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

Optimal. Leaf size=38 \[ -\frac {2 \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{x \left (k^2 x-1\right )} \]

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Rubi [A]  time = 0.77, antiderivative size = 26, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6718, 21, 1614, 8} \begin {gather*} \frac {2 (1-x)}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[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 1614

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 6718

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*} \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 {\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*}

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Mathematica [A]  time = 0.21, size = 21, normalized size = 0.55 \begin {gather*} -\frac {2 (x-1)}{\sqrt {(x-1) x \left (k^2 x-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[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)]

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IntegrateAlgebraic [A]  time = 0.17, size = 38, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{x \left (-1+k^2 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(x*(-1 + k^2*x))

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fricas [A]  time = 0.47, size = 36, normalized size = 0.95 \begin {gather*} -\frac {2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x}}{k^{2} x^{2} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.13, size = 20, normalized size = 0.53

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

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.29, size = 28, normalized size = 0.74 \begin {gather*} -\frac {2\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}}{x\,\left (k^2\,x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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