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

Optimal. Leaf size=51 \[ \frac{2 \sqrt{a x^2+b x^3}}{x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0490906, antiderivative size = 51, 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 = {2021, 2008, 206} \[ \frac{2 \sqrt{a x^2+b x^3}}{x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^2 + b*x^3]/x^2,x]

[Out]

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

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*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{\sqrt{a x^2+b x^3}}{x^2} \, dx &=\frac{2 \sqrt{a x^2+b x^3}}{x}+a \int \frac{1}{\sqrt{a x^2+b x^3}} \, dx\\ &=\frac{2 \sqrt{a x^2+b x^3}}{x}-(2 a) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^3}}\right )\\ &=\frac{2 \sqrt{a x^2+b x^3}}{x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0278593, size = 53, normalized size = 1.04 \[ \frac{2 x \left (-\sqrt{a} \sqrt{a+b x} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+a+b x\right )}{\sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^2 + b*x^3]/x^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 52, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{b{x}^{3}+a{x}^{2}}}{x\sqrt{bx+a}} \left ( \sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) -\sqrt{bx+a} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^3 + a*x^2)/x^2, x)

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Fricas [A]  time = 0.850497, size = 247, normalized size = 4.84 \begin{align*} \left [\frac{\sqrt{a} x \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}}}{x}, \frac{2 \,{\left (\sqrt{-a} x \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) + \sqrt{b x^{3} + a x^{2}}\right )}}{x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21336, size = 88, normalized size = 1.73 \begin{align*} 2 \,{\left (\frac{a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x + a}\right )} \mathrm{sgn}\left (x\right ) - \frac{2 \,{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )} \mathrm{sgn}\left (x\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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