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

Optimal. Leaf size=24 \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

[Out]

(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3

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Rubi [A]  time = 0.0087115, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {15, 266, 63, 207} \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^2]/Sqrt[1 + x^3],x]

[Out]

(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0039839, size = 24, normalized size = 1. \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^2]/Sqrt[1 + x^3],x]

[Out]

(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3

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Maple [A]  time = 0.006, size = 19, normalized size = 0.8 \begin{align*} -{\frac{2\,x}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{\frac{a}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.03697, size = 177, normalized size = 7.38 \begin{align*} \left [\frac{1}{3} \, x \sqrt{\frac{a}{x^{2}}} \log \left (\frac{x^{3} - 2 \, \sqrt{x^{3} + 1} + 2}{x^{3}}\right ), \frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{x^{3} + 1} \sqrt{-a} x \sqrt{\frac{a}{x^{2}}}}{a x^{3} + a}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16676, size = 42, normalized size = 1.75 \begin{align*} -\frac{1}{3} \, \sqrt{a}{\left (\log \left (\sqrt{x^{3} + 1} + 1\right ) - \log \left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right )\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(a)*(log(sqrt(x^3 + 1) + 1) - log(abs(sqrt(x^3 + 1) - 1)))*sgn(x)