3.1983 \(\int \frac {1}{(a+\frac {b}{x^3})^2} \, dx\)

Optimal. Leaf size=144 \[ \frac {2 \sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{7/3}}-\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{7/3}}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (a x^3+b\right )} \]

[Out]

4/3*x/a^2-1/3*x^4/a/(a*x^3+b)-4/9*b^(1/3)*ln(b^(1/3)+a^(1/3)*x)/a^(7/3)+2/9*b^(1/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)
*x+a^(2/3)*x^2)/a^(7/3)+4/9*b^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)/b^(1/3)*3^(1/2))/a^(7/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {193, 288, 321, 200, 31, 634, 617, 204, 628} \[ \frac {2 \sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{7/3}}-\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{7/3}}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^3)^(-2),x]

[Out]

(4*x)/(3*a^2) - x^4/(3*a*(b + a*x^3)) + (4*b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[
3]*a^(7/3)) - (4*b^(1/3)*Log[b^(1/3) + a^(1/3)*x])/(9*a^(7/3)) + (2*b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x +
a^(2/3)*x^2])/(9*a^(7/3))

Rule 31

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

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[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 288

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

Rule 321

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

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} \, dx &=\int \frac {x^6}{\left (b+a x^3\right )^2} \, dx\\ &=-\frac {x^4}{3 a \left (b+a x^3\right )}+\frac {4 \int \frac {x^3}{b+a x^3} \, dx}{3 a}\\ &=\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (b+a x^3\right )}-\frac {(4 b) \int \frac {1}{b+a x^3} \, dx}{3 a^2}\\ &=\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (b+a x^3\right )}-\frac {\left (4 \sqrt [3]{b}\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 a^2}-\frac {\left (4 \sqrt [3]{b}\right ) \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 a^2}\\ &=\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (b+a x^3\right )}-\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{7/3}}+\frac {\left (2 \sqrt [3]{b}\right ) \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 a^{7/3}}-\frac {\left (2 b^{2/3}\right ) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a^2}\\ &=\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (b+a x^3\right )}-\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{7/3}}-\frac {\left (4 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 a^{7/3}}\\ &=\frac {4 x}{3 a^2}-\frac {x^4}{3 a \left (b+a x^3\right )}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 a^{7/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 127, normalized size = 0.88 \[ \frac {2 \sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+\frac {3 \sqrt [3]{a} b x}{a x^3+b}-4 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+4 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+9 \sqrt [3]{a} x}{9 a^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^3)^(-2),x]

[Out]

(9*a^(1/3)*x + (3*a^(1/3)*b*x)/(b + a*x^3) + 4*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] - 4
*b^(1/3)*Log[b^(1/3) + a^(1/3)*x] + 2*b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(9*a^(7/3))

________________________________________________________________________________________

fricas [A]  time = 1.06, size = 146, normalized size = 1.01 \[ \frac {9 \, a x^{4} + 4 \, \sqrt {3} {\left (a x^{3} + b\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 2 \, {\left (a x^{3} + b\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 4 \, {\left (a x^{3} + b\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 12 \, b x}{9 \, {\left (a^{3} x^{3} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(9*a*x^4 + 4*sqrt(3)*(a*x^3 + b)*(-b/a)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b/a)^(2/3) - sqrt(3)*b)/b) - 2*(
a*x^3 + b)*(-b/a)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3)) + 4*(a*x^3 + b)*(-b/a)^(1/3)*log(x - (-b/a)^(
1/3)) + 12*b*x)/(a^3*x^3 + a^2*b)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 127, normalized size = 0.88 \[ \frac {4 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {x}{a^{2}} + \frac {b x}{3 \, {\left (a x^{3} + b\right )} a^{2}} - \frac {4 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {2 \, \left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/a^2 + x/a^2 + 1/3*b*x/((a*x^3 + b)*a^2) - 4/9*sqrt(3)*(-a^2*b)^(1/
3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/a^3 - 2/9*(-a^2*b)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (
-b/a)^(2/3))/a^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 115, normalized size = 0.80 \[ \frac {b x}{3 \left (a \,x^{3}+b \right ) a^{2}}+\frac {x}{a^{2}}-\frac {4 \sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {b}{a}\right )^{\frac {2}{3}} a^{3}}-\frac {4 b \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {b}{a}\right )^{\frac {2}{3}} a^{3}}+\frac {2 b \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {b}{a}\right )^{\frac {2}{3}} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*x+1/3/a^2*b*x/(a*x^3+b)-4/9/a^3*b/(1/a*b)^(2/3)*ln(x+(1/a*b)^(1/3))+2/9/a^3*b/(1/a*b)^(2/3)*ln(x^2-(1/a*
b)^(1/3)*x+(1/a*b)^(2/3))-4/9/a^3*b/(1/a*b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/a*b)^(1/3)*x-1))

________________________________________________________________________________________

maxima [A]  time = 1.98, size = 125, normalized size = 0.87 \[ \frac {b x}{3 \, {\left (a^{3} x^{3} + a^{2} b\right )}} + \frac {x}{a^{2}} - \frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {2 \, b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {4 \, b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.32, size = 132, normalized size = 0.92 \[ \frac {x}{a^2}+\frac {4\,{\left (-b\right )}^{1/3}\,\ln \left ({\left (-b\right )}^{4/3}+a^{1/3}\,b\,x\right )}{9\,a^{7/3}}+\frac {b\,x}{3\,\left (a^3\,x^3+b\,a^2\right )}-\frac {4\,{\left (-b\right )}^{1/3}\,\ln \left (4\,b\,x-\frac {4\,{\left (-b\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{7/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (4\,b\,x+\frac {9\,{\left (-b\right )}^{4/3}\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{a^{1/3}}\right )\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{a^{7/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^3)^2,x)

[Out]

x/a^2 + (4*(-b)^(1/3)*log((-b)^(4/3) + a^(1/3)*b*x))/(9*a^(7/3)) + (b*x)/(3*(a^2*b + a^3*x^3)) - (4*(-b)^(1/3)
*log(4*b*x - (4*(-b)^(4/3)*((3^(1/2)*1i)/2 + 1/2))/a^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*a^(7/3)) + ((-b)^(1/3)*
log(4*b*x + (9*(-b)^(4/3)*((3^(1/2)*2i)/9 - 2/9))/a^(1/3))*((3^(1/2)*2i)/9 - 2/9))/a^(7/3)

________________________________________________________________________________________

sympy [A]  time = 0.34, size = 48, normalized size = 0.33 \[ \frac {b x}{3 a^{3} x^{3} + 3 a^{2} b} + \operatorname {RootSum} {\left (729 t^{3} a^{7} + 64 b, \left (t \mapsto t \log {\left (- \frac {9 t a^{2}}{4} + x \right )} \right )\right )} + \frac {x}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x/(3*a**3*x**3 + 3*a**2*b) + RootSum(729*_t**3*a**7 + 64*b, Lambda(_t, _t*log(-9*_t*a**2/4 + x))) + x/a**2

________________________________________________________________________________________