Integrand size = 26, antiderivative size = 74 \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \] Output:
1/2*arctan(1/2*b^(1/2)*x*2^(1/2)/(-b*x^2-1)^(1/4))*2^(1/2)/b^(3/2)-1/2*arc tanh(1/2*b^(1/2)*x*2^(1/2)/(-b*x^2-1)^(1/4))*2^(1/2)/b^(3/2)
Time = 1.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \] Input:
Integrate[x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)),x]
Output:
-((-ArcTan[(Sqrt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))] + ArcTanh[(Sqrt[b]*x) /(Sqrt[2]*(-1 - b*x^2)^(1/4))])/(Sqrt[2]*b^(3/2)))
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, 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.
\(\displaystyle \int \frac {x^2}{\left (-b x^2-2\right ) \left (-b x^2-1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 351 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt {2} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt {2} b^{3/2}}\) |
Input:
Int[x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)),x]
Output:
ArcTan[(Sqrt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2)) - ArcTa nh[(Sqrt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2))
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]
\[\int \frac {x^{2}}{\left (-b \,x^{2}-2\right ) \left (-b \,x^{2}-1\right )^{\frac {3}{4}}}d x\]
Input:
int(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x)
Output:
int(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (57) = 114\).
Time = 0.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.86 \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\left [-\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {b} x}{2 \, {\left (b x^{2} + 1\right )}}\right ) - \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} + 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} + 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}, -\frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {-b} x}{2 \, {\left (b x^{2} + 1\right )}}\right ) + \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} - 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} - 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {-b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}\right ] \] Input:
integrate(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x, algorithm="fricas")
Output:
[-1/4*(2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(-b*x^2 - 1)^(3/4)*sqrt(b)*x/( b*x^2 + 1)) - sqrt(2)*sqrt(b)*log(-(b^2*x^4 + 4*sqrt(-b*x^2 - 1)*b*x^2 - 4 *b*x^2 - 2*sqrt(2)*((-b*x^2 - 1)^(1/4)*b*x^3 + 2*(-b*x^2 - 1)^(3/4)*x)*sqr t(b) - 4)/(b^2*x^4 + 4*b*x^2 + 4)))/b^2, -1/4*(2*sqrt(2)*sqrt(-b)*arctan(1 /2*sqrt(2)*(-b*x^2 - 1)^(3/4)*sqrt(-b)*x/(b*x^2 + 1)) + sqrt(2)*sqrt(-b)*l og(-(b^2*x^4 - 4*sqrt(-b*x^2 - 1)*b*x^2 - 4*b*x^2 - 2*sqrt(2)*((-b*x^2 - 1 )^(1/4)*b*x^3 - 2*(-b*x^2 - 1)^(3/4)*x)*sqrt(-b) - 4)/(b^2*x^4 + 4*b*x^2 + 4)))/b^2]
\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=- \int \frac {x^{2}}{b x^{2} \left (- b x^{2} - 1\right )^{\frac {3}{4}} + 2 \left (- b x^{2} - 1\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(x**2/(-b*x**2-2)/(-b*x**2-1)**(3/4),x)
Output:
-Integral(x**2/(b*x**2*(-b*x**2 - 1)**(3/4) + 2*(-b*x**2 - 1)**(3/4)), x)
\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x, algorithm="maxima")
Output:
-integrate(x^2/((b*x^2 + 2)*(-b*x^2 - 1)^(3/4)), x)
\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x, algorithm="giac")
Output:
integrate(-x^2/((b*x^2 + 2)*(-b*x^2 - 1)^(3/4)), x)
Timed out. \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=-\int \frac {x^2}{{\left (-b\,x^2-1\right )}^{3/4}\,\left (b\,x^2+2\right )} \,d x \] Input:
int(-x^2/((- b*x^2 - 1)^(3/4)*(b*x^2 + 2)),x)
Output:
-int(x^2/((- b*x^2 - 1)^(3/4)*(b*x^2 + 2)), x)
\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\left (\int \frac {x^{2}}{\left (b \,x^{2}+1\right )^{\frac {3}{4}} b \,x^{2}+2 \left (b \,x^{2}+1\right )^{\frac {3}{4}}}d x \right ) \left (-1\right )^{\frac {1}{4}} \] Input:
int(x^2/(-b*x^2-2)/(-b*x^2-1)^(3/4),x)
Output:
( - int(x**2/((b*x**2 + 1)**(3/4)*b*x**2 + 2*(b*x**2 + 1)**(3/4)),x))/( - 1)**(3/4)