Integrand size = 34, antiderivative size = 87 \[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=-\frac {2 \sqrt {x} \sqrt {a^2-\left (1+a^2\right ) x+x^2} \arctan \left (\frac {(1-a) \sqrt {x}}{\sqrt {a^2-\left (1+a^2\right ) x+x^2}}\right )}{(1-a) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \]
-2*arctan((1-a)*x^(1/2)/(a^2-(a^2+1)*x+x^2)^(1/2))*x^(1/2)*(a^2-(a^2+1)*x+ x^2)^(1/2)/(1-a)/(a^2*x-(a^2+1)*x^2+x^3)^(1/2)
Time = 10.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.33 \[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=-\frac {2 \arctan \left (\frac {(-1+a) x}{\sqrt {(-1+x) x \left (-a^2+x\right )}}\right )}{-1+a} \]
Time = 0.50 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2467, 25, 2035, 2212, 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.
\(\displaystyle \int \frac {a+x}{(x-a) \sqrt {-\left (a^2+1\right ) x^2+a^2 x+x^3}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {-\left (a^2+1\right ) x+a^2+x^2} \int -\frac {a+x}{(a-x) \sqrt {x} \sqrt {a^2+x^2-\left (a^2+1\right ) x}}dx}{\sqrt {-\left (a^2+1\right ) x^2+a^2 x+x^3}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {-\left (a^2+1\right ) x+a^2+x^2} \int \frac {a+x}{(a-x) \sqrt {x} \sqrt {a^2+x^2-\left (a^2+1\right ) x}}dx}{\sqrt {-\left (a^2+1\right ) x^2+a^2 x+x^3}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-\left (a^2+1\right ) x+a^2+x^2} \int \frac {a+x}{(a-x) \sqrt {a^2+x^2-\left (a^2+1\right ) x}}d\sqrt {x}}{\sqrt {-\left (a^2+1\right ) x^2+a^2 x+x^3}}\) |
\(\Big \downarrow \) 2212 |
\(\displaystyle -\frac {2 a \sqrt {x} \sqrt {-\left (a^2+1\right ) x+a^2+x^2} \int \frac {1}{a x (1-a)^2+a}d\frac {\sqrt {x}}{\sqrt {a^2+x^2-\left (a^2+1\right ) x}}}{\sqrt {-\left (a^2+1\right ) x^2+a^2 x+x^3}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-\left (a^2+1\right ) x+a^2+x^2} \arctan \left (\frac {(1-a) \sqrt {x}}{\sqrt {-\left (a^2+1\right ) x+a^2+x^2}}\right )}{(1-a) \sqrt {-\left (a^2+1\right ) x^2+a^2 x+x^3}}\) |
(-2*Sqrt[x]*Sqrt[a^2 - (1 + a^2)*x + x^2]*ArcTan[((1 - a)*Sqrt[x])/Sqrt[a^ 2 - (1 + a^2)*x + x^2]])/((1 - a)*Sqrt[a^2*x - (1 + a^2)*x^2 + x^3])
3.1.81.3.1 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[(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[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Simp[A Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B}, x] & & EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
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]
Time = 0.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.38
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\sqrt {-\left (a^{2}-x \right ) x \left (-1+x \right )}}{x \left (a -1\right )}\right )}{a -1}\) | \(33\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {-\left (a^{2}-x \right ) x \left (-1+x \right )}}{x \left (a -1\right )}\right )}{a -1}\) | \(33\) |
