3.220 \(\int \frac{\sqrt{-5+x} \sqrt{3+x}}{(-1+x) (-25+x^2)} \, dx\)

Optimal. Leaf size=54 \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{4} \sqrt{x-5} \sqrt{x+3}\right )+\frac{\tanh ^{-1}\left (\frac{\sqrt{5} \sqrt{x+3}}{\sqrt{x-5}}\right )}{3 \sqrt{5}} \]

[Out]

ArcTan[(Sqrt[-5 + x]*Sqrt[3 + x])/4]/6 + ArcTanh[(Sqrt[5]*Sqrt[3 + x])/Sqrt[-5 + x]]/(3*Sqrt[5])

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Rubi [A]  time = 0.0959866, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1586, 178, 92, 203, 93, 206} \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{4} \sqrt{x-5} \sqrt{x+3}\right )+\frac{\tanh ^{-1}\left (\frac{\sqrt{5} \sqrt{x+3}}{\sqrt{x-5}}\right )}{3 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-5 + x]*Sqrt[3 + x])/((-1 + x)*(-25 + x^2)),x]

[Out]

ArcTan[(Sqrt[-5 + x]*Sqrt[3 + x])/4]/6 + ArcTanh[(Sqrt[5]*Sqrt[3 + x])/Sqrt[-5 + x]]/(3*Sqrt[5])

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 178

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbo
l] :> Dist[(b*e - a*f)/(b*c - a*d), Int[((e + f*x)^(p - 1)*(g + h*x)^q)/(a + b*x), x], x] - Dist[(d*e - c*f)/(
b*c - a*d), Int[((e + f*x)^(p - 1)*(g + h*x)^q)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] &&
LtQ[0, p, 1]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{-5+x} \sqrt{3+x}}{(-1+x) \left (-25+x^2\right )} \, dx &=\int \frac{\sqrt{3+x}}{\sqrt{-5+x} (-1+x) (5+x)} \, dx\\ &=\frac{1}{3} \int \frac{1}{\sqrt{-5+x} \sqrt{3+x} (5+x)} \, dx+\frac{2}{3} \int \frac{1}{\sqrt{-5+x} (-1+x) \sqrt{3+x}} \, dx\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{2-10 x^2} \, dx,x,\frac{\sqrt{3+x}}{\sqrt{-5+x}}\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{16+x^2} \, dx,x,\sqrt{-5+x} \sqrt{3+x}\right )\\ &=\frac{1}{6} \tan ^{-1}\left (\frac{1}{4} \sqrt{-5+x} \sqrt{3+x}\right )+\frac{\tanh ^{-1}\left (\frac{\sqrt{5} \sqrt{3+x}}{\sqrt{-5+x}}\right )}{3 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.102016, size = 47, normalized size = 0.87 \[ \frac{1}{15} \left (5 \tan ^{-1}\left (\sqrt{\frac{x-5}{x+3}}\right )+\sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{\frac{x-5}{x+3}}}{\sqrt{5}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[-5 + x]*Sqrt[3 + x])/((-1 + x)*(-25 + x^2)),x]

[Out]

(5*ArcTan[Sqrt[(-5 + x)/(3 + x)]] + Sqrt[5]*ArcTanh[Sqrt[(-5 + x)/(3 + x)]/Sqrt[5]])/15

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Maple [A]  time = 0.023, size = 64, normalized size = 1.2 \begin{align*}{\frac{1}{30}\sqrt{-5+x}\sqrt{3+x} \left ( \sqrt{5}{\it Artanh} \left ({\frac{ \left ( 5+3\,x \right ) \sqrt{5}}{5}{\frac{1}{\sqrt{{x}^{2}-2\,x-15}}}} \right ) -5\,\arctan \left ( 4\,{\frac{1}{\sqrt{{x}^{2}-2\,x-15}}} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-2\,x-15}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5+x)^(1/2)*(3+x)^(1/2)/(-1+x)/(x^2-25),x)

[Out]

1/30*(-5+x)^(1/2)*(3+x)^(1/2)*(5^(1/2)*arctanh(1/5*(5+3*x)*5^(1/2)/(x^2-2*x-15)^(1/2))-5*arctan(4/(x^2-2*x-15)
^(1/2)))/(x^2-2*x-15)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x + 3} \sqrt{x - 5}}{{\left (x^{2} - 25\right )}{\left (x - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+x)^(1/2)*(3+x)^(1/2)/(-1+x)/(x^2-25),x, algorithm="maxima")

[Out]

integrate(sqrt(x + 3)*sqrt(x - 5)/((x^2 - 25)*(x - 1)), x)

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Fricas [A]  time = 1.82628, size = 211, normalized size = 3.91 \begin{align*} \frac{1}{30} \, \sqrt{5} \log \left (\frac{\sqrt{x + 3} \sqrt{x - 5}{\left (3 \, \sqrt{5} + 5\right )} + \sqrt{5}{\left (3 \, x + 5\right )} + 9 \, x + 15}{x + 5}\right ) + \frac{1}{3} \, \arctan \left (\frac{1}{4} \, \sqrt{x + 3} \sqrt{x - 5} - \frac{1}{4} \, x + \frac{1}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+x)^(1/2)*(3+x)^(1/2)/(-1+x)/(x^2-25),x, algorithm="fricas")

[Out]

1/30*sqrt(5)*log((sqrt(x + 3)*sqrt(x - 5)*(3*sqrt(5) + 5) + sqrt(5)*(3*x + 5) + 9*x + 15)/(x + 5)) + 1/3*arcta
n(1/4*sqrt(x + 3)*sqrt(x - 5) - 1/4*x + 1/4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x + 3}}{\sqrt{x - 5} \left (x - 1\right ) \left (x + 5\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+x)**(1/2)*(3+x)**(1/2)/(-1+x)/(x**2-25),x)

[Out]

Integral(sqrt(x + 3)/(sqrt(x - 5)*(x - 1)*(x + 5)), x)

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Giac [B]  time = 1.08187, size = 100, normalized size = 1.85 \begin{align*} -\frac{1}{30} \, \sqrt{5} \log \left (\frac{{\left (\sqrt{x + 3} - \sqrt{x - 5}\right )}^{2} - 4 \, \sqrt{5} + 12}{{\left (\sqrt{x + 3} - \sqrt{x - 5}\right )}^{2} + 4 \, \sqrt{5} + 12}\right ) - \frac{1}{3} \, \arctan \left (\frac{1}{8} \,{\left (\sqrt{x + 3} - \sqrt{x - 5}\right )}^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+x)^(1/2)*(3+x)^(1/2)/(-1+x)/(x^2-25),x, algorithm="giac")

[Out]

-1/30*sqrt(5)*log(((sqrt(x + 3) - sqrt(x - 5))^2 - 4*sqrt(5) + 12)/((sqrt(x + 3) - sqrt(x - 5))^2 + 4*sqrt(5)
+ 12)) - 1/3*arctan(1/8*(sqrt(x + 3) - sqrt(x - 5))^2)