3.256 \(\int \frac{1}{\sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}} \]

[Out]

(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/Sqrt[a]

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Rubi [A]  time = 0.0105473, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2008, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*x^2 + b*x^3],x]

[Out]

(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/Sqrt[a]

Rule 2008

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

Rubi steps

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

Mathematica [A]  time = 0.0089751, size = 46, normalized size = 1.53 \[ -\frac{2 x \sqrt{a+b x} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*x^2 + b*x^3],x]

[Out]

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

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Maple [A]  time = 0.004, size = 39, normalized size = 1.3 \begin{align*} -2\,{\frac{x\sqrt{bx+a}}{\sqrt{b{x}^{3}+a{x}^{2}}\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(b*x^3+a*x^2)^(1/2)*x*(b*x+a)^(1/2)/a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{3} + a x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*x^3 + a*x^2), x)

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Fricas [A]  time = 0.803761, size = 171, normalized size = 5.7 \begin{align*} \left [\frac{\log \left (\frac{b x^{2} + 2 \, a x - 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right )}{\sqrt{a}}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[log((b*x^2 + 2*a*x - 2*sqrt(b*x^3 + a*x^2)*sqrt(a))/x^2)/sqrt(a), 2*sqrt(-a)*arctan(sqrt(b*x^3 + a*x^2)*sqrt(
-a)/(a*x))/a]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*x**2 + b*x**3), x)

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Giac [A]  time = 1.16818, size = 61, normalized size = 2.03 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{-a}} + \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(a)/sqrt(-a))*sgn(x)/sqrt(-a) + 2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*sgn(x))