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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 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])

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 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 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]

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 = 0.01, 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)

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 127, normalized size = 2.89 \[ \left [-\sqrt {a} \arctan \left (\frac {\sqrt {a x^{3}}}{\sqrt {a} x}\right ) + \frac {1}{2} \, \sqrt {a} \log \left (\frac {a x^{2} + a x + 2 \, \sqrt {a x^{3}} \sqrt {a}}{x^{2} - x}\right ), -\sqrt {-a} \arctan \left (\frac {\sqrt {a x^{3}} \sqrt {-a}}{a x}\right ) + \frac {1}{2} \, \sqrt {-a} \log \left (\frac {a x^{2} - a x - 2 \, \sqrt {a x^{3}} \sqrt {-a}}{x^{2} + x}\right )\right ] \]

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))]

________________________________________________________________________________________

giac [A]  time = 0.21, size = 43, normalized size = 0.98 \[ -\frac {{\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a x}}{\sqrt {-a}}\right )}{\sqrt {-a}} + a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a x}}{\sqrt {a}}\right )\right )} \mathrm {sgn}\relax (x)}{a} \]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 43, normalized size = 0.98 \[ \frac {\sqrt {a \,x^{3}}\, \left (\arctanh \left (\frac {\sqrt {a x}}{\sqrt {a}}\right )-\arctan \left (\frac {\sqrt {a x}}{\sqrt {a}}\right )\right ) \sqrt {a}}{\sqrt {a x}\, x} \]

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)

________________________________________________________________________________________

maxima [A]  time = 2.18, size = 32, normalized size = 0.73 \[ -\sqrt {a} \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {a} \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \sqrt {a} \log \left (\sqrt {x} - 1\right ) \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a\,x^3}}{x-x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3)^(1/2)/(x - x^3),x)

[Out]

int((a*x^3)^(1/2)/(x - x^3), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {a x^{3}}}{x^{3} - x}\, dx \]

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)

________________________________________________________________________________________