\(\int \frac {x^2}{(-2 b+a x^2) (-b+a x^2)^{3/4}} \, dx\) [1356]

Optimal result

Integrand size = 28, antiderivative size = 98 \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \] Output:

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

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.98, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {351}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 351

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 351

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))*ArcTanh[( 
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] && NegQ[b^2/a]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.02 \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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