\(\int \frac {x^2}{(-2 a-b x^2) (-a-b x^2)^{3/4}} \, dx\) [1080]

Optimal result

Integrand size = 30, antiderivative size = 98 \[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {453} \[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]

[In]

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

[Out]

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

Rule 453

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-b/(Sqrt[2]*a*d*Rt[-b^2/a,
4]^3))*ArcTan[(Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] + Simp[(b/(Sqrt[2]*a*d*Rt[-b^2/a, 4]^3))*ArcT
anh[(Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && Neg
Q[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 )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}{\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{2}}{\left (-b \,x^{2}-2 a \right ) \left (-b \,x^{2}-a \right )^{\frac {3}{4}}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.06 \[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} - {\left (-b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

integrate(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x, algorithm="fricas")

[Out]

-1/2*(1/4)^(1/4)*(1/(a*b^6))^(1/4)*log(((1/4)^(1/4)*b^2*x*(1/(a*b^6))^(1/4) + (-b*x^2 - a)^(1/4))/x) + 1/2*(1/
4)^(1/4)*(1/(a*b^6))^(1/4)*log(-((1/4)^(1/4)*b^2*x*(1/(a*b^6))^(1/4) - (-b*x^2 - a)^(1/4))/x) - 1/2*I*(1/4)^(1
/4)*(1/(a*b^6))^(1/4)*log((I*(1/4)^(1/4)*b^2*x*(1/(a*b^6))^(1/4) + (-b*x^2 - a)^(1/4))/x) + 1/2*I*(1/4)^(1/4)*
(1/(a*b^6))^(1/4)*log((-I*(1/4)^(1/4)*b^2*x*(1/(a*b^6))^(1/4) + (-b*x^2 - a)^(1/4))/x)

Sympy [F]

\[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=- \int \frac {x^{2}}{2 a \left (- a - b x^{2}\right )^{\frac {3}{4}} + b x^{2} \left (- a - b x^{2}\right )^{\frac {3}{4}}}\, dx \]

[In]

integrate(x**2/(-b*x**2-2*a)/(-b*x**2-a)**(3/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2/(-b*x^2-2*a)/(-b*x^2-a)^(3/4),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx=-\int \frac {x^2}{{\left (-b\,x^2-a\right )}^{3/4}\,\left (b\,x^2+2\,a\right )} \,d x \]

[In]

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

[Out]

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