Integrand size = 41, antiderivative size = 75 \[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=\frac {2 \sqrt {a^2-b^2 x} \arctan \left (\frac {\sqrt {a^2+b^2 x}}{\sqrt {a^2-b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}} \] Output:
2*(-b^2*x+a^2)^(1/2)*arctan((b^2*x+a^2)^(1/2)/(-b^2*x+a^2)^(1/2))/b^2/(a-b *x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)
Time = 10.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=-\frac {2 \sqrt {a^2-b^2 x} \arctan \left (\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}} \] Input:
Integrate[1/(Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]]*Sqrt[a^2 + b^2*x]),x]
Output:
(-2*Sqrt[a^2 - b^2*x]*ArcTan[Sqrt[a^2 - b^2*x]/Sqrt[a^2 + b^2*x]])/(b^2*Sq rt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]])
Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2038, 45, 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.
\(\displaystyle \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx\) |
\(\Big \downarrow \) 2038 |
\(\displaystyle \frac {\sqrt {a^2-b^2 x} \int \frac {1}{\sqrt {a^2-b^2 x} \sqrt {a^2+b^2 x}}dx}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {2 \sqrt {a^2-b^2 x} \int \frac {1}{-\frac {\left (a^2-b^2 x\right ) b^2}{a^2+b^2 x}-b^2}d\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {2 \sqrt {a^2-b^2 x} \arctan \left (\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\) |
Input:
Int[1/(Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]]*Sqrt[a^2 + b^2*x]),x]
Output:
(-2*Sqrt[a^2 - b^2*x]*ArcTan[Sqrt[a^2 - b^2*x]/Sqrt[a^2 + b^2*x]])/(b^2*Sq rt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
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[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p _)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) ^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] ) Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && !(Eq Q[n, 2] && IGtQ[q, 0])
\[\int \frac {1}{\sqrt {a -b \sqrt {x}}\, \sqrt {a +b \sqrt {x}}\, \sqrt {b^{2} x +a^{2}}}d x\]
Input:
int(1/(a-b*x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)/(b^2*x+a^2)^(1/2),x)
Output:
int(1/(a-b*x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)/(b^2*x+a^2)^(1/2),x)
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=-\frac {2 \, \arctan \left (-\frac {a^{2} - \sqrt {b^{2} x + a^{2}} \sqrt {b \sqrt {x} + a} \sqrt {-b \sqrt {x} + a}}{b^{2} x}\right )}{b^{2}} \] Input:
integrate(1/(a-b*x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)/(b^2*x+a^2)^(1/2),x, a lgorithm="fricas")
Output:
-2*arctan(-(a^2 - sqrt(b^2*x + a^2)*sqrt(b*sqrt(x) + a)*sqrt(-b*sqrt(x) + a))/(b^2*x))/b^2
\[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=\int \frac {1}{\sqrt {a - b \sqrt {x}} \sqrt {a + b \sqrt {x}} \sqrt {a^{2} + b^{2} x}}\, dx \] Input:
integrate(1/(a-b*x**(1/2))**(1/2)/(a+b*x**(1/2))**(1/2)/(b**2*x+a**2)**(1/ 2),x)
Output:
Integral(1/(sqrt(a - b*sqrt(x))*sqrt(a + b*sqrt(x))*sqrt(a**2 + b**2*x)), x)
\[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=\int { \frac {1}{\sqrt {b^{2} x + a^{2}} \sqrt {b \sqrt {x} + a} \sqrt {-b \sqrt {x} + a}} \,d x } \] Input:
integrate(1/(a-b*x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)/(b^2*x+a^2)^(1/2),x, a lgorithm="maxima")
Output:
integrate(1/(sqrt(b^2*x + a^2)*sqrt(b*sqrt(x) + a)*sqrt(-b*sqrt(x) + a)), x)
\[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=\int { \frac {1}{\sqrt {b^{2} x + a^{2}} \sqrt {b \sqrt {x} + a} \sqrt {-b \sqrt {x} + a}} \,d x } \] Input:
integrate(1/(a-b*x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)/(b^2*x+a^2)^(1/2),x, a lgorithm="giac")
Output:
integrate(1/(sqrt(b^2*x + a^2)*sqrt(b*sqrt(x) + a)*sqrt(-b*sqrt(x) + a)), x)
Timed out. \[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=\int \frac {1}{\sqrt {a+b\,\sqrt {x}}\,\sqrt {a-b\,\sqrt {x}}\,\sqrt {a^2+x\,b^2}} \,d x \] Input:
int(1/((a + b*x^(1/2))^(1/2)*(a - b*x^(1/2))^(1/2)*(b^2*x + a^2)^(1/2)),x)
Output:
int(1/((a + b*x^(1/2))^(1/2)*(a - b*x^(1/2))^(1/2)*(b^2*x + a^2)^(1/2)), x )
\[ \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx=\int \frac {\sqrt {\sqrt {x}\, b +a}\, \sqrt {-\sqrt {x}\, b +a}\, \sqrt {b^{2} x +a^{2}}}{-b^{4} x^{2}+a^{4}}d x \] Input:
int(1/(a-b*x^(1/2))^(1/2)/(a+b*x^(1/2))^(1/2)/(b^2*x+a^2)^(1/2),x)
Output:
int((sqrt(sqrt(x)*b + a)*sqrt( - sqrt(x)*b + a)*sqrt(a**2 + b**2*x))/(a**4 - b**4*x**2),x)