3.10.89 \(\int \frac {1}{\sqrt {\frac {a x^2+b x^3}{x}}} \, dx\) [989]

Optimal. Leaf size=28 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}} \]

[Out]

2*arctanh(x*b^(1/2)/(b*x^2+a*x)^(1/2))/b^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2004, 634, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a*x^2 + b*x^3)/x],x]

[Out]

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

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 2004

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 55, normalized size = 1.96 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b} \sqrt {x (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 29, normalized size = 1.04

method	result	size
default	\(\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x^3+a*x^2)/x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))/b^(1/2)

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Maxima [A]
time = 0.30, size = 27, normalized size = 0.96 \begin {gather*} \frac {\log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/sqrt(b)

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Fricas [A]
time = 0.32, size = 62, normalized size = 2.21 \begin {gather*} \left [\frac {\log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right )}{b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\frac {a x^{2} + b x^{3}}{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt((a*x**2 + b*x**3)/x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
time = 2.80, size = 61, normalized size = 2.18 \begin {gather*} \frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (2 \, x + \frac {a}{b}\right )} + \frac {a^{2} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{8 \, b^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(b*x^2 + a*x)*(2*x + a/b) + 1/8*a^2*log(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(3/2)

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Mupad [B]
time = 3.52, size = 28, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x^2 + b*x^3)/x)^(1/2),x)

[Out]

log((a/2 + b*x)/b^(1/2) + (a*x + b*x^2)^(1/2))/b^(1/2)

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