3.81 \(\int \frac{a+x}{(-a+x) \sqrt{a^2 x-(1+a^2) x^2+x^3}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{2 \sqrt{x} \sqrt{-\left (a^2+1\right ) x+a^2+x^2} \tan ^{-1}\left (\frac{(1-a) \sqrt{x}}{\sqrt{-\left (a^2+1\right ) x+a^2+x^2}}\right )}{(1-a) \sqrt{-\left (a^2+1\right ) x^2+a^2 x+x^3}} \]

[Out]

(-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)*Sq
rt[a^2*x - (1 + a^2)*x^2 + x^3])

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Rubi [A]  time = 0.873863, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2056, 6733, 1698, 205} \[ -\frac{2 \sqrt{x} \sqrt{-\left (a^2+1\right ) x+a^2+x^2} \tan ^{-1}\left (\frac{(1-a) \sqrt{x}}{\sqrt{-\left (a^2+1\right ) x+a^2+x^2}}\right )}{(1-a) \sqrt{-\left (a^2+1\right ) x^2+a^2 x+x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + x)/((-a + x)*Sqrt[a^2*x - (1 + a^2)*x^2 + x^3]),x]

[Out]

(-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)*Sq
rt[a^2*x - (1 + a^2)*x^2 + x^3])

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 1698

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{a+x}{(-a+x) \sqrt{a^2 x-\left (1+a^2\right ) x^2+x^3}} \, dx &=\frac{\left (\sqrt{x} \sqrt{a^2-\left (1+a^2\right ) x+x^2}\right ) \int \frac{a+x}{\sqrt{x} (-a+x) \sqrt{a^2-\left (1+a^2\right ) x+x^2}} \, dx}{\sqrt{a^2 x-\left (1+a^2\right ) x^2+x^3}}\\ &=\frac{\left (2 \sqrt{x} \sqrt{a^2-\left (1+a^2\right ) x+x^2}\right ) \operatorname{Subst}\left (\int \frac{a+x^2}{\left (-a+x^2\right ) \sqrt{a^2+\left (-1-a^2\right ) x^2+x^4}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2 x-\left (1+a^2\right ) x^2+x^3}}\\ &=\frac{\left (2 a \sqrt{x} \sqrt{a^2-\left (1+a^2\right ) x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a-\left (-2 a^2-a \left (-1-a^2\right )\right ) x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a^2-\left (1+a^2\right ) x+x^2}}\right )}{\sqrt{a^2 x-\left (1+a^2\right ) x^2+x^3}}\\ &=-\frac{2 \sqrt{x} \sqrt{a^2-\left (1+a^2\right ) x+x^2} \tan ^{-1}\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}}\\ \end{align*}

Mathematica [C]  time = 0.915921, size = 159, normalized size = 1.83 \[ -\frac{2 i \left (a^2-x\right )^{3/2} \sqrt{\frac{x-1}{x-a^2}} \sqrt{\frac{x}{x-a^2}} \left ((a+1) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-a^2}}{\sqrt{a^2-x}}\right ),1-\frac{1}{a^2}\right )-2 \Pi \left (\frac{a-1}{a};i \sinh ^{-1}\left (\frac{\sqrt{-a^2}}{\sqrt{a^2-x}}\right )|1-\frac{1}{a^2}\right )\right )}{(a-1) \sqrt{-a^2} \sqrt{(x-1) x \left (x-a^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + x)/((-a + x)*Sqrt[a^2*x - (1 + a^2)*x^2 + x^3]),x]

[Out]

((-2*I)*(a^2 - x)^(3/2)*Sqrt[(-1 + x)/(-a^2 + x)]*Sqrt[x/(-a^2 + x)]*((1 + a)*EllipticF[I*ArcSinh[Sqrt[-a^2]/S
qrt[a^2 - x]], 1 - a^(-2)] - 2*EllipticPi[(-1 + a)/a, I*ArcSinh[Sqrt[-a^2]/Sqrt[a^2 - x]], 1 - a^(-2)]))/((-1
+ a)*Sqrt[-a^2]*Sqrt[(-1 + x)*x*(-a^2 + x)])

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Maple [C]  time = 0.034, size = 206, normalized size = 2.4 \begin{align*} -2\,{\frac{{a}^{2}}{\sqrt{-{a}^{2}{x}^{2}+{a}^{2}x+{x}^{3}-{x}^{2}}}\sqrt{-{\frac{-{a}^{2}+x}{{a}^{2}}}}\sqrt{{\frac{-1+x}{{a}^{2}-1}}}\sqrt{{\frac{x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-{a}^{2}+x}{{a}^{2}}}},\sqrt{{\frac{{a}^{2}}{{a}^{2}-1}}} \right ) }-4\,{\frac{{a}^{3}}{\sqrt{-{a}^{2}{x}^{2}+{a}^{2}x+{x}^{3}-{x}^{2}} \left ({a}^{2}-a \right ) }\sqrt{-{\frac{-{a}^{2}+x}{{a}^{2}}}}\sqrt{{\frac{-1+x}{{a}^{2}-1}}}\sqrt{{\frac{x}{{a}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{-{a}^{2}+x}{{a}^{2}}}},{\frac{{a}^{2}}{{a}^{2}-a}},\sqrt{{\frac{{a}^{2}}{{a}^{2}-1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+x)/(-a+x)/(a^2*x-(a^2+1)*x^2+x^3)^(1/2),x)

[Out]

-2*a^2*(-(-a^2+x)/a^2)^(1/2)*((-1+x)/(a^2-1))^(1/2)*(x/a^2)^(1/2)/(-a^2*x^2+a^2*x+x^3-x^2)^(1/2)*EllipticF((-(
-a^2+x)/a^2)^(1/2),(a^2/(a^2-1))^(1/2))-4*a^3*(-(-a^2+x)/a^2)^(1/2)*((-1+x)/(a^2-1))^(1/2)*(x/a^2)^(1/2)/(-a^2
*x^2+a^2*x+x^3-x^2)^(1/2)/(a^2-a)*EllipticPi((-(-a^2+x)/a^2)^(1/2),a^2/(a^2-a),(a^2/(a^2-1))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{a + x}{\sqrt{a^{2} x -{\left (a^{2} + 1\right )} x^{2} + x^{3}}{\left (a - x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+x)/(-a+x)/(a^2*x-(a^2+1)*x^2+x^3)^(1/2),x, algorithm="maxima")

[Out]

-integrate((a + x)/(sqrt(a^2*x - (a^2 + 1)*x^2 + x^3)*(a - x)), x)

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Fricas [A]  time = 2.35953, size = 193, normalized size = 2.22 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+x)/(-a+x)/(a^2*x-(a^2+1)*x^2+x^3)^(1/2),x, algorithm="fricas")

[Out]

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + x}{\sqrt{x \left (- a^{2} + x\right ) \left (x - 1\right )} \left (- a + x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+x)/(-a+x)/(a**2*x-(a**2+1)*x**2+x**3)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{a + x}{\sqrt{a^{2} x -{\left (a^{2} + 1\right )} x^{2} + x^{3}}{\left (a - x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+x)/(-a+x)/(a^2*x-(a^2+1)*x^2+x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(-(a + x)/(sqrt(a^2*x - (a^2 + 1)*x^2 + x^3)*(a - x)), x)