\(\int \frac {1}{(a+\frac {b}{x^3}) x^2} \, dx\) [1972]

Optimal result

Integrand size = 13, antiderivative size = 115 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \]

[Out]

-1/3*ln(b^(1/3)+a^(1/3)*x)/a^(2/3)/b^(1/3)+1/6*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(2/3)/b^(1/3)-1/3*a
rctan(1/3*(b^(1/3)-2*a^(1/3)*x)/b^(1/3)*3^(1/2))/a^(2/3)/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {269, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}} \]

[In]

Int[1/((a + b/x^3)*x^2),x]

[Out]

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

Rule 31

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

Rule 210

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

Rule 269

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 298

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 631

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 642

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 648

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \]

[In]

Integrate[1/((a + b/x^3)*x^2),x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{3 a}\)	\(27\)
default	\(-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\)	\(91\)

[In]

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

[Out]

1/3/a*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*a+b))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.64 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (-a^{2} b\right )^{\frac {2}{3}} x}{a x^{3} + b}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} a x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, a x + \left (-a^{2} b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} a x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b + 1, \left ( t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )} \]

[In]

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

[Out]

RootSum(27*_t**3*a**2*b + 1, Lambda(_t, _t*log(9*_t**2*a*b + x)))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=-\frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.80 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {\ln \left (a^{1/3}\,x-{\left (-b\right )}^{1/3}\right )}{3\,a^{2/3}\,{\left (-b\right )}^{1/3}}+\frac {\ln \left (a\,x-\frac {a^{2/3}\,{\left (-b\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}}-\frac {\ln \left (a\,x-\frac {a^{2/3}\,{\left (-b\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}} \]

[In]

int(1/(x^2*(a + b/x^3)),x)

[Out]

log(a^(1/3)*x - (-b)^(1/3))/(3*a^(2/3)*(-b)^(1/3)) + (log(a*x - (a^(2/3)*(-b)^(1/3)*(3^(1/2)*1i - 1)^2)/4)*(3^
(1/2)*1i - 1))/(6*a^(2/3)*(-b)^(1/3)) - (log(a*x - (a^(2/3)*(-b)^(1/3)*(3^(1/2)*1i + 1)^2)/4)*(3^(1/2)*1i + 1)
)/(6*a^(2/3)*(-b)^(1/3))