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

Optimal. Leaf size=54 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}}-\frac{\sqrt{a x^2+b x^3}}{a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0489975, antiderivative size = 54, normalized size of antiderivative = 1., 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 = {2025, 2008, 206} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}}-\frac{\sqrt{a x^2+b x^3}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a*x^2 + b*x^3]),x]

[Out]

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

Rule 2025

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

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

Mathematica [A]  time = 0.0445731, size = 66, normalized size = 1.22 \[ \frac{2 b x (a+b x) \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )}{2 \sqrt{\frac{b x}{a}+1}}-\frac{a}{2 b x}\right )}{a^2 \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x*(a + b*x)*(-a/(2*b*x) + ArcTanh[Sqrt[1 + (b*x)/a]]/(2*Sqrt[1 + (b*x)/a])))/(a^2*Sqrt[x^2*(a + b*x)])

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Maple [A]  time = 0.007, size = 55, normalized size = 1. \begin{align*} -{\sqrt{bx+a} \left ({a}^{{\frac{3}{2}}}\sqrt{bx+a}-{\it Artanh} \left ({\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ) xab \right ){\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}{a}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(1/2)*(a^(3/2)*(b*x+a)^(1/2)-arctanh((b*x+a)^(1/2)/a^(1/2))*x*a*b)/(b*x^3+a*x^2)^(1/2)/a^(5/2)

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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}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.828951, size = 289, normalized size = 5.35 \begin{align*} \left [\frac{\sqrt{a} b x^{2} \log \left (\frac{b x^{2} + 2 \, a x + 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right ) - 2 \, \sqrt{b x^{3} + a x^{2}} a}{2 \, a^{2} x^{2}}, -\frac{\sqrt{-a} b x^{2} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) + \sqrt{b x^{3} + a x^{2}} a}{a^{2} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(a)*b*x^2*log((b*x^2 + 2*a*x + 2*sqrt(b*x^3 + a*x^2)*sqrt(a))/x^2) - 2*sqrt(b*x^3 + a*x^2)*a)/(a^2*x
^2), -(sqrt(-a)*b*x^2*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-a)/(a*x)) + sqrt(b*x^3 + a*x^2)*a)/(a^2*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError