3.4.73 \(\int \frac {(a x^6)^{3/2}}{x (1-x^4)} \, dx\) [373]

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

[Out]

-a*(a*x^6)^(1/2)/x^2-1/5*a*x^2*(a*x^6)^(1/2)+1/2*a*arctan(x)*(a*x^6)^(1/2)/x^3+1/2*a*arctanh(x)*(a*x^6)^(1/2)/
x^3

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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, 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, 308, 218, 212, 209} \begin {gather*} \frac {a \sqrt {a x^6} \text {ArcTan}(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} \end {gather*}

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 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 308

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]

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*}

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Mathematica [A]
time = 0.01, size = 44, normalized size = 0.62 \begin {gather*} -\frac {a \sqrt {a x^6} \left (20 x+4 x^5-10 \tan ^{-1}(x)+5 \log (1-x)-5 \log (1+x)\right )}{20 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.21, size = 38, normalized size = 0.54

method	result	size
default	\(-\frac {\left (a \,x^{6}\right )^{\frac {3}{2}} \left (4 x^{5}+5 \ln \left (-1+x \right )-5 \ln \left (1+x \right )-10 \arctan \left (x \right )+20 x \right )}{20 x^{9}}\)	\(38\)
meijerg	\(-\frac {\left (a \,x^{6}\right )^{\frac {3}{2}} \left (-1\right )^{\frac {3}{4}} \left (-\frac {4 x \left (-1\right )^{\frac {1}{4}} \left (9 x^{4}+45\right )}{45}-\frac {x \left (-1\right )^{\frac {1}{4}} \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{4 x^{9}}\)	\(70\)
risch	\(-\frac {a \,x^{2} \sqrt {a \,x^{6}}}{5}-\frac {a \sqrt {a \,x^{6}}}{x^{2}}-\frac {a \sqrt {a \,x^{6}}\, \ln \left (-1+x \right )}{4 x^{3}}-\frac {i a \sqrt {a \,x^{6}}\, \ln \left (x -i\right )}{4 x^{3}}+\frac {i a \sqrt {a \,x^{6}}\, \ln \left (x +i\right )}{4 x^{3}}+\frac {a \sqrt {a \,x^{6}}\, \ln \left (1+x \right )}{4 x^{3}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.60, size = 40, normalized size = 0.56 \begin {gather*} -\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 {gather*}

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 = 0.35, size = 41, normalized size = 0.58 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\left (a x^{6}\right )^{\frac {3}{2}}}{x^{5} - x}\, dx \end {gather*}

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 = 4.09, size = 42, normalized size = 0.59 \begin {gather*} -\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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (a\,x^6\right )}^{3/2}}{x\,\left (x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x^6)^(3/2)/(x*(x^4 - 1)),x)

[Out]

-int((a*x^6)^(3/2)/(x*(x^4 - 1)), x)

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