3.125 \(\int \frac{1}{x^5 (a^3+x^3)} \, dx\)

Optimal. Leaf size=73 \[ -\frac{1}{4 a^3 x^4}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7}+\frac{1}{a^6 x}-\frac{\log (a+x)}{3 a^7}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a^7} \]

[Out]

-1/(4*a^3*x^4) + 1/(a^6*x) - ArcTan[(a - 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7) - Log[a + x]/(3*a^7) + Log[a^2 - a*x
+ x^2]/(6*a^7)

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Rubi [A]  time = 0.0440198, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {325, 292, 31, 634, 617, 204, 628} \[ -\frac{1}{4 a^3 x^4}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7}+\frac{1}{a^6 x}-\frac{\log (a+x)}{3 a^7}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a^3 + x^3)),x]

[Out]

-1/(4*a^3*x^4) + 1/(a^6*x) - ArcTan[(a - 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7) - Log[a + x]/(3*a^7) + Log[a^2 - a*x
+ x^2]/(6*a^7)

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (a^3+x^3\right )} \, dx &=-\frac{1}{4 a^3 x^4}-\frac{\int \frac{1}{x^2 \left (a^3+x^3\right )} \, dx}{a^3}\\ &=-\frac{1}{4 a^3 x^4}+\frac{1}{a^6 x}+\frac{\int \frac{x}{a^3+x^3} \, dx}{a^6}\\ &=-\frac{1}{4 a^3 x^4}+\frac{1}{a^6 x}-\frac{\int \frac{1}{a+x} \, dx}{3 a^7}+\frac{\int \frac{a+x}{a^2-a x+x^2} \, dx}{3 a^7}\\ &=-\frac{1}{4 a^3 x^4}+\frac{1}{a^6 x}-\frac{\log (a+x)}{3 a^7}+\frac{\int \frac{-a+2 x}{a^2-a x+x^2} \, dx}{6 a^7}+\frac{\int \frac{1}{a^2-a x+x^2} \, dx}{2 a^6}\\ &=-\frac{1}{4 a^3 x^4}+\frac{1}{a^6 x}-\frac{\log (a+x)}{3 a^7}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{a}\right )}{a^7}\\ &=-\frac{1}{4 a^3 x^4}+\frac{1}{a^6 x}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a^7}-\frac{\log (a+x)}{3 a^7}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7}\\ \end{align*}

Mathematica [A]  time = 0.0126279, size = 74, normalized size = 1.01 \[ -\frac{1}{4 a^3 x^4}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7}+\frac{1}{a^6 x}-\frac{\log (a+x)}{3 a^7}+\frac{\tan ^{-1}\left (\frac{2 x-a}{\sqrt{3} a}\right )}{\sqrt{3} a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a^3 + x^3)),x]

[Out]

-1/(4*a^3*x^4) + 1/(a^6*x) + ArcTan[(-a + 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7) - Log[a + x]/(3*a^7) + Log[a^2 - a*x
+ x^2]/(6*a^7)

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Maple [A]  time = 0.009, size = 67, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{3}{x}^{4}}}+{\frac{1}{{a}^{6}x}}+{\frac{\ln \left ({a}^{2}-ax+{x}^{2} \right ) }{6\,{a}^{7}}}+{\frac{\sqrt{3}}{3\,{a}^{7}}\arctan \left ({\frac{ \left ( 2\,x-a \right ) \sqrt{3}}{3\,a}} \right ) }-{\frac{\ln \left ( a+x \right ) }{3\,{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(a^3+x^3),x)

[Out]

-1/4/a^3/x^4+1/a^6/x+1/6*ln(a^2-a*x+x^2)/a^7+1/3/a^7*3^(1/2)*arctan(1/3*(2*x-a)*3^(1/2)/a)-1/3*ln(a+x)/a^7

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Maxima [A]  time = 1.41137, size = 89, normalized size = 1.22 \begin{align*} \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac{\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac{\log \left (a + x\right )}{3 \, a^{7}} - \frac{a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a^3+x^3),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a^7 + 1/6*log(a^2 - a*x + x^2)/a^7 - 1/3*log(a + x)/a^7 - 1/4*(a^
3 - 4*x^3)/(a^6*x^4)

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Fricas [A]  time = 2.08795, size = 178, normalized size = 2.44 \begin{align*} \frac{4 \, \sqrt{3} x^{4} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right ) + 2 \, x^{4} \log \left (a^{2} - a x + x^{2}\right ) - 4 \, x^{4} \log \left (a + x\right ) - 3 \, a^{4} + 12 \, a x^{3}}{12 \, a^{7} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a^3+x^3),x, algorithm="fricas")

[Out]

1/12*(4*sqrt(3)*x^4*arctan(-1/3*sqrt(3)*(a - 2*x)/a) + 2*x^4*log(a^2 - a*x + x^2) - 4*x^4*log(a + x) - 3*a^4 +
12*a*x^3)/(a^7*x^4)

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Sympy [C]  time = 0.375662, size = 90, normalized size = 1.23 \begin{align*} \frac{- a^{3} + 4 x^{3}}{4 a^{6} x^{4}} + \frac{- \frac{\log{\left (a + x \right )}}{3} + \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right ) \log{\left (9 a \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right )^{2} + x \right )} + \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right ) \log{\left (9 a \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right )^{2} + x \right )}}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(a**3+x**3),x)

[Out]

(-a**3 + 4*x**3)/(4*a**6*x**4) + (-log(a + x)/3 + (1/6 - sqrt(3)*I/6)*log(9*a*(1/6 - sqrt(3)*I/6)**2 + x) + (1
/6 + sqrt(3)*I/6)*log(9*a*(1/6 + sqrt(3)*I/6)**2 + x))/a**7

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Giac [A]  time = 1.07286, size = 90, normalized size = 1.23 \begin{align*} \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac{\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac{\log \left ({\left | a + x \right |}\right )}{3 \, a^{7}} - \frac{a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a^3+x^3),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a^7 + 1/6*log(a^2 - a*x + x^2)/a^7 - 1/3*log(abs(a + x))/a^7 - 1/
4*(a^3 - 4*x^3)/(a^6*x^4)