3.126 \(\int \frac{1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx\)

Optimal. Leaf size=204 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) + ArcT
an[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(2*2^(2/
3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sq
rt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(2
*2^(2/3)*a^(5/6)*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.127058, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) + ArcT
an[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(2*2^(2/
3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sq
rt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(2
*2^(2/3)*a^(5/6)*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 73.7032, size = 355, normalized size = 1.74 \[ \frac{\sqrt [3]{2} \sqrt [3]{1 - \frac{b x^{2}}{a}} \log{\left (\sqrt [3]{2} \sqrt [3]{1 - \frac{\sqrt{b} x}{\sqrt{a}}} + \left (1 + \frac{\sqrt{b} x}{\sqrt{a}}\right )^{\frac{2}{3}} \right )}}{8 \sqrt{a} \sqrt{b} \sqrt [3]{a - b x^{2}}} - \frac{\sqrt [3]{2} \sqrt [3]{1 - \frac{b x^{2}}{a}} \log{\left (\left (1 - \frac{\sqrt{b} x}{\sqrt{a}}\right )^{\frac{2}{3}} + \sqrt [3]{2} \sqrt [3]{1 + \frac{\sqrt{b} x}{\sqrt{a}}} \right )}}{8 \sqrt{a} \sqrt{b} \sqrt [3]{a - b x^{2}}} - \frac{\sqrt [3]{2} \sqrt{3} \sqrt [3]{1 - \frac{b x^{2}}{a}} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (1 + \frac{\sqrt{b} x}{\sqrt{a}}\right )^{\frac{2}{3}}}{3 \sqrt [3]{1 - \frac{\sqrt{b} x}{\sqrt{a}}}} \right )}}{12 \sqrt{a} \sqrt{b} \sqrt [3]{a - b x^{2}}} - \frac{\sqrt [3]{2} \sqrt{3} \sqrt [3]{1 - \frac{b x^{2}}{a}} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (1 - \frac{\sqrt{b} x}{\sqrt{a}}\right )^{\frac{2}{3}}}{3 \sqrt [3]{1 + \frac{\sqrt{b} x}{\sqrt{a}}}} - \frac{\sqrt{3}}{3} \right )}}{12 \sqrt{a} \sqrt{b} \sqrt [3]{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**2+a)**(1/3)/(b*x**2+3*a),x)

[Out]

2**(1/3)*(1 - b*x**2/a)**(1/3)*log(2**(1/3)*(1 - sqrt(b)*x/sqrt(a))**(1/3) + (1
+ sqrt(b)*x/sqrt(a))**(2/3))/(8*sqrt(a)*sqrt(b)*(a - b*x**2)**(1/3)) - 2**(1/3)*
(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))/(8*sqrt(a)*sqrt(b)*(a - b*x**2)**(1/3)) - 2**(1/3)*sqrt(3)*(
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 - sqrt(b)*x/sqrt(a))**(1/3)))/(12*sqrt(a)*sqrt(b)*(a - b*x**2)**(1/3
)) - 2**(1/3)*sqrt(3)*(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)/(12*sqrt(a)*sqr
t(b)*(a - b*x**2)**(1/3))

_______________________________________________________________________________________

Mathematica [C]  time = 0.063994, size = 162, normalized size = 0.79 \[ \frac{9 a x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(9*a*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)])/((a - b*x^2)^(1/3)
*(3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b*
x^2*(-AppellF1[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)])))

_______________________________________________________________________________________

Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{2}+3\,a}{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(1/3)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**2+a)**(1/3)/(b*x**2+3*a),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(1/3)),x, algorithm="giac")

[Out]

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