Integrand size = 13, antiderivative size = 115 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \] Output:
-1/3*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3^(1/2)/b^(1/3))*3^(1/2)/a^(2/3)/b^( 1/3)-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)
Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \] Input:
Integrate[1/((a + b/x^3)*x^2),x]
Output:
(-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))
Time = 0.45 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {795, 821, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^2 \left (a+\frac {b}{x^3}\right )} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {x}{a x^3+b}dx\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
Input:
Int[1/((a + b/x^3)*x^2),x]
Output:
-1/3*Log[b^(1/3) + a^(1/3)*x]/(a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]])/a^(1/3)) + Log[b^(2/3) - a^(1/3)*b^(1/3)* x + a^(2/3)*x^2]/(2*a^(1/3)))/(3*a^(1/3)*b^(1/3))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (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 | \(-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\) | \(91\) |
Input:
int(1/(a+b/x^3)/x^2,x,method=_RETURNVERBOSE)
Output:
1/3/a*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*a+b))
Time = 0.09 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.64 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (-a^{2} b\right )^{\frac {2}{3}} x}{a x^{3} + b}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} a x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, a x + \left (-a^{2} b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} a x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b}\right ] \] Input:
integrate(1/(a+b/x^3)/x^2,x, algorithm="fricas")
Output:
[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)*lo g(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)]
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b + 1, \left ( t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )} \] Input:
integrate(1/(a+b/x**3)/x**2,x)
Output:
RootSum(27*_t**3*a**2*b + 1, Lambda(_t, _t*log(9*_t**2*a*b + x)))
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} \] Input:
integrate(1/(a+b/x^3)/x^2,x, algorithm="maxima")
Output:
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*l og(x + (b/a)^(1/3))/(a*(b/a)^(1/3))
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=-\frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} \] Input:
integrate(1/(a+b/x^3)/x^2,x, algorithm="giac")
Output:
-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)
Time = 0.54 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {\ln \left (a^{1/3}\,x-{\left (-b\right )}^{1/3}\right )}{3\,a^{2/3}\,{\left (-b\right )}^{1/3}}+\frac {\ln \left (a\,x-\frac {a^{2/3}\,{\left (-b\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}}-\frac {\ln \left (a\,x-\frac {a^{2/3}\,{\left (-b\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}} \] Input:
int(1/(x^2*(a + b/x^3)),x)
Output:
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))
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right )+\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right )-2 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}} a^{\frac {2}{3}}} \] Input:
int(1/(a+b/x^3)/x^2,x)
Output:
(2*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3))) + log(a**(2/ 3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3)) - 2*log(a**(1/3)*x + b**(1/3)))/ (6*b**(1/3)*a**(2/3))