Integrand size = 13, antiderivative size = 122 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=-\frac {1}{b x}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} b^{4/3}}+\frac {\sqrt [3]{a} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{4/3}} \] Output:
-1/b/x+1/3*a^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3^(1/2)/b^(1/3))*3^(1/ 2)/b^(4/3)+1/3*a^(1/3)*ln(b^(1/3)+a^(1/3)*x)/b^(4/3)-1/6*a^(1/3)*ln(b^(2/3 )-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/b^(4/3)
Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=\frac {-6 \sqrt [3]{b}+2 \sqrt {3} \sqrt [3]{a} x \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 \sqrt [3]{a} x \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-\sqrt [3]{a} x \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{4/3} x} \] Input:
Integrate[1/((a + b/x^3)*x^5),x]
Output:
(-6*b^(1/3) + 2*Sqrt[3]*a^(1/3)*x*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[ 3]] + 2*a^(1/3)*x*Log[b^(1/3) + a^(1/3)*x] - a^(1/3)*x*Log[b^(2/3) - a^(1/ 3)*b^(1/3)*x + a^(2/3)*x^2])/(6*b^(4/3)*x)
Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {795, 847, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+\frac {b}{x^3}\right )} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {1}{x^2 \left (a x^3+b\right )}dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {a \int \frac {x}{a x^3+b}dx}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {a \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\) |
Input:
Int[1/((a + b/x^3)*x^5),x]
Output:
-(1/(b*x)) - (a*(-1/3*Log[b^(1/3) + a^(1/3)*x]/(a^(2/3)*b^(1/3)) + (-((Sqr t[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))))/b
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*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(-\frac {1}{x b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{4} \textit {\_Z}^{3}-a \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} b^{4}+3 a \right ) x -b^{3} \textit {\_R}^{2}\right )\right )}{3}\) | \(53\) |
| default | \(-\frac {1}{x b}-\frac {\left (-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right ) a}{b}\) | \(106\) |
Input:
int(1/(a+b/x^3)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/x/b+1/3*sum(_R*ln((-4*_R^3*b^4+3*a)*x-b^3*_R^2),_R=RootOf(_Z^3*b^4-a))
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=-\frac {2 \, \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a}{b}\right )^{\frac {2}{3}} + b \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 6}{6 \, b x} \] Input:
integrate(1/(a+b/x^3)/x^5,x, algorithm="fricas")
Output:
-1/6*(2*sqrt(3)*x*(a/b)^(1/3)*arctan(2/3*sqrt(3)*x*(a/b)^(1/3) - 1/3*sqrt( 3)) + x*(a/b)^(1/3)*log(a*x^2 - b*x*(a/b)^(2/3) + b*(a/b)^(1/3)) - 2*x*(a/ b)^(1/3)*log(a*x + b*(a/b)^(2/3)) + 6)/(b*x)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=\operatorname {RootSum} {\left (27 t^{3} b^{4} - a, \left ( t \mapsto t \log {\left (\frac {9 t^{2} b^{3}}{a} + x \right )} \right )\right )} - \frac {1}{b x} \] Input:
integrate(1/(a+b/x**3)/x**5,x)
Output:
RootSum(27*_t**3*b**4 - a, Lambda(_t, _t*log(9*_t**2*b**3/a + x))) - 1/(b* x)
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {1}{b x} \] Input:
integrate(1/(a+b/x^3)/x^5,x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/3))/(b*(b/a)^ (1/3)) - 1/6*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3))/(b*(b/a)^(1/3)) + 1/3* log(x + (b/a)^(1/3))/(b*(b/a)^(1/3)) - 1/(b*x)
Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=\frac {a \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b^{2}} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} - \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}} - \frac {1}{b x} \] Input:
integrate(1/(a+b/x^3)/x^5,x, algorithm="giac")
Output:
1/3*a*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/b^2 + 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*b^2) - 1/6*( -a^2*b)^(2/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/(a*b^2) - 1/(b*x)
Time = 0.52 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=\frac {a^{1/3}\,\ln \left (a^{1/3}\,x+b^{1/3}\right )}{3\,b^{4/3}}-\frac {1}{b\,x}-\frac {a^{1/3}\,\ln \left (4\,a^{1/3}\,x-2\,b^{1/3}+\sqrt {3}\,b^{1/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{4/3}}+\frac {a^{1/3}\,\ln \left (4\,a^{1/3}\,x-2\,b^{1/3}-\sqrt {3}\,b^{1/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{4/3}} \] Input:
int(1/(x^5*(a + b/x^3)),x)
Output:
(a^(1/3)*log(a^(1/3)*x + b^(1/3)))/(3*b^(4/3)) - 1/(b*x) - (a^(1/3)*log(3^ (1/2)*b^(1/3)*2i + 4*a^(1/3)*x - 2*b^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(3*b^( 4/3)) + (a^(1/3)*log(4*a^(1/3)*x - 3^(1/2)*b^(1/3)*2i - 2*b^(1/3))*((3^(1/ 2)*1i)/6 - 1/6))/b^(4/3)
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a x -6 b^{\frac {1}{3}} a^{\frac {2}{3}}-\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) a x +2 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) a x}{6 b^{\frac {4}{3}} a^{\frac {2}{3}} x} \] Input:
int(1/(a+b/x^3)/x^5,x)
Output:
( - 2*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))*a*x - 6*b **(1/3)*a**(2/3) - log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3))*a*x + 2*log(a**(1/3)*x + b**(1/3))*a*x)/(6*b**(1/3)*a**(2/3)*b*x)