3.138 \(\int \frac{3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=590 \[ -\frac{3\ 3^{3/4} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{2} a^{2/3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8 a^{2/3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{9 x}{4 a \sqrt [3]{a-b x^2}}+\frac{9 x}{4 a \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{3 x}{2 \left (a-b x^2\right )^{4/3}} \]

[Out]

(3*x)/(2*(a - b*x^2)^(4/3)) + (9*x)/(4*a*(a - b*x^2)^(1/3)) + (9*x)/(4*a*((1 - S
qrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (9*3^(1/4)*Sqrt[2 + Sqrt[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]])/(8*a^(2/3)*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)]) - (3*3^(3/4)*(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]])
/(2*Sqrt[2]*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqr
t[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

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Rubi [A]  time = 0.790696, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3\ 3^{3/4} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{2} a^{2/3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8 a^{2/3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{9 x}{4 a \sqrt [3]{a-b x^2}}+\frac{9 x}{4 a \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{3 x}{2 \left (a-b x^2\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]  Int[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]

[Out]

(3*x)/(2*(a - b*x^2)^(4/3)) + (9*x)/(4*a*(a - b*x^2)^(1/3)) + (9*x)/(4*a*((1 - S
qrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (9*3^(1/4)*Sqrt[2 + Sqrt[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]])/(8*a^(2/3)*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)]) - (3*3^(3/4)*(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]])
/(2*Sqrt[2]*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqr
t[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 42.61, size = 466, normalized size = 0.79 \[ \frac{3 x}{2 \left (a - b x^{2}\right )^{\frac{4}{3}}} - \frac{9 x}{4 a \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )} + \frac{9 x}{4 a \sqrt [3]{a - b x^{2}}} + \frac{9 \sqrt [4]{3} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{8 a^{\frac{2}{3}} b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} - \frac{3 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{4 a^{\frac{2}{3}} b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+3*a)/(-b*x**2+a)**(7/3),x)

[Out]

3*x/(2*(a - b*x**2)**(4/3)) - 9*x/(4*a*(a**(1/3)*(-1 + sqrt(3)) + (a - b*x**2)**
(1/3))) + 9*x/(4*a*(a - b*x**2)**(1/3)) + 9*3**(1/4)*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))/(8*a**(2/3)*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)) - 3*sqrt
(2)*3**(3/4)*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))/(4*a**(2/3)*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))

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Mathematica [C]  time = 0.0871279, size = 74, normalized size = 0.13 \[ \frac{-3 x \left (a-b x^2\right ) \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+15 a x-9 b x^3}{4 a \left (a-b x^2\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]

[Out]

(15*a*x - 9*b*x^3 - 3*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3,
1/2, 3/2, (b*x^2)/a])/(4*a*(a - b*x^2)^(4/3))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{(b{x}^{2}+3\,a) \left ( -b{x}^{2}+a \right ) ^{-{\frac{7}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+3*a)/(-b*x^2+a)^(7/3),x)

[Out]

int((b*x^2+3*a)/(-b*x^2+a)^(7/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + 3 \, a}{{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x^{2} + 3 \, a}{{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3),x, algorithm="fricas")

[Out]

integral((b*x^2 + 3*a)/((b^2*x^4 - 2*a*b*x^2 + a^2)*(-b*x^2 + a)^(1/3)), x)

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Sympy [A]  time = 36.9502, size = 60, normalized size = 0.1 \[ \frac{3 x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac{4}{3}}} + \frac{b x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{3} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+3*a)/(-b*x**2+a)**(7/3),x)

[Out]

3*x*hyper((1/2, 7/3), (3/2,), b*x**2*exp_polar(2*I*pi)/a)/a**(4/3) + b*x**3*hype
r((3/2, 7/3), (5/2,), b*x**2*exp_polar(2*I*pi)/a)/(3*a**(7/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + 3 \, a}{{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3), x)