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