3.1.6 \(\int \frac {\sqrt {b x+c x^2}}{x^2} \, dx\)

Optimal. Leaf size=47 \[ 2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 \sqrt {b x+c x^2}}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {662, 620, 206} \begin {gather*} 2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 \sqrt {b x+c x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/x^2,x]

[Out]

(-2*Sqrt[b*x + c*x^2])/x + 2*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]]

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 620

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

Rule 662

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 + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\sqrt {b x+c x^2}}{x^2} \, dx &=-\frac {2 \sqrt {b x+c x^2}}{x}+c \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 \sqrt {b x+c x^2}}{x}+(2 c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {2 \sqrt {b x+c x^2}}{x}+2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 63, normalized size = 1.34 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (\frac {\sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {\frac {c x}{b}+1}}-1\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x + c*x^2]/x^2,x]

[Out]

(2*Sqrt[x*(b + c*x)]*(-1 + (Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/
x

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.13, size = 53, normalized size = 1.13 \begin {gather*} -\frac {2 \sqrt {b x+c x^2}}{x}-\sqrt {c} \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b*x + c*x^2]/x^2,x]

[Out]

(-2*Sqrt[b*x + c*x^2])/x - Sqrt[c]*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]]

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 95, normalized size = 2.02 \begin {gather*} \left [\frac {\sqrt {c} x \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, \sqrt {c x^{2} + b x}}{x}, -\frac {2 \, {\left (\sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c x^{2} + b x}\right )}}{x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^2,x, algorithm="fricas")

[Out]

[(sqrt(c)*x*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*sqrt(c*x^2 + b*x))/x, -2*(sqrt(-c)*x*arctan(sqrt(
c*x^2 + b*x)*sqrt(-c)/(c*x)) + sqrt(c*x^2 + b*x))/x]

________________________________________________________________________________________

giac [A]  time = 0.31, size = 60, normalized size = 1.28 \begin {gather*} -\sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \frac {2 \, b}{\sqrt {c} x - \sqrt {c x^{2} + b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^2,x, algorithm="giac")

[Out]

-sqrt(c)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)) + 2*b/(sqrt(c)*x - sqrt(c*x^2 + b*x))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 66, normalized size = 1.40 \begin {gather*} \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )+\frac {2 \sqrt {c \,x^{2}+b x}\, c}{b}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{b \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/x^2,x)

[Out]

-2/b/x^2*(c*x^2+b*x)^(3/2)+2/b*c*(c*x^2+b*x)^(1/2)+c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))

________________________________________________________________________________________

maxima [A]  time = 1.05, size = 44, normalized size = 0.94 \begin {gather*} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {2 \, \sqrt {c x^{2} + b x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^2,x, algorithm="maxima")

[Out]

sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*sqrt(c*x^2 + b*x)/x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(1/2)/x^2,x)

[Out]

int((b*x + c*x^2)^(1/2)/x^2, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(1/2)/x**2,x)

[Out]

Integral(sqrt(x*(b + c*x))/x**2, x)

________________________________________________________________________________________