elliptic | \(-\frac {2 a^{2} \sqrt {-\frac {-a^{2}+x}{a^{2}}}\, \sqrt {\frac {-1+x}{a^{2}-1}}\, \sqrt {\frac {x}{a^{2}}}\, F\left (\sqrt {-\frac {-a^{2}+x}{a^{2}}}, \sqrt {\frac {a^{2}}{a^{2}-1}}\right )}{\sqrt {-a^{2} x^{2}+a^{2} x +x^{3}-x^{2}}}-\frac {4 a^{3} \sqrt {-\frac {-a^{2}+x}{a^{2}}}\, \sqrt {\frac {-1+x}{a^{2}-1}}\, \sqrt {\frac {x}{a^{2}}}\, \Pi \left (\sqrt {-\frac {-a^{2}+x}{a^{2}}}, \frac {a^{2}}{a^{2}-a}, \sqrt {\frac {a^{2}}{a^{2}-1}}\right )}{\sqrt {-a^{2} x^{2}+a^{2} x +x^{3}-x^{2}}\, \left (a^{2}-a \right )}\) | \(206\) |
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {a^{2} x - {\left (a^{2} + 1\right )} x^{2} + x^{3}} {\left (a^{2} - 2 \, {\left (a^{2} - a + 1\right )} x + x^{2}\right )}}{2 \, {\left ({\left (a - 1\right )} x^{3} - {\left (a^{3} - a^{2} + a - 1\right )} x^{2} + {\left (a^{3} - a^{2}\right )} x\right )}}\right )}{a - 1} \]
arctan(1/2*sqrt(a^2*x - (a^2 + 1)*x^2 + x^3)*(a^2 - 2*(a^2 - a + 1)*x + x^ 2)/((a - 1)*x^3 - (a^3 - a^2 + a - 1)*x^2 + (a^3 - a^2)*x))/(a - 1)
\[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=\int \frac {a + x}{\sqrt {x \left (- a^{2} + x\right ) \left (x - 1\right )} \left (- a + x\right )}\, dx \]
\[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=\int { -\frac {a + x}{\sqrt {a^{2} x - {\left (a^{2} + 1\right )} x^{2} + x^{3}} {\left (a - x\right )}} \,d x } \]
\[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=\int { -\frac {a + x}{\sqrt {a^{2} x - {\left (a^{2} + 1\right )} x^{2} + x^{3}} {\left (a - x\right )}} \,d x } \]
Time = 0.16 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.49 \[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=\frac {4\,a\,\left (a^2-1\right )\,\sqrt {\frac {x}{a^2}}\,\sqrt {\frac {x-1}{a^2-1}}\,\sqrt {-\frac {x-a^2}{a^2-1}}\,\Pi \left (-\frac {a^2-1}{a-a^2};\mathrm {asin}\left (\sqrt {-\frac {x-a^2}{a^2-1}}\right )\middle |\frac {a^2-1}{a^2}\right )}{\left (a-a^2\right )\,\sqrt {a^2\,x-x^2\,\left (a^2+1\right )+x^3}}-\frac {2\,\left (a^2-1\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-a^2}{a^2-1}}\right )\middle |\frac {a^2-1}{a^2}\right )\,\sqrt {\frac {x}{a^2}}\,\sqrt {\frac {x-1}{a^2-1}}\,\sqrt {-\frac {x-a^2}{a^2-1}}}{\sqrt {a^2\,x-x^2\,\left (a^2+1\right )+x^3}} \]
(4*a*(a^2 - 1)*(x/a^2)^(1/2)*((x - 1)/(a^2 - 1))^(1/2)*(-(x - a^2)/(a^2 - 1))^(1/2)*ellipticPi(-(a^2 - 1)/(a - a^2), asin((-(x - a^2)/(a^2 - 1))^(1/ 2)), (a^2 - 1)/a^2))/((a - a^2)*(a^2*x - x^2*(a^2 + 1) + x^3)^(1/2)) - (2* (a^2 - 1)*ellipticF(asin((-(x - a^2)/(a^2 - 1))^(1/2)), (a^2 - 1)/a^2)*(x/ a^2)^(1/2)*((x - 1)/(a^2 - 1))^(1/2)*(-(x - a^2)/(a^2 - 1))^(1/2))/(a^2*x - x^2*(a^2 + 1) + x^3)^(1/2)
\[ \int \frac {a+x}{(-a+x) \sqrt {a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx=\left (\int \frac {\sqrt {x}\, \sqrt {x -1}\, \sqrt {-a^{2}+x}}{a^{3} x^{2}-a^{2} x^{3}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}+a \,x^{2}-x^{3}}d x \right ) a +\int \frac {\sqrt {x}\, \sqrt {x -1}\, \sqrt {-a^{2}+x}}{a^{3} x -a^{2} x^{2}-a^{3}+a^{2} x -a \,x^{2}+x^{3}+a x -x^{2}}d x \]