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

Optimal. Leaf size=561 \[ -\frac{3 \sqrt{2} 3^{3/4} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{6 x}{\sqrt [3]{a-b x^2}}+\frac{9 x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}} \]

[Out]

(6*x)/(a - b*x^2)^(1/3) + (9*x)/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (9
*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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*Sqrt[3]])/(2*b*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)]
) - (3*Sqrt[2]*3^(3/4)*a^(1/3)*(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 - Sq
rt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(b*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 = 0.773649, antiderivative size = 561, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{3 \sqrt{2} 3^{3/4} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{6 x}{\sqrt [3]{a-b x^2}}+\frac{9 x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(6*x)/(a - b*x^2)^(1/3) + (9*x)/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (9
*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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*Sqrt[3]])/(2*b*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)]
) - (3*Sqrt[2]*3^(3/4)*a^(1/3)*(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 - Sq
rt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(b*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 = 32.7853, size = 442, normalized size = 0.79 \[ \frac{9 \sqrt [4]{3} \sqrt [3]{a} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{2 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} - \frac{3 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{a} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} - \frac{9 x}{\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}} + \frac{6 x}{\sqrt [3]{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*3**(1/4)*a**(1/3)*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)
)/(2*b*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)) - 3*sqrt(2)*3**(3/4)*a**(1/3)*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))/(b*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)) - 9*x/(a**(1/3)*(-1 + sqrt(3))
+ (a - b*x**2)**(1/3)) + 6*x/(a - b*x**2)**(1/3)

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Mathematica [C]  time = 0.0507215, size = 51, normalized size = 0.09 \[ -\frac{3 x \left (\sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-2\right )}{\sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{b x^{2} + 3 \, a}{{\left (b x^{2} - a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(b*x^2 + 3*a)/((b*x^2 - a)*(-b*x^2 + a)^(1/3)), x)

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Sympy [A]  time = 9.81075, size = 60, normalized size = 0.11 \[ \frac{3 x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt [3]{a}} + \frac{b x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x*hyper((1/2, 4/3), (3/2,), b*x**2*exp_polar(2*I*pi)/a)/a**(1/3) + b*x**3*hype
r((4/3, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a)/(3*a**(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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