\(\int \frac {1}{(-2 a-b x^2) \sqrt [4]{-a-b x^2}} \, dx\) [319]

Optimal result

Integrand size = 27, antiderivative size = 103 \[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]

[Out]

-1/4*arctan(1/2*x*b^(1/2)/a^(1/4)/(-b*x^2-a)^(1/4)*2^(1/2))/a^(3/4)*2^(1/2)/b^(1/2)-1/4*arctanh(1/2*x*b^(1/2)/
a^(1/4)/(-b*x^2-a)^(1/4)*2^(1/2))/a^(3/4)*2^(1/2)/b^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {407} \[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]

[In]

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

[Out]

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

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}{\sqrt {b} x}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}{\sqrt {b} x}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {1}{\left (-b \,x^{2}-2 a \right ) \left (-b \,x^{2}-a \right )^{\frac {1}{4}}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 20.42 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.55 \[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} + i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {-2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) \]

[In]

integrate(1/(-b*x^2-2*a)/(-b*x^2-a)^(1/4),x, algorithm="fricas")

[Out]

1/4*(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*log(-(2*(1/4)^(3/4)*sqrt(-b*x^2 - a)*a^2*b^2*x*(1/(a^3*b^2))^(3/4) + (-b*x
^2 - a)^(1/4)*a^2*b*sqrt(1/(a^3*b^2)) - (1/4)^(1/4)*a*b*x*(1/(a^3*b^2))^(1/4) - (-b*x^2 - a)^(3/4))/(b*x^2 + 2
*a)) - 1/4*(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*log((2*(1/4)^(3/4)*sqrt(-b*x^2 - a)*a^2*b^2*x*(1/(a^3*b^2))^(3/4) -
(-b*x^2 - a)^(1/4)*a^2*b*sqrt(1/(a^3*b^2)) - (1/4)^(1/4)*a*b*x*(1/(a^3*b^2))^(1/4) + (-b*x^2 - a)^(3/4))/(b*x
^2 + 2*a)) + 1/4*I*(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*log((2*I*(1/4)^(3/4)*sqrt(-b*x^2 - a)*a^2*b^2*x*(1/(a^3*b^2
))^(3/4) + (-b*x^2 - a)^(1/4)*a^2*b*sqrt(1/(a^3*b^2)) + I*(1/4)^(1/4)*a*b*x*(1/(a^3*b^2))^(1/4) + (-b*x^2 - a)
^(3/4))/(b*x^2 + 2*a)) - 1/4*I*(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*log((-2*I*(1/4)^(3/4)*sqrt(-b*x^2 - a)*a^2*b^2*
x*(1/(a^3*b^2))^(3/4) + (-b*x^2 - a)^(1/4)*a^2*b*sqrt(1/(a^3*b^2)) - I*(1/4)^(1/4)*a*b*x*(1/(a^3*b^2))^(1/4) +
(-b*x^2 - a)^(3/4))/(b*x^2 + 2*a))

Sympy [F]

\[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=- \int \frac {1}{2 a \sqrt [4]{- a - b x^{2}} + b x^{2} \sqrt [4]{- a - b x^{2}}}\, dx \]

[In]

integrate(1/(-b*x**2-2*a)/(-b*x**2-a)**(1/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(-b*x^2-2*a)/(-b*x^2-a)^(1/4),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(-b*x^2-2*a)/(-b*x^2-a)^(1/4),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx=-\int \frac {1}{{\left (-b\,x^2-a\right )}^{1/4}\,\left (b\,x^2+2\,a\right )} \,d x \]

[In]

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

[Out]

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