Integrand size = 28, antiderivative size = 125 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {x \left (a-b x^2\right )}{4 a^2 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a-b x^2} \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
1/4*x*(-b*x^2+a)/a^2/(b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2)+3/8*arctan(x*2^( 1/2)*b^(1/2)/(-b*x^2+a)^(1/2))*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a^2*2^(1/2 )/b^(1/2)/(-b^2*x^4+a^2)^(1/2)
Time = 2.33 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {\sqrt {a^2-b^2 x^4} \left (2 \sqrt {b} x \sqrt {a-b x^2}+3 \sqrt {2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )\right )}{8 a^2 \sqrt {b} \sqrt {a-b x^2} \left (a+b x^2\right )^{3/2}} \]
(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a - b*x^2] + 3*Sqrt[2]*(a + b*x^2)* ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]]))/(8*a^2*Sqrt[b]*Sqrt[a - b*x^ 2]*(a + b*x^2)^(3/2))
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1396, 296, 291, 218}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \int \frac {1}{\sqrt {a-b x^2} \left (b x^2+a\right )^2}dx}{\sqrt {a^2-b^2 x^4}}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \left (\frac {3 \int \frac {1}{\sqrt {a-b x^2} \left (b x^2+a\right )}dx}{4 a}+\frac {x \sqrt {a-b x^2}}{4 a^2 \left (a+b x^2\right )}\right )}{\sqrt {a^2-b^2 x^4}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \left (\frac {3 \int \frac {1}{\frac {2 a b x^2}{a-b x^2}+a}d\frac {x}{\sqrt {a-b x^2}}}{4 a}+\frac {x \sqrt {a-b x^2}}{4 a^2 \left (a+b x^2\right )}\right )}{\sqrt {a^2-b^2 x^4}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \left (\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b}}+\frac {x \sqrt {a-b x^2}}{4 a^2 \left (a+b x^2\right )}\right )}{\sqrt {a^2-b^2 x^4}}\) |
(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*((x*Sqrt[a - b*x^2])/(4*a^2*(a + b*x^2)) + (3*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]])/(4*Sqrt[2]*a^2*Sqrt[b])) )/Sqrt[a^2 - b^2*x^4]
3.3.3.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(101)=202\).
Time = 0.23 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.90
| method | result | size |
| default | \(-\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, b^{\frac {5}{2}} \left (3 \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}-\sqrt {-a b}\, x +a \right )}{b x -\sqrt {-a b}}\right ) \sqrt {2}\, b^{\frac {3}{2}} x^{2} \sqrt {a}-3 \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}+\sqrt {-a b}\, x +a \right )}{b x +\sqrt {-a b}}\right ) \sqrt {2}\, b^{\frac {3}{2}} x^{2} \sqrt {a}+3 \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}-\sqrt {-a b}\, x +a \right )}{b x -\sqrt {-a b}}\right ) \sqrt {2}\, a^{\frac {3}{2}} \sqrt {b}-3 \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}+\sqrt {-a b}\, x +a \right )}{b x +\sqrt {-a b}}\right ) \sqrt {2}\, a^{\frac {3}{2}} \sqrt {b}+4 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) b \,x^{2} \sqrt {-a b}-4 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right ) b \,x^{2} \sqrt {-a b}-4 \sqrt {-b \,x^{2}+a}\, \sqrt {b}\, \sqrt {-a b}\, x +4 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) a \sqrt {-a b}-4 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right ) a \sqrt {-a b}\right )}{4 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \left (-\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (b x +\sqrt {-a b}\right ) \left (b x -\sqrt {-a b}\right ) \sqrt {-a b}}\) | \(488\) |
-1/4*(-b^2*x^4+a^2)^(1/2)*b^(5/2)*(3*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1 /2)-(-a*b)^(1/2)*x+a)/(b*x-(-a*b)^(1/2)))*2^(1/2)*b^(3/2)*x^2*a^(1/2)-3*ln (2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x+(-a*b)^(1/2) ))*2^(1/2)*b^(3/2)*x^2*a^(1/2)+3*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)- (-a*b)^(1/2)*x+a)/(b*x-(-a*b)^(1/2)))*2^(1/2)*a^(3/2)*b^(1/2)-3*ln(2*b*(2^ (1/2)*a^(1/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x+(-a*b)^(1/2)))*2^(1/ 2)*a^(3/2)*b^(1/2)+4*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))*b*x^2*(-a*b)^(1/2) -4*arctan(b^(1/2)*x/(1/b*(-b*x+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2))*b*x^ 2*(-a*b)^(1/2)-4*(-b*x^2+a)^(1/2)*b^(1/2)*(-a*b)^(1/2)*x+4*arctan(b^(1/2)* x/(-b*x^2+a)^(1/2))*a*(-a*b)^(1/2)-4*arctan(b^(1/2)*x/(1/b*(-b*x+(a*b)^(1/ 2))*(b*x+(a*b)^(1/2)))^(1/2))*a*(-a*b)^(1/2))/(b*x^2+a)^(1/2)/(-b*x^2+a)^( 1/2)/(-(-a*b)^(1/2)+(a*b)^(1/2))^2/((-a*b)^(1/2)+(a*b)^(1/2))^2/(b*x+(-a*b )^(1/2))/(b*x-(-a*b)^(1/2))/(-a*b)^(1/2)
Time = 0.25 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\left [\frac {4 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} b x - 3 \, \sqrt {2} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {-b} \log \left (-\frac {3 \, b^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {-b} x - a^{2}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{16 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}, \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} b x - 3 \, \sqrt {2} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {b}}{2 \, {\left (b^{2} x^{3} + a b x\right )}}\right )}{8 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}\right ] \]
[1/16*(4*sqrt(-b^2*x^4 + a^2)*sqrt(b*x^2 + a)*b*x - 3*sqrt(2)*(b^2*x^4 + 2 *a*b*x^2 + a^2)*sqrt(-b)*log(-(3*b^2*x^4 + 2*a*b*x^2 - 2*sqrt(2)*sqrt(-b^2 *x^4 + a^2)*sqrt(b*x^2 + a)*sqrt(-b)*x - a^2)/(b^2*x^4 + 2*a*b*x^2 + a^2)) )/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b), 1/8*(2*sqrt(-b^2*x^4 + a^2)*sqrt( b*x^2 + a)*b*x - 3*sqrt(2)*(b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b)*arctan(1/2* sqrt(2)*sqrt(-b^2*x^4 + a^2)*sqrt(b*x^2 + a)*sqrt(b)/(b^2*x^3 + a*b*x)))/( a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)]
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {a^2-b^2\,x^4}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]