3.120 \(\int \frac{\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=775 \[ \frac{11 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{11 \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac{11 x}{3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}} \]

[Out]

(2*x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)) - (11*x)/(3*((1 - Sqrt[3])*a^(1/3) - (
a - b*x^2)^(1/3))) - (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(Sq
rt[3]*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a
- b*x^2)^(1/3)))/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) + (2^(1/3)*a^(1/6)*ArcTanh[(Sqr
t[b]*x)/Sqrt[a]])/(3*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a
^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/Sqrt[b] - (11*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*3^(3/4)*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)]) + (11*Sqrt[2]*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 + S
qrt[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]])/(3*3^(1/4)*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 = 1.05508, antiderivative size = 775, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{11 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{11 \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{2 x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )}-\frac{11 x}{3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a - b*x^2)^(5/3)/(3*a + b*x^2)^2,x]

[Out]

(2*x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)) - (11*x)/(3*((1 - Sqrt[3])*a^(1/3) - (
a - b*x^2)^(1/3))) - (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(Sq
rt[3]*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a
- b*x^2)^(1/3)))/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) + (2^(1/3)*a^(1/6)*ArcTanh[(Sqr
t[b]*x)/Sqrt[a]])/(3*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a
^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/Sqrt[b] - (11*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*3^(3/4)*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)]) + (11*Sqrt[2]*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 + S
qrt[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]])/(3*3^(1/4)*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 = 158.993, size = 814, normalized size = 1.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(5/3)/(b*x**2+3*a)**2,x)

[Out]

-11*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))/(6*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)) + 11*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))/(9*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)) - 2**(1/3)*sqrt(a)*(1 - b
*x**2/a)**(1/3)*log(2**(1/3)*(1 - sqrt(b)*x/sqrt(a))**(1/3) + (1 + sqrt(b)*x/sqr
t(a))**(2/3))/(2*sqrt(b)*(a - b*x**2)**(1/3)) + 2**(1/3)*sqrt(a)*(1 - b*x**2/a)*
*(1/3)*log((1 - sqrt(b)*x/sqrt(a))**(2/3) + 2**(1/3)*(1 + sqrt(b)*x/sqrt(a))**(1
/3))/(2*sqrt(b)*(a - b*x**2)**(1/3)) + 2**(1/3)*sqrt(3)*sqrt(a)*(1 - b*x**2/a)**
(1/3)*atan(sqrt(3)/3 - 2**(2/3)*sqrt(3)*(1 + sqrt(b)*x/sqrt(a))**(2/3)/(3*(1 - s
qrt(b)*x/sqrt(a))**(1/3)))/(3*sqrt(b)*(a - b*x**2)**(1/3)) + 2**(1/3)*sqrt(3)*sq
rt(a)*(1 - b*x**2/a)**(1/3)*atan(2**(2/3)*sqrt(3)*(1 - sqrt(b)*x/sqrt(a))**(2/3)
/(3*(1 + sqrt(b)*x/sqrt(a))**(1/3)) - sqrt(3)/3)/(3*sqrt(b)*(a - b*x**2)**(1/3))
+ 2*x*(a - b*x**2)**(2/3)/(3*(3*a + b*x**2)) + 11*x/(3*(a**(1/3)*(-1 + sqrt(3))
+ (a - b*x**2)**(1/3)))

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Mathematica [C]  time = 0.227754, size = 320, normalized size = 0.41 \[ \frac{x \left (-\frac{27 a^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}+\frac{55 a b x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+15 a F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}+6 a-6 b x^2\right )}{9 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a - b*x^2)^(5/3)/(3*a + b*x^2)^2,x]

[Out]

(x*(6*a - 6*b*x^2 - (27*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)
])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b*x^2*(-Appell
F1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + AppellF1[3/2, 4/3, 1, 5/2, (b*
x^2)/a, -(b*x^2)/(3*a)])) + (55*a*b*x^2*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(
b*x^2)/(3*a)])/(15*a*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b
*x^2*(-AppellF1[5/2, 1/3, 2, 7/2, (b*x^2)/a, -(b*x^2)/(3*a)] + AppellF1[5/2, 4/3
, 1, 7/2, (b*x^2)/a, -(b*x^2)/(3*a)]))))/(9*(a - b*x^2)^(1/3)*(3*a + b*x^2))

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( b{x}^{2}+3\,a \right ) ^{2}} \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(5/3)/(b*x^2+3*a)^2,x)

[Out]

int((-b*x^2+a)^(5/3)/(b*x^2+3*a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{5}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^2,x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^2, x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a - b x^{2}\right )^{\frac{5}{3}}}{\left (3 a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(5/3)/(b*x**2+3*a)**2,x)

[Out]

Integral((a - b*x**2)**(5/3)/(3*a + b*x**2)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{5}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^2,x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^2, x)