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[ArcSi
n[((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]])/(7*Sqrt[2]*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 = 38.3236, size = 478, normalized size = 0.83 \[ - \frac{81 \sqrt [4]{3} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{28 b^{3} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} + \frac{27 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{14 b^{3} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} - \frac{81 a x}{14 b^{2} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )} - \frac{3 x^{3}}{2 b \sqrt [3]{a + b x^{2}}} + \frac{27 x \left (a + b x^{2}\right )^{\frac{2}{3}}}{14 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81*3**(1/4)*a**(4/3)*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))/(28*b**3*x*sqrt(-a**(1/3)*(a**(1/3) - (a + b*x**2)**(1/3))/(a**(1/3)*(-1 + s
qrt(3)) + (a + b*x**2)**(1/3))**2)) + 27*sqrt(2)*3**(3/4)*a**(4/3)*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)*(a**(1/3) - (a + b*x**2)**(1/3))*elliptic_f(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))/(14*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)) - 81*a*x/(14*b**2
*(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))) - 3*x**3/(2*b*(a + b*x**2)**(1
/3)) + 27*x*(a + b*x**2)**(2/3)/(14*b**2)

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Mathematica [C]  time = 0.0617138, size = 65, normalized size = 0.11 \[ \frac{3 x \left (-9 a \sqrt [3]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+9 a+2 b x^2\right )}{14 b^2 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*(9*a + 2*b*x^2 - 9*a*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2,
-((b*x^2)/a)]))/(14*b^2*(a + b*x^2)^(1/3))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(4/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^2 + a)^(4/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(b*x^2 + a)^(4/3), x)

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Sympy [A]  time = 2.58212, size = 27, normalized size = 0.05 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^2 + a)^(4/3), x)