3.1076 \(\int \frac{x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]

[Out]

ArcTan[(Sqrt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2)) - ArcTanh[(Sq
rt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0990811, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)),x]

[Out]

ArcTan[(Sqrt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2)) - ArcTanh[(Sq
rt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.5794, size = 46, normalized size = 0.62 \[ \frac{x^{3} \sqrt [4]{- b x^{2} - 1} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- b x^{2},- \frac{b x^{2}}{2} \right )}}{6 \sqrt [4]{b x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-b*x**2-2)/(-b*x**2-1)**(3/4),x)

[Out]

x**3*(-b*x**2 - 1)**(1/4)*appellf1(3/2, 3/4, 1, 5/2, -b*x**2, -b*x**2/2)/(6*(b*x
**2 + 1)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.242502, size = 143, normalized size = 1.93 \[ \frac{10 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )}{3 \left (-b x^2-1\right )^{3/4} \left (b x^2+2\right ) \left (b x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-b x^2,-\frac{b x^2}{2}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-b x^2,-\frac{b x^2}{2}\right )\right )-10 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)),x]

[Out]

(10*x^3*AppellF1[3/2, 3/4, 1, 5/2, -(b*x^2), -(b*x^2)/2])/(3*(-1 - b*x^2)^(3/4)*
(2 + b*x^2)*(-10*AppellF1[3/2, 3/4, 1, 5/2, -(b*x^2), -(b*x^2)/2] + b*x^2*(2*App
ellF1[5/2, 3/4, 2, 7/2, -(b*x^2), -(b*x^2)/2] + 3*AppellF1[5/2, 7/4, 1, 7/2, -(b
*x^2), -(b*x^2)/2])))

_______________________________________________________________________________________

Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{-b{x}^{2}-2} \left ( -b{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x)

[Out]

int(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2}}{{\left (b x^{2} + 2\right )}{\left (-b x^{2} - 1\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((b*x^2 + 2)*(-b*x^2 - 1)^(3/4)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.233858, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}}}{\sqrt{b} x}\right ) - \log \left (\frac{4 \,{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} b x - \sqrt{2}{\left (b x^{2} + 2 \, \sqrt{-b x^{2} - 1}\right )} \sqrt{b}}{b x^{2} - 2 \, \sqrt{-b x^{2} - 1}}\right )\right )}}{4 \, b^{\frac{3}{2}}}, -\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b}}{b x}\right ) - \log \left (-\frac{4 \,{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} b x + \sqrt{2}{\left (b x^{2} - 2 \, \sqrt{-b x^{2} - 1}\right )} \sqrt{-b}}{b x^{2} + 2 \, \sqrt{-b x^{2} - 1}}\right )\right )}}{4 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((b*x^2 + 2)*(-b*x^2 - 1)^(3/4)),x, algorithm="fricas")

[Out]

[-1/4*sqrt(2)*(2*arctan(sqrt(2)*(-b*x^2 - 1)^(1/4)/(sqrt(b)*x)) - log((4*(-b*x^2
- 1)^(1/4)*b*x - sqrt(2)*(b*x^2 + 2*sqrt(-b*x^2 - 1))*sqrt(b))/(b*x^2 - 2*sqrt(
-b*x^2 - 1))))/b^(3/2), -1/4*sqrt(2)*(2*arctan(sqrt(2)*(-b*x^2 - 1)^(1/4)*sqrt(-
b)/(b*x)) - log(-(4*(-b*x^2 - 1)^(1/4)*b*x + sqrt(2)*(b*x^2 - 2*sqrt(-b*x^2 - 1)
)*sqrt(-b))/(b*x^2 + 2*sqrt(-b*x^2 - 1))))/(sqrt(-b)*b)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{b x^{2} \left (- b x^{2} - 1\right )^{\frac{3}{4}} + 2 \left (- b x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-b*x**2-2)/(-b*x**2-1)**(3/4),x)

[Out]

-Integral(x**2/(b*x**2*(-b*x**2 - 1)**(3/4) + 2*(-b*x**2 - 1)**(3/4)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2}}{{\left (b x^{2} + 2\right )}{\left (-b x^{2} - 1\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((b*x^2 + 2)*(-b*x^2 - 1)^(3/4)),x, algorithm="giac")

[Out]

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