Integrand size = 45, antiderivative size = 110 \[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \left (39 a b+32 a^3 x\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{105 b^3 x^2}-\frac {4 \left (15 b-32 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{105 b^3 x^3} \]
4/105*(32*a^3*x+39*a*b)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b^3/x^2-4/105* (-32*a^2*x+15*b)*(a^2*x^2-b*x)^(1/2)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b ^3/x^3
Time = 3.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )^2 \left (15 b^2+32 a^3 x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )+a b \left (-47 a x+39 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{105 b^3 \sqrt {x \left (-b+a^2 x\right )} \left (x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )\right )^{3/2}} \]
(4*(a*x + Sqrt[x*(-b + a^2*x)])^2*(15*b^2 + 32*a^3*x*(a*x + Sqrt[x*(-b + a ^2*x)]) + a*b*(-47*a*x + 39*Sqrt[x*(-b + a^2*x)])))/(105*b^3*Sqrt[x*(-b + a^2*x)]*(x*(a*x + Sqrt[x*(-b + a^2*x)]))^(3/2))
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.
\(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b x} \left (x \sqrt {a^2 x^2-b x}+a x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x-b} \int \frac {1}{\sqrt {x} \sqrt {a^2 x-b} \left (a x^2+\sqrt {a^2 x^2-b x} x\right )^{3/2}}dx}{\sqrt {a^2 x^2-b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {a^2 x-b} \int \frac {1}{\sqrt {a^2 x-b} \left (a x^2+\sqrt {a^2 x^2-b x} x\right )^{3/2}}d\sqrt {x}}{\sqrt {a^2 x^2-b x}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {a^2 x-b} \int \frac {1}{\sqrt {a^2 x-b} \left (a x^2+\sqrt {a^2 x^2-b x} x\right )^{3/2}}d\sqrt {x}}{\sqrt {a^2 x^2-b x}}\) |
3.17.30.3.1 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]
\[\int \frac {1}{\sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \, {\left (32 \, a^{3} x^{2} + 39 \, a b x + \sqrt {a^{2} x^{2} - b x} {\left (32 \, a^{2} x - 15 \, b\right )}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{105 \, b^{3} x^{3}} \]
4/105*(32*a^3*x^2 + 39*a*b*x + sqrt(a^2*x^2 - b*x)*(32*a^2*x - 15*b))*sqrt (a*x^2 + sqrt(a^2*x^2 - b*x)*x)/(b^3*x^3)
\[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \sqrt {x \left (a^{2} x - b\right )}}\, dx \]
\[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {a^2\,x^2-b\,x}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]