3.376 \(\int \frac{\sqrt{a x^3}}{x-x^3} \, dx\)

Optimal. Leaf size=44 \[ \frac{\sqrt{a x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{a x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0148215, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {15, 1584, 329, 298, 203, 206} \[ \frac{\sqrt{a x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{a x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^3]/(x - x^3),x]

[Out]

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

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 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

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^3}}{x-x^3} \, dx &=\frac{\sqrt{a x^3} \int \frac{x^{3/2}}{x-x^3} \, dx}{x^{3/2}}\\ &=\frac{\sqrt{a x^3} \int \frac{\sqrt{x}}{1-x^2} \, dx}{x^{3/2}}\\ &=\frac{\left (2 \sqrt{a x^3}\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\sqrt{x}\right )}{x^{3/2}}\\ &=\frac{\sqrt{a x^3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{a x^3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )}{x^{3/2}}\\ &=-\frac{\sqrt{a x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\frac{\sqrt{a x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0091698, size = 30, normalized size = 0.68 \[ \frac{\sqrt{a x^3} \left (\tanh ^{-1}\left (\sqrt{x}\right )-\tan ^{-1}\left (\sqrt{x}\right )\right )}{x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^3]/(x - x^3),x]

[Out]

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

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Maple [A]  time = 0.011, size = 43, normalized size = 1. \begin{align*}{\frac{1}{x}\sqrt{a{x}^{3}}\sqrt{a} \left ({\it Artanh} \left ({\sqrt{ax}{\frac{1}{\sqrt{a}}}} \right ) -\arctan \left ({\sqrt{ax}{\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.71386, size = 43, normalized size = 0.98 \begin{align*} -\sqrt{a} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \sqrt{a} \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \sqrt{a} \log \left (\sqrt{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*arctan(sqrt(x)) + 1/2*sqrt(a)*log(sqrt(x) + 1) - 1/2*sqrt(a)*log(sqrt(x) - 1)

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Fricas [A]  time = 1.3382, size = 308, normalized size = 7. \begin{align*} \left [-\sqrt{a} \arctan \left (\frac{\sqrt{a x^{3}}}{\sqrt{a} x}\right ) + \frac{1}{2} \, \sqrt{a} \log \left (\frac{a x^{2} + a x + 2 \, \sqrt{a x^{3}} \sqrt{a}}{x^{2} - x}\right ), -\sqrt{-a} \arctan \left (\frac{\sqrt{a x^{3}} \sqrt{-a}}{a x}\right ) + \frac{1}{2} \, \sqrt{-a} \log \left (\frac{a x^{2} - a x - 2 \, \sqrt{a x^{3}} \sqrt{-a}}{x^{2} + x}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(a*x**3)/(x**3 - x), x)

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Giac [A]  time = 1.12935, size = 51, normalized size = 1.16 \begin{align*} -{\left (\frac{a \arctan \left (\frac{\sqrt{a x}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{a} \arctan \left (\frac{\sqrt{a x}}{\sqrt{a}}\right )\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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