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

Optimal. Leaf size=202 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}} \]

[Out]

-ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTan[(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]) - Arc
Tanh[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTan
h[(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])

_______________________________________________________________________________________

Rubi [A]  time = 0.103011, antiderivative size = 202, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTan[(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]) - Arc
Tanh[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTan
h[(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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.6769, size = 348, normalized size = 1.72 \[ \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} - \frac{\sqrt{b} x}{\sqrt{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} + \frac{\sqrt{b} x}{\sqrt{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} + \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} b - \frac{b^{\frac{3}{2}} x}{\sqrt{a}} - \frac{\sqrt [3]{2} \sqrt{3} b \sqrt [3]{a + b x^{2}}}{\sqrt [3]{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} b + \frac{b^{\frac{3}{2}} x}{\sqrt{a}} - \frac{\sqrt [3]{2} \sqrt{3} b \sqrt [3]{a + b x^{2}}}{\sqrt [3]{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} - \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \left (\sqrt{3} \sqrt{a} - \sqrt{b} x\right )}{3 \sqrt [6]{a} \sqrt [3]{a + b x^{2}}} \right )}}{12 a^{\frac{5}{6}} \sqrt{b}} + \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \left (\sqrt{3} \sqrt{a} + \sqrt{b} x\right )}{3 \sqrt [6]{a} \sqrt [3]{a + b x^{2}}} \right )}}{12 a^{\frac{5}{6}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*sqrt(3)*log(sqrt(3) - sqrt(b)*x/sqrt(a))/(24*a**(5/6)*sqrt(b)) - 2**(1/
3)*sqrt(3)*log(sqrt(3) + sqrt(b)*x/sqrt(a))/(24*a**(5/6)*sqrt(b)) + 2**(1/3)*sqr
t(3)*log(sqrt(3)*b - b**(3/2)*x/sqrt(a) - 2**(1/3)*sqrt(3)*b*(a + b*x**2)**(1/3)
/a**(1/3))/(24*a**(5/6)*sqrt(b)) - 2**(1/3)*sqrt(3)*log(sqrt(3)*b + b**(3/2)*x/s
qrt(a) - 2**(1/3)*sqrt(3)*b*(a + b*x**2)**(1/3)/a**(1/3))/(24*a**(5/6)*sqrt(b))
- 2**(1/3)*atan(sqrt(3)/3 + 2**(2/3)*(sqrt(3)*sqrt(a) - sqrt(b)*x)/(3*a**(1/6)*(
a + b*x**2)**(1/3)))/(12*a**(5/6)*sqrt(b)) + 2**(1/3)*atan(sqrt(3)/3 + 2**(2/3)*
(sqrt(3)*sqrt(a) + sqrt(b)*x)/(3*a**(1/6)*(a + b*x**2)**(1/3)))/(12*a**(5/6)*sqr
t(b))

_______________________________________________________________________________________

Mathematica [C]  time = 0.233952, size = 166, normalized size = 0.82 \[ \frac{9 a x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2} \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(9*a*x*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), (b*x^2)/(3*a)])/((3*a - b*x^2)*(
a + b*x^2)^(1/3)*(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 = 0.064, 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+3*a)/(b*x^2+a)^(1/3),x)

[Out]

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

_______________________________________________________________________________________

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

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, algorithm="maxima")

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

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)

[Out]

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

_______________________________________________________________________________________

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

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, algorithm="giac")

[Out]

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