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

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

[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 = 0.0374281, 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 \[ \frac{a \sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{a \sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{1}{5} a x^2 \sqrt{a x^6}-\frac{a \sqrt{a x^6}}{x^2} \]

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)

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Rubi in Sympy [A]  time = 15.0726, size = 65, normalized size = 0.92 \[ - \frac{a x^{2} \sqrt{a x^{6}}}{5} - \frac{a \sqrt{a x^{6}}}{x^{2}} + \frac{a \sqrt{a x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} + \frac{a \sqrt{a x^{6}} \operatorname{atanh}{\left (x \right )}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**6)**(3/2)/x/(-x**4+1),x)

[Out]

-a*x**2*sqrt(a*x**6)/5 - a*sqrt(a*x**6)/x**2 + a*sqrt(a*x**6)*atan(x)/(2*x**3) +
a*sqrt(a*x**6)*atanh(x)/(2*x**3)

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Mathematica [A]  time = 0.037031, 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 = 0.014, size = 38, normalized size = 0.5 \[ -{\frac{4\,{x}^{5}+5\,\ln \left ( -1+x \right ) -5\,\ln \left ( 1+x \right ) -10\,\arctan \left ( x \right ) +20\,x}{20\,{x}^{9}} \left ( a{x}^{6} \right ) ^{{\frac{3}{2}}}} \]

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(-1+x)-5*ln(1+x)-10*arctan(x)+20*x)/x^9

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Maxima [A]  time = 0.774762, size = 54, normalized size = 0.76 \[ -\frac{1}{5} \, a^{\frac{3}{2}} x^{5} - a^{\frac{3}{2}} x + \frac{1}{2} \, a^{\frac{3}{2}} \arctan \left (x\right ) + \frac{1}{4} \, a^{\frac{3}{2}} \log \left (x + 1\right ) - \frac{1}{4} \, a^{\frac{3}{2}} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(a*x^6)^(3/2)/((x^4 - 1)*x),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 = 0.293685, size = 55, normalized size = 0.77 \[ -\frac{\sqrt{a x^{6}}{\left (4 \, a x^{5} + 20 \, a x - 10 \, a \arctan \left (x\right ) - 5 \, a \log \left (\frac{x + 1}{x - 1}\right )\right )}}{20 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(a*x^6)^(3/2)/((x^4 - 1)*x),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. \[ - \int \frac{\left (a x^{6}\right )^{\frac{3}{2}}}{x^{5} - x}\, dx \]

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/XCAS [A]  time = 0.264066, size = 57, normalized size = 0.8 \[ -\frac{1}{20} \,{\left (4 \, x^{5}{\rm sign}\left (x\right ) + 20 \, x{\rm sign}\left (x\right ) - 10 \, \arctan \left (x\right ){\rm sign}\left (x\right ) - 5 \,{\rm ln}\left ({\left | x + 1 \right |}\right ){\rm sign}\left (x\right ) + 5 \,{\rm ln}\left ({\left | x - 1 \right |}\right ){\rm sign}\left (x\right )\right )} a^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(4*x^5*sign(x) + 20*x*sign(x) - 10*arctan(x)*sign(x) - 5*ln(abs(x + 1))*si
gn(x) + 5*ln(abs(x - 1))*sign(x))*a^(3/2)