Integrand size = 45, antiderivative size = 200 \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\frac {\left (-9 b+8 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{12 a b x}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (\frac {19 b-8 a^2 x}{12 b}+\frac {3 \sqrt {b} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{4 \sqrt {2} a^{3/2} x}\right ) \]
1/12*(8*a^2*x-9*b)*(a^2*x^2-b*x)^(1/2)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2) /a/b/x+(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)*(1/12*(-8*a^2*x+19*b)/b+3/8*b^( 1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)*arctan(2^(1/2)*a^(1/2)*(-a*x+(a^2*x^ 2-b*x)^(1/2))^(1/2)/b^(1/2))*2^(1/2)/a^(3/2)/x)
Time = 4.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (-2 \sqrt {a} x \left (-9 b^2+a b \left (17 a x-19 \sqrt {x \left (-b+a^2 x\right )}\right )+8 a^3 x \left (-a x+\sqrt {x \left (-b+a^2 x\right )}\right )\right )+9 \sqrt {2} b^{3/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{24 a^{3/2} b x \sqrt {x \left (-b+a^2 x\right )}} \]
(Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(-2*Sqrt[a]*x*(-9*b^2 + a*b*(17*a*x - 19*Sqrt[x*(-b + a^2*x)]) + 8*a^3*x*(-(a*x) + Sqrt[x*(-b + a^2*x)])) + 9* Sqrt[2]*b^(3/2)*Sqrt[x*(-b + a^2*x)]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]*A rcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]])/Sqrt[b]]))/(24 *a^(3/2)*b*x*Sqrt[x*(-b + a^2*x)])
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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b x}}{\sqrt {x \sqrt {a^2 x^2-b x}+a x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {a^2 x^2-b x} \int \frac {\sqrt {x} \sqrt {a^2 x-b}}{\sqrt {a x^2+\sqrt {a^2 x^2-b x} x}}dx}{\sqrt {x} \sqrt {a^2 x-b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {a^2 x^2-b x} \int \frac {x \sqrt {a^2 x-b}}{\sqrt {a x^2+\sqrt {a^2 x^2-b x} x}}d\sqrt {x}}{\sqrt {x} \sqrt {a^2 x-b}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {2 \sqrt {a^2 x^2-b x} \int \frac {x \sqrt {a^2 x-b}}{\sqrt {a x^2+\sqrt {a^2 x^2-b x} x}}d\sqrt {x}}{\sqrt {x} \sqrt {a^2 x-b}}\) |
3.25.66.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
\[\int \frac {\sqrt {a^{2} x^{2}-b x}}{\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}}}d x\]
Time = 0.27 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\left [\frac {9 \, \sqrt {2} \sqrt {a} b^{2} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (8 \, a^{4} x^{2} - 19 \, a^{2} b x - {\left (8 \, a^{3} x - 9 \, a b\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{48 \, a^{2} b x}, \frac {9 \, \sqrt {2} \sqrt {-a} b^{2} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (8 \, a^{4} x^{2} - 19 \, a^{2} b x - {\left (8 \, a^{3} x - 9 \, a b\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{24 \, a^{2} b x}\right ] \]
[1/48*(9*sqrt(2)*sqrt(a)*b^2*x*log(-(4*a^2*x^2 + 4*sqrt(a^2*x^2 - b*x)*a*x - b*x - 2*(sqrt(2)*a^(3/2)*x + sqrt(2)*sqrt(a^2*x^2 - b*x)*sqrt(a))*sqrt( a*x^2 + sqrt(a^2*x^2 - b*x)*x))/x) - 4*(8*a^4*x^2 - 19*a^2*b*x - (8*a^3*x - 9*a*b)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^2*b* x), 1/24*(9*sqrt(2)*sqrt(-a)*b^2*x*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^ 2*x^2 - b*x)*x)*sqrt(-a)/(a*x)) - 2*(8*a^4*x^2 - 19*a^2*b*x - (8*a^3*x - 9 *a*b)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^2*b*x)]
\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int \frac {\sqrt {x \left (a^{2} x - b\right )}}{\sqrt {x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )}}\, dx \]
\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b x}}{\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}} \,d x } \]
\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b x}}{\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b\,x}}{\sqrt {a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}}} \,d x \]