3.2364 \(\int \frac{\sqrt{-2-3 x+5 x^2}}{x} \, dx\)

Optimal. Leaf size=88 \[ \sqrt{5 x^2-3 x-2}+\sqrt{2} \tan ^{-1}\left (\frac{3 x+4}{2 \sqrt{2} \sqrt{5 x^2-3 x-2}}\right )+\frac{3 \tanh ^{-1}\left (\frac{3-10 x}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right )}{2 \sqrt{5}} \]

[Out]

Sqrt[-2 - 3*x + 5*x^2] + Sqrt[2]*ArcTan[(4 + 3*x)/(2*Sqrt[2]*Sqrt[-2 - 3*x + 5*x^2])] + (3*ArcTanh[(3 - 10*x)/
(2*Sqrt[5]*Sqrt[-2 - 3*x + 5*x^2])])/(2*Sqrt[5])

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Rubi [A]  time = 0.0487672, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {734, 843, 621, 206, 724, 204} \[ \sqrt{5 x^2-3 x-2}+\sqrt{2} \tan ^{-1}\left (\frac{3 x+4}{2 \sqrt{2} \sqrt{5 x^2-3 x-2}}\right )+\frac{3 \tanh ^{-1}\left (\frac{3-10 x}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-2 - 3*x + 5*x^2]/x,x]

[Out]

Sqrt[-2 - 3*x + 5*x^2] + Sqrt[2]*ArcTan[(4 + 3*x)/(2*Sqrt[2]*Sqrt[-2 - 3*x + 5*x^2])] + (3*ArcTanh[(3 - 10*x)/
(2*Sqrt[5]*Sqrt[-2 - 3*x + 5*x^2])])/(2*Sqrt[5])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\sqrt{-2-3 x+5 x^2}}{x} \, dx &=\sqrt{-2-3 x+5 x^2}-\frac{1}{2} \int \frac{4+3 x}{x \sqrt{-2-3 x+5 x^2}} \, dx\\ &=\sqrt{-2-3 x+5 x^2}-\frac{3}{2} \int \frac{1}{\sqrt{-2-3 x+5 x^2}} \, dx-2 \int \frac{1}{x \sqrt{-2-3 x+5 x^2}} \, dx\\ &=\sqrt{-2-3 x+5 x^2}-3 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-3+10 x}{\sqrt{-2-3 x+5 x^2}}\right )+4 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,\frac{-4-3 x}{\sqrt{-2-3 x+5 x^2}}\right )\\ &=\sqrt{-2-3 x+5 x^2}+\sqrt{2} \tan ^{-1}\left (\frac{4+3 x}{2 \sqrt{2} \sqrt{-2-3 x+5 x^2}}\right )+\frac{3 \tanh ^{-1}\left (\frac{3-10 x}{2 \sqrt{5} \sqrt{-2-3 x+5 x^2}}\right )}{2 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0359788, size = 84, normalized size = 0.95 \[ \sqrt{5 x^2-3 x-2}-\sqrt{2} \tan ^{-1}\left (\frac{-3 x-4}{2 \sqrt{10 x^2-6 x-4}}\right )-\frac{3 \tanh ^{-1}\left (\frac{10 x-3}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-2 - 3*x + 5*x^2]/x,x]

[Out]

Sqrt[-2 - 3*x + 5*x^2] - Sqrt[2]*ArcTan[(-4 - 3*x)/(2*Sqrt[-4 - 6*x + 10*x^2])] - (3*ArcTanh[(-3 + 10*x)/(2*Sq
rt[5]*Sqrt[-2 - 3*x + 5*x^2])])/(2*Sqrt[5])

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Maple [A]  time = 0.042, size = 71, normalized size = 0.8 \begin{align*} \sqrt{5\,{x}^{2}-3\,x-2}-{\frac{3\,\sqrt{5}}{10}\ln \left ({\frac{\sqrt{5}}{5} \left ( -{\frac{3}{2}}+5\,x \right ) }+\sqrt{5\,{x}^{2}-3\,x-2} \right ) }-\sqrt{2}\arctan \left ({\frac{ \left ( -3\,x-4 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{5\,{x}^{2}-3\,x-2}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*x^2-3*x-2)^(1/2)-3/10*ln(1/5*(-3/2+5*x)*5^(1/2)+(5*x^2-3*x-2)^(1/2))*5^(1/2)-2^(1/2)*arctan(1/4*(-3*x-4)*2^
(1/2)/(5*x^2-3*x-2)^(1/2))

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Maxima [A]  time = 1.51522, size = 81, normalized size = 0.92 \begin{align*} \sqrt{2} \arcsin \left (\frac{3 \, x}{7 \,{\left | x \right |}} + \frac{4}{7 \,{\left | x \right |}}\right ) - \frac{3}{10} \, \sqrt{5} \log \left (2 \, \sqrt{5} \sqrt{5 \, x^{2} - 3 \, x - 2} + 10 \, x - 3\right ) + \sqrt{5 \, x^{2} - 3 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*arcsin(3/7*x/abs(x) + 4/7/abs(x)) - 3/10*sqrt(5)*log(2*sqrt(5)*sqrt(5*x^2 - 3*x - 2) + 10*x - 3) + sqr
t(5*x^2 - 3*x - 2)

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Fricas [A]  time = 2.96689, size = 232, normalized size = 2.64 \begin{align*} \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (3 \, x + 4\right )}}{4 \, \sqrt{5 \, x^{2} - 3 \, x - 2}}\right ) + \frac{3}{20} \, \sqrt{5} \log \left (-4 \, \sqrt{5} \sqrt{5 \, x^{2} - 3 \, x - 2}{\left (10 \, x - 3\right )} + 200 \, x^{2} - 120 \, x - 31\right ) + \sqrt{5 \, x^{2} - 3 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*arctan(1/4*sqrt(2)*(3*x + 4)/sqrt(5*x^2 - 3*x - 2)) + 3/20*sqrt(5)*log(-4*sqrt(5)*sqrt(5*x^2 - 3*x - 2
)*(10*x - 3) + 200*x^2 - 120*x - 31) + sqrt(5*x^2 - 3*x - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((x - 1)*(5*x + 2))/x, x)

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Giac [A]  time = 1.10228, size = 104, normalized size = 1.18 \begin{align*} -2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} - 3 \, x - 2}\right )}\right ) + \frac{3}{10} \, \sqrt{5} \log \left ({\left | -10 \, \sqrt{5} x + 3 \, \sqrt{5} + 10 \, \sqrt{5 \, x^{2} - 3 \, x - 2} \right |}\right ) + \sqrt{5 \, x^{2} - 3 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(5)*x - sqrt(5*x^2 - 3*x - 2))) + 3/10*sqrt(5)*log(abs(-10*sqrt(5)*x + 3*s
qrt(5) + 10*sqrt(5*x^2 - 3*x - 2))) + sqrt(5*x^2 - 3*x - 2)