3.1.6 \(\int \frac {\sqrt {b x+c x^2}}{x^2} \, dx\) [6]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {676, 634, 212} \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 212

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

Rule 634

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 676

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) \text {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*}

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Mathematica [A]
time = 0.06, size = 61, normalized size = 1.30 \begin {gather*} -\frac {2 \left (b+c x+\sqrt {c} \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{\sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x + c*x^2]/x^2,x]

[Out]

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

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Maple [A]
time = 0.50, size = 69, normalized size = 1.47

method	result	size
risch	\(-\frac {2 \left (c x +b \right )}{\sqrt {x \left (c x +b \right )}}+\sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )\)	\(46\)
default	\(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{b \,x^{2}}+\frac {2 c \left (\sqrt {c \,x^{2}+b x}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 \sqrt {c}}\right )}{b}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, 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

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Fricas [A]
time = 1.46, 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]

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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)

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Giac [A]
time = 2.26, 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))

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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)

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