3.373 \(\int \frac{(a x^6)^{3/2}}{x (1-x^4)} \, dx\)

Optimal. Leaf size=71 \[ -\frac{1}{5} a x^2 \sqrt{a x^6}-\frac{a \sqrt{a x^6}}{x^2}+\frac{a \sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{a \sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3} \]

[Out]

-((a*Sqrt[a*x^6])/x^2) - (a*x^2*Sqrt[a*x^6])/5 + (a*Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + (a*Sqrt[a*x^6]*ArcTanh[x]
)/(2*x^3)

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Rubi [A]  time = 0.0146736, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {15, 302, 212, 206, 203} \[ -\frac{1}{5} a x^2 \sqrt{a x^6}-\frac{a \sqrt{a x^6}}{x^2}+\frac{a \sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{a \sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^6)^(3/2)/(x*(1 - x^4)),x]

[Out]

-((a*Sqrt[a*x^6])/x^2) - (a*x^2*Sqrt[a*x^6])/5 + (a*Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + (a*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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 212

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

Rubi steps

\begin{align*} \int \frac{\left (a x^6\right )^{3/2}}{x \left (1-x^4\right )} \, dx &=\frac{\left (a \sqrt{a x^6}\right ) \int \frac{x^8}{1-x^4} \, dx}{x^3}\\ &=\frac{\left (a \sqrt{a x^6}\right ) \int \left (-1-x^4+\frac{1}{1-x^4}\right ) \, dx}{x^3}\\ &=-\frac{a \sqrt{a x^6}}{x^2}-\frac{1}{5} a x^2 \sqrt{a x^6}+\frac{\left (a \sqrt{a x^6}\right ) \int \frac{1}{1-x^4} \, dx}{x^3}\\ &=-\frac{a \sqrt{a x^6}}{x^2}-\frac{1}{5} a x^2 \sqrt{a x^6}+\frac{\left (a \sqrt{a x^6}\right ) \int \frac{1}{1-x^2} \, dx}{2 x^3}+\frac{\left (a \sqrt{a x^6}\right ) \int \frac{1}{1+x^2} \, dx}{2 x^3}\\ &=-\frac{a \sqrt{a x^6}}{x^2}-\frac{1}{5} a x^2 \sqrt{a x^6}+\frac{a \sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{a \sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0154545, size = 44, normalized size = 0.62 \[ -\frac{a \sqrt{a x^6} \left (4 x^5+20 x+5 \log (1-x)-5 \log (x+1)-10 \tan ^{-1}(x)\right )}{20 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^6)^(3/2)/(x*(1 - x^4)),x]

[Out]

-(a*Sqrt[a*x^6]*(20*x + 4*x^5 - 10*ArcTan[x] + 5*Log[1 - x] - 5*Log[1 + x]))/(20*x^3)

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Maple [A]  time = 0.009, size = 38, normalized size = 0.5 \begin{align*} -{\frac{4\,{x}^{5}+5\,\ln \left ( x-1 \right ) -5\,\ln \left ( 1+x \right ) -10\,\arctan \left ( x \right ) +20\,x}{20\,{x}^{9}} \left ( a{x}^{6} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6)^(3/2)/x/(-x^4+1),x)

[Out]

-1/20*(a*x^6)^(3/2)*(4*x^5+5*ln(x-1)-5*ln(1+x)-10*arctan(x)+20*x)/x^9

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Maxima [A]  time = 1.67262, size = 54, normalized size = 0.76 \begin{align*} -\frac{1}{5} \, a^{\frac{3}{2}} x^{5} - a^{\frac{3}{2}} x + \frac{1}{2} \, a^{\frac{3}{2}} \arctan \left (x\right ) + \frac{1}{4} \, a^{\frac{3}{2}} \log \left (x + 1\right ) - \frac{1}{4} \, a^{\frac{3}{2}} \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6)^(3/2)/x/(-x^4+1),x, algorithm="maxima")

[Out]

-1/5*a^(3/2)*x^5 - a^(3/2)*x + 1/2*a^(3/2)*arctan(x) + 1/4*a^(3/2)*log(x + 1) - 1/4*a^(3/2)*log(x - 1)

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Fricas [A]  time = 1.30996, size = 116, normalized size = 1.63 \begin{align*} -\frac{\sqrt{a x^{6}}{\left (4 \, a x^{5} + 20 \, a x - 10 \, a \arctan \left (x\right ) - 5 \, a \log \left (\frac{x + 1}{x - 1}\right )\right )}}{20 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6)^(3/2)/x/(-x^4+1),x, algorithm="fricas")

[Out]

-1/20*sqrt(a*x^6)*(4*a*x^5 + 20*a*x - 10*a*arctan(x) - 5*a*log((x + 1)/(x - 1)))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\left (a x^{6}\right )^{\frac{3}{2}}}{x^{5} - x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**6)**(3/2)/x/(-x**4+1),x)

[Out]

-Integral((a*x**6)**(3/2)/(x**5 - x), x)

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Giac [A]  time = 1.12712, size = 57, normalized size = 0.8 \begin{align*} -\frac{1}{20} \,{\left (4 \, x^{5} \mathrm{sgn}\left (x\right ) + 20 \, x \mathrm{sgn}\left (x\right ) - 10 \, \arctan \left (x\right ) \mathrm{sgn}\left (x\right ) - 5 \, \log \left ({\left | x + 1 \right |}\right ) \mathrm{sgn}\left (x\right ) + 5 \, \log \left ({\left | x - 1 \right |}\right ) \mathrm{sgn}\left (x\right )\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6)^(3/2)/x/(-x^4+1),x, algorithm="giac")

[Out]

-1/20*(4*x^5*sgn(x) + 20*x*sgn(x) - 10*arctan(x)*sgn(x) - 5*log(abs(x + 1))*sgn(x) + 5*log(abs(x - 1))*sgn(x))
*a^(3/2)