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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0649497, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1979, 2007, 2029, 206} \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(a + b*x^3)/x^2],x]

[Out]

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

Rule 1979

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

Rule 2007

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(p*(n - j)), x] + Dist
[a, Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[
Simplify[j*p + 1], 0]

Rule 2029

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

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

Mathematica [A]  time = 0.0366667, size = 66, normalized size = 1.29 \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x} \left (\sqrt{a+b x^3}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\right )}{3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(a + b*x^3)/x^2],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 55, normalized size = 1.1 \begin{align*}{\frac{2\,x}{3}\sqrt{{\frac{b{x}^{3}+a}{{x}^{2}}}} \left ( -\sqrt{a}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) +\sqrt{b{x}^{3}+a} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{b x^{3} + a}{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, algorithm="maxima")

[Out]

integrate(sqrt((b*x^3 + a)/x^2), x)

________________________________________________________________________________________

Fricas [A]  time = 0.846674, size = 261, normalized size = 5.12 \begin{align*} \left [\frac{2}{3} \, x \sqrt{\frac{b x^{3} + a}{x^{2}}} + \frac{1}{3} \, \sqrt{a} \log \left (\frac{b x^{3} - 2 \, \sqrt{a} x \sqrt{\frac{b x^{3} + a}{x^{2}}} + 2 \, a}{x^{3}}\right ), \frac{2}{3} \, x \sqrt{\frac{b x^{3} + a}{x^{2}}} + \frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x \sqrt{\frac{b x^{3} + a}{x^{2}}}}{a}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.13836, size = 93, normalized size = 1.82 \begin{align*} \frac{2}{3} \,{\left (\frac{a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x^{3} + 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 )}{3 \, \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, algorithm="giac")

[Out]

2/3*(a*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + sqrt(b*x^3 + a))*sgn(x) - 2/3*(a*arctan(sqrt(a)/sqrt(-a)) +
sqrt(-a)*sqrt(a))*sgn(x)/sqrt(-a)