\(\int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x (1-k^2 x)} (-1+k^2 x)} \, dx\) [475]

Optimal result

Integrand size = 47, antiderivative size = 38 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2 \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{x \left (-1+k^2 x\right )} \]

[Out]

-2*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/x/(k^2*x-1)

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6850, 21, 1628, 8} \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\frac {2 (1-x)}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \]

[In]

Int[(1 - 2*k^2*x + k^2*x^2)/(x*Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^2*x)),x]

[Out]

(2*(1 - x))/Sqrt[(1 - x)*x*(1 - k^2*x)]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 1628

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*
(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 6850

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n*z^q)^FracP
art[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p]))), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {1-x} x^{3/2} \sqrt {1-k^2 x} \left (-1+k^2 x\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = -\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {1-x} x^{3/2} \left (1-k^2 x\right )^{3/2}} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 (1-x)}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int 0 \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 (1-x)}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.55 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2 (-1+x)}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}} \]

[In]

Integrate[(1 - 2*k^2*x + k^2*x^2)/(x*Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^2*x)),x]

[Out]

(-2*(-1 + x))/Sqrt[(-1 + x)*x*(-1 + k^2*x)]

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53

method	result	size
gosper	\(-\frac {2 \left (x -1\right )}{\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}\)	\(20\)
trager	\(-\frac {2 \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}{x \left (k^{2} x -1\right )}\)	\(39\)
risch	\(\frac {2 \left (x -1\right ) \left (k^{2} x -1\right )}{\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}-\frac {2 x \left (x -1\right ) k^{2}}{\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}\)	\(51\)
elliptic	\(\frac {2 k^{2} x^{2}-2 k^{2} x -2 x +2}{\sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}-\frac {2 \left (k^{2} x^{2}-k^{2} x \right )}{\sqrt {\left (x -\frac {1}{k^{2}}\right ) \left (k^{2} x^{2}-k^{2} x \right )}}\)	\(84\)
default	\(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 k^{2} x^{2}-2 k^{2} x -2 x +2}{\sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}+\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \left (\left (\frac {1}{k^{2}}-1\right ) \operatorname {EllipticE}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )+\operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )\right )}{\sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\left (-k^{2}+1\right ) \left (\frac {2 k^{2} x^{2}-2 k^{2} x}{\left (k^{2}-1\right ) \sqrt {\left (x -\frac {1}{k^{2}}\right ) \left (k^{2} x^{2}-k^{2} x \right )}}-\frac {2 \left (-1+\frac {k^{2}}{k^{2}-1}\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \left (\left (\frac {1}{k^{2}}-1\right ) \operatorname {EllipticE}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )+\operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}\right )\)	\(548\)

[In]

int((k^2*x^2-2*k^2*x+1)/x/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x-1),x,method=_RETURNVERBOSE)

[Out]

-2*(x-1)/((x-1)*x*(k^2*x-1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x}}{k^{2} x^{2} - x} \]

[In]

integrate((k^2*x^2-2*k^2*x+1)/x/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x-1),x, algorithm="fricas")

[Out]

-2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)/(k^2*x^2 - x)

Sympy [F]

\[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\int \frac {k^{2} x^{2} - 2 k^{2} x + 1}{x \sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k^{2} x - 1\right )}\, dx \]

[In]

integrate((k**2*x**2-2*k**2*x+1)/x/((1-x)*x*(-k**2*x+1))**(1/2)/(k**2*x-1),x)

[Out]

Integral((k**2*x**2 - 2*k**2*x + 1)/(x*sqrt(x*(x - 1)*(k**2*x - 1))*(k**2*x - 1)), x)

Maxima [F]

\[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{{\left (k^{2} x - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} x} \,d x } \]

[In]

integrate((k^2*x^2-2*k^2*x+1)/x/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x-1),x, algorithm="maxima")

[Out]

integrate((k^2*x^2 - 2*k^2*x + 1)/((k^2*x - 1)*sqrt((k^2*x - 1)*(x - 1)*x)*x), x)

Giac [F]

\[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{{\left (k^{2} x - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} x} \,d x } \]

[In]

integrate((k^2*x^2-2*k^2*x+1)/x/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x-1),x, algorithm="giac")

[Out]

integrate((k^2*x^2 - 2*k^2*x + 1)/((k^2*x - 1)*sqrt((k^2*x - 1)*(x - 1)*x)*x), x)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}}{x\,\left (k^2\,x-1\right )} \]

[In]

int((k^2*x^2 - 2*k^2*x + 1)/(x*(k^2*x - 1)*(x*(k^2*x - 1)*(x - 1))^(1/2)),x)

[Out]

-(2*(x*(k^2*x - 1)*(x - 1))^(1/2))/(x*(k^2*x - 1))