3.372 \(\int \frac{\sqrt{a x^6}}{x-x^5} \, dx\)

Optimal. Leaf size=37 \[ \frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3} \]

[Out]

-(Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + (Sqrt[a*x^6]*ArcTanh[x])/(2*x^3)

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Rubi [A]  time = 0.012713, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {15, 1584, 298, 203, 206} \[ \frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^6]/(x - x^5),x]

[Out]

-(Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + (Sqrt[a*x^6]*ArcTanh[x])/(2*x^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 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 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^6}}{x-x^5} \, dx &=\frac{\sqrt{a x^6} \int \frac{x^3}{x-x^5} \, dx}{x^3}\\ &=\frac{\sqrt{a x^6} \int \frac{x^2}{1-x^4} \, dx}{x^3}\\ &=\frac{\sqrt{a x^6} \int \frac{1}{1-x^2} \, dx}{2 x^3}-\frac{\sqrt{a x^6} \int \frac{1}{1+x^2} \, dx}{2 x^3}\\ &=-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0045869, size = 33, normalized size = 0.89 \[ -\frac{\sqrt{a x^6} \left (\log (1-x)-\log (x+1)+2 \tan ^{-1}(x)\right )}{4 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^6]/(x - x^5),x]

[Out]

-(Sqrt[a*x^6]*(2*ArcTan[x] + Log[1 - x] - Log[1 + x]))/(4*x^3)

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Maple [A]  time = 0.006, size = 28, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( x-1 \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{a{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6)^(1/2)/(-x^5+x),x)

[Out]

-1/4*(a*x^6)^(1/2)*(ln(x-1)-ln(1+x)+2*arctan(x))/x^3

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Maxima [A]  time = 1.68832, size = 35, normalized size = 0.95 \begin{align*} -\frac{1}{2} \, \sqrt{a} \arctan \left (x\right ) + \frac{1}{4} \, \sqrt{a} \log \left (x + 1\right ) - \frac{1}{4} \, \sqrt{a} \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6)^(1/2)/(-x^5+x),x, algorithm="maxima")

[Out]

-1/2*sqrt(a)*arctan(x) + 1/4*sqrt(a)*log(x + 1) - 1/4*sqrt(a)*log(x - 1)

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Fricas [A]  time = 1.28832, size = 80, normalized size = 2.16 \begin{align*} -\frac{\sqrt{a x^{6}}{\left (2 \, \arctan \left (x\right ) - \log \left (\frac{x + 1}{x - 1}\right )\right )}}{4 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6)^(1/2)/(-x^5+x),x, algorithm="fricas")

[Out]

-1/4*sqrt(a*x^6)*(2*arctan(x) - log((x + 1)/(x - 1)))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt{a x^{6}}}{x^{5} - x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**6)**(1/2)/(-x**5+x),x)

[Out]

-Integral(sqrt(a*x**6)/(x**5 - x), x)

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Giac [A]  time = 1.11676, size = 39, normalized size = 1.05 \begin{align*} -\frac{1}{4} \,{\left (2 \, \arctan \left (x\right ) \mathrm{sgn}\left (x\right ) - \log \left ({\left | x + 1 \right |}\right ) \mathrm{sgn}\left (x\right ) + \log \left ({\left | x - 1 \right |}\right ) \mathrm{sgn}\left (x\right )\right )} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6)^(1/2)/(-x^5+x),x, algorithm="giac")

[Out]

-1/4*(2*arctan(x)*sgn(x) - log(abs(x + 1))*sgn(x) + log(abs(x - 1))*sgn(x))*sqrt(a)