Integrand size = 45, antiderivative size = 117 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=-\frac {4 \left (3 b c+32 a c x-8 c^3 x\right ) \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{105 b^2 x^2}+\frac {4 \left (15 b+20 a x+4 c^2 x\right ) \sqrt {-b x+a x^2} \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{105 b^2 x^3} \]
-4/105*(-8*c^3*x+32*a*c*x+3*b*c)*(-x*(-c*x+(a*x^2-b*x)^(1/2)))^(1/2)/b^2/x ^2+4/105*(4*c^2*x+20*a*x+15*b)*(a*x^2-b*x)^(1/2)*(-x*(-c*x+(a*x^2-b*x)^(1/ 2)))^(1/2)/b^2/x^3
\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx \]
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.
\(\displaystyle \int \frac {\sqrt {c x^2-x \sqrt {a x^2-b x}}}{x^3 \sqrt {a x^2-b x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x-b} \int \frac {\sqrt {c x^2-x \sqrt {a x^2-b x}}}{x^{7/2} \sqrt {a x-b}}dx}{\sqrt {a x^2-b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {a x-b} \int \frac {\sqrt {c x^2-x \sqrt {a x^2-b x}}}{x^3 \sqrt {a x-b}}d\sqrt {x}}{\sqrt {a x^2-b x}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {a x-b} \int \frac {\sqrt {c x^2-x \sqrt {a x^2-b x}}}{x^3 \sqrt {a x-b}}d\sqrt {x}}{\sqrt {a x^2-b x}}\) |
3.18.43.3.1 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]
\[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3} \sqrt {a \,x^{2}-b x}}d x\]
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=-\frac {4 \, {\left (3 \, b c x - 8 \, {\left (c^{3} - 4 \, a c\right )} x^{2} - \sqrt {a x^{2} - b x} {\left (4 \, {\left (c^{2} + 5 \, a\right )} x + 15 \, b\right )}\right )} \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{105 \, b^{2} x^{3}} \]
-4/105*(3*b*c*x - 8*(c^3 - 4*a*c)*x^2 - sqrt(a*x^2 - b*x)*(4*(c^2 + 5*a)*x + 15*b))*sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)/(b^2*x^3)
\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {x \left (c x - \sqrt {a x^{2} - b x}\right )}}{x^{3} \sqrt {x \left (a x - b\right )}}\, dx \]
\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int { \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{\sqrt {a x^{2} - b x} x^{3}} \,d x } \]
\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int { \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{\sqrt {a x^{2} - b x} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {c\,x^2-x\,\sqrt {a\,x^2-b\,x}}}{x^3\,\sqrt {a\,x^2-b\,x}} \,d x \]