Integrand size = 11, antiderivative size = 124 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {x^2}{2 a}+\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}} \] Output:
1/2*x^2/a+1/3*b^(2/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3^(1/2)/b^(1/3))*3^ (1/2)/a^(5/3)+1/3*b^(2/3)*ln(b^(1/3)+a^(1/3)*x)/a^(5/3)-1/6*b^(2/3)*ln(b^( 2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(5/3)
Time = 0.01 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {3 a^{2/3} x^2+2 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}} \] Input:
Integrate[x/(a + b/x^3),x]
Output:
(3*a^(2/3)*x^2 + 2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt [3]] + 2*b^(2/3)*Log[b^(1/3) + a^(1/3)*x] - b^(2/3)*Log[b^(2/3) - a^(1/3)* b^(1/3)*x + a^(2/3)*x^2])/(6*a^(5/3))
Time = 0.48 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {795, 843, 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.
| \(\displaystyle \int \frac {x}{a+\frac {b}{x^3}} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {x^4}{a x^3+b}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \int \frac {x}{a x^3+b}dx}{a}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^2}{2 a}-\frac {b \left (\frac {\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\) |
Input:
Int[x/(a + b/x^3),x]
Output:
x^2/(2*a) - (b*(-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))))/a
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[((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] - Simp[ 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]
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 = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(\frac {x^{2}}{2 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{3 a^{2}}\) | \(37\) |
| default | \(\frac {x^{2}}{2 a}-\frac {\left (-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(106\) |
Input:
int(x/(a+b/x^3),x,method=_RETURNVERBOSE)
Output:
1/2/a*x^2-1/3/a^2*b*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*a+b))
Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {3 \, x^{2} - 2 \, \sqrt {3} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right )}{6 \, a} \] Input:
integrate(x/(a+b/x^3),x, algorithm="fricas")
Output:
1/6*(3*x^2 - 2*sqrt(3)*(b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(b^2/a^2) ^(1/3) - sqrt(3)*b)/b) - (b^2/a^2)^(1/3)*log(b*x^2 - a*x*(b^2/a^2)^(2/3) + b*(b^2/a^2)^(1/3)) + 2*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^2)^(2/3)))/a
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.26 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{5} - b^{2}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{3}}{b} + x \right )} \right )\right )} + \frac {x^{2}}{2 a} \] Input:
integrate(x/(a+b/x**3),x)
Output:
RootSum(27*_t**3*a**5 - b**2, Lambda(_t, _t*log(9*_t**2*a**3/b + x))) + x* *2/(2*a)
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.88 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {x^{2}}{2 \, a} - \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} \] Input:
integrate(x/(a+b/x^3),x, algorithm="maxima")
Output:
1/2*x^2/a - 1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/ 3))/(a^2*(b/a)^(1/3)) - 1/6*b*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3))/(a^2* (b/a)^(1/3)) + 1/3*b*log(x + (b/a)^(1/3))/(a^2*(b/a)^(1/3))
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {x^{2}}{2 \, a} + \frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} - \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3}} \] Input:
integrate(x/(a+b/x^3),x, algorithm="giac")
Output:
1/2*x^2/a + 1/3*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/a + 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^3 - 1 /6*(-a^2*b)^(2/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^3
Time = 0.48 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {x^2}{2\,a}+\frac {b^{2/3}\,\ln \left (\frac {b^{7/3}}{a^{4/3}}+\frac {b^2\,x}{a}\right )}{3\,a^{5/3}}-\frac {b^{2/3}\,\ln \left (\frac {b^2\,x}{a}+\frac {b^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{a^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{5/3}}+\frac {b^{2/3}\,\ln \left (\frac {b^2\,x}{a}+\frac {9\,b^{7/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{a^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{5/3}} \] Input:
int(x/(a + b/x^3),x)
Output:
x^2/(2*a) + (b^(2/3)*log(b^(7/3)/a^(4/3) + (b^2*x)/a))/(3*a^(5/3)) - (b^(2 /3)*log((b^2*x)/a + (b^(7/3)*((3^(1/2)*1i)/2 + 1/2)^2)/a^(4/3))*((3^(1/2)* 1i)/2 + 1/2))/(3*a^(5/3)) + (b^(2/3)*log((b^2*x)/a + (9*b^(7/3)*((3^(1/2)* 1i)/6 - 1/6)^2)/a^(4/3))*((3^(1/2)*1i)/6 - 1/6))/a^(5/3)
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67 \[ \int \frac {x}{a+\frac {b}{x^3}} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) b +3 b^{\frac {1}{3}} a^{\frac {2}{3}} x^{2}-\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) b +2 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) b}{6 b^{\frac {1}{3}} a^{\frac {5}{3}}} \] Input:
int(x/(a+b/x^3),x)
Output:
( - 2*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))*b + 3*b** (1/3)*a**(2/3)*x**2 - log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3))* b + 2*log(a**(1/3)*x + b**(1/3))*b)/(6*b**(1/3)*a**(2/3)*a)