3.1989 \(\int \frac {1}{(a+\frac {b}{x^3})^2 x^6} \, dx\)

Optimal. Leaf size=134 \[ -\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3}}+\frac {x}{3 b \left (a x^3+b\right )} \]

[Out]

1/3*x/b/(a*x^3+b)+2/9*ln(b^(1/3)+a^(1/3)*x)/a^(1/3)/b^(5/3)-1/9*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(1
/3)/b^(5/3)-2/9*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)/b^(1/3)*3^(1/2))/a^(1/3)/b^(5/3)*3^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {263, 199, 200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3}}+\frac {x}{3 b \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)^2*x^6),x]

[Out]

x/(3*b*(b + a*x^3)) - (2*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(5/3)) + (2*L
og[b^(1/3) + a^(1/3)*x])/(9*a^(1/3)*b^(5/3)) - Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(9*a^(1/3)*b^(5/
3))

Rule 31

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 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]

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]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^6} \, dx &=\int \frac {1}{\left (b+a x^3\right )^2} \, dx\\ &=\frac {x}{3 b \left (b+a x^3\right )}+\frac {2 \int \frac {1}{b+a x^3} \, dx}{3 b}\\ &=\frac {x}{3 b \left (b+a x^3\right )}+\frac {2 \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 b^{5/3}}+\frac {2 \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 b^{5/3}}\\ &=\frac {x}{3 b \left (b+a x^3\right )}+\frac {2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 \sqrt [3]{a} b^{5/3}}+\frac {\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 b^{4/3}}\\ &=\frac {x}{3 b \left (b+a x^3\right )}+\frac {2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{a} b^{5/3}}\\ &=\frac {x}{3 b \left (b+a x^3\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3}}+\frac {2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 118, normalized size = 0.88 \[ \frac {-\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{\sqrt [3]{a}}+\frac {3 b^{2/3} x}{a x^3+b}+\frac {2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{9 b^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)^2*x^6),x]

[Out]

((3*b^(2/3)*x)/(b + a*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]])/a^(1/3) + (2*Log[b^(1/3)
+ a^(1/3)*x])/a^(1/3) - Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/a^(1/3))/(9*b^(5/3))

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fricas [A]  time = 0.85, size = 389, normalized size = 2.90 \[ \left [\frac {3 \, a b^{2} x + 3 \, \sqrt {\frac {1}{3}} {\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b x - b^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a x^{3} + b}\right ) - {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{2} b^{3} x^{3} + a b^{4}\right )}}, \frac {3 \, a b^{2} x + 6 \, \sqrt {\frac {1}{3}} {\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) - {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{2} b^{3} x^{3} + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/9*(3*a*b^2*x + 3*sqrt(1/3)*(a^2*b*x^3 + a*b^2)*sqrt(-(a*b^2)^(1/3)/a)*log((2*a*b*x^3 - 3*(a*b^2)^(1/3)*b*x
- b^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a*b^2)^(2/3)*x - (a*b^2)^(1/3)*b)*sqrt(-(a*b^2)^(1/3)/a))/(a*x^3 + b)) - (a*
x^3 + b)*(a*b^2)^(2/3)*log(a*b*x^2 - (a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b) + 2*(a*x^3 + b)*(a*b^2)^(2/3)*log(a*b*
x + (a*b^2)^(2/3)))/(a^2*b^3*x^3 + a*b^4), 1/9*(3*a*b^2*x + 6*sqrt(1/3)*(a^2*b*x^3 + a*b^2)*sqrt((a*b^2)^(1/3)
/a)*arctan(sqrt(1/3)*(2*(a*b^2)^(2/3)*x - (a*b^2)^(1/3)*b)*sqrt((a*b^2)^(1/3)/a)/b^2) - (a*x^3 + b)*(a*b^2)^(2
/3)*log(a*b*x^2 - (a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b) + 2*(a*x^3 + b)*(a*b^2)^(2/3)*log(a*b*x + (a*b^2)^(2/3)))
/(a^2*b^3*x^3 + a*b^4)]

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giac [A]  time = 0.17, size = 127, normalized size = 0.95 \[ -\frac {2 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, b^{2}} + \frac {x}{3 \, {\left (a x^{3} + b\right )} b} + \frac {2 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{2}} + \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/b^2 + 1/3*x/((a*x^3 + b)*b) + 2/9*sqrt(3)*(-a^2*b)^(1/3)*arctan(1
/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/(a*b^2) + 1/9*(-a^2*b)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(
2/3))/(a*b^2)

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maple [A]  time = 0.00, size = 115, normalized size = 0.86 \[ \frac {x}{3 \left (a \,x^{3}+b \right ) b}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {b}{a}\right )^{\frac {2}{3}} a b}+\frac {2 \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {b}{a}\right )^{\frac {2}{3}} a b}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {b}{a}\right )^{\frac {2}{3}} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^3)^2/x^6,x)

[Out]

1/3*x/b/(a*x^3+b)+2/9/b/a/(1/a*b)^(2/3)*ln(x+(1/a*b)^(1/3))-1/9/b/a/(1/a*b)^(2/3)*ln(x^2-(1/a*b)^(1/3)*x+(1/a*
b)^(2/3))+2/9/b/a/(1/a*b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/a*b)^(1/3)*x-1))

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maxima [A]  time = 1.89, size = 122, normalized size = 0.91 \[ \frac {x}{3 \, {\left (a b x^{3} + b^{2}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a b \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a b \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x/(a*b*x^3 + b^2) + 2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/3))/(a*b*(b/a)^(2/3)) - 1/
9*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3))/(a*b*(b/a)^(2/3)) + 2/9*log(x + (b/a)^(1/3))/(a*b*(b/a)^(2/3))

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mupad [B]  time = 0.20, size = 128, normalized size = 0.96 \[ \frac {x}{3\,b\,\left (a\,x^3+b\right )}+\frac {2\,\ln \left (\frac {2\,a^{5/3}}{b^{2/3}}+\frac {2\,a^2\,x}{b}\right )}{9\,a^{1/3}\,b^{5/3}}+\frac {\ln \left (\frac {2\,a^2\,x}{b}+\frac {a^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{1/3}\,b^{5/3}}-\frac {\ln \left (\frac {2\,a^2\,x}{b}-\frac {a^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{1/3}\,b^{5/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(a + b/x^3)^2),x)

[Out]

x/(3*b*(b + a*x^3)) + (2*log((2*a^(5/3))/b^(2/3) + (2*a^2*x)/b))/(9*a^(1/3)*b^(5/3)) + (log((2*a^2*x)/b + (a^(
5/3)*(3^(1/2)*1i - 1))/b^(2/3))*(3^(1/2)*1i - 1))/(9*a^(1/3)*b^(5/3)) - (log((2*a^2*x)/b - (a^(5/3)*(3^(1/2)*1
i + 1))/b^(2/3))*(3^(1/2)*1i + 1))/(9*a^(1/3)*b^(5/3))

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sympy [A]  time = 0.33, size = 39, normalized size = 0.29 \[ \frac {x}{3 a b x^{3} + 3 b^{2}} + \operatorname {RootSum} {\left (729 t^{3} a b^{5} - 8, \left (t \mapsto t \log {\left (\frac {9 t b^{2}}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(3*a*b*x**3 + 3*b**2) + RootSum(729*_t**3*a*b**5 - 8, Lambda(_t, _t*log(9*_t*b**2/2 + x)))

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