\(\int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx\) [568]

Optimal result

Integrand size = 19, antiderivative size = 143 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{2/3} \sqrt [3]{b} n} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.22 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=\frac {3 x^{n/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},-\frac {b x^n}{a}\right )}{a n} \] Input:

Integrate[x^(-1 + n/3)/(a + b*x^n),x]
 

Output:

(3*x^(n/3)*Hypergeometric2F1[1/3, 1, 4/3, -((b*x^n)/a)])/(a*n)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {868, 750, 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.

\(\displaystyle \int \frac {x^{\frac {n}{3}-1}}{a+b x^n} \, dx\)

\(\Big \downarrow \) 868

\(\displaystyle \frac {3 \int \frac {1}{b x^n+a}dx^{n/3}}{n}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {3 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x^{n/3}+\sqrt [3]{a}}dx^{n/3}}{3 a^{2/3}}\right )}{n}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}\right )}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}\right )}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}+\frac {3 \int \frac {1}{-x^{2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{n}\)

Input:

Int[x^(-1 + n/3)/(a + b*x^n),x]
 

Output:

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

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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 868

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) 
 Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[ 
{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

Maple [C] (verified)

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

Time = 0.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.22

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{2} b \,n^{3} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (a n \textit {\_R} +x^{\frac {n}{3}}\right )\)	\(31\)

Input:

int(x^(-1+1/3*n)/(a+b*x^n),x,method=_RETURNVERBOSE)
 

Output:

sum(_R*ln(a*n*_R+x^(1/3*n)),_R=RootOf(_Z^3*a^2*b*n^3-1))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.80 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=\left [\frac {\sqrt {3} a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} x^{n - 3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x x^{\frac {1}{3} \, n - 1} - a^{2} + \sqrt {3} {\left (2 \, a b x^{2} x^{\frac {2}{3} \, n - 2} + \left (a^{2} b\right )^{\frac {2}{3}} x x^{\frac {1}{3} \, n - 1} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} x^{n - 3} + a}\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (\frac {a b x x^{\frac {1}{3} \, n - 1} + \left (a^{2} b\right )^{\frac {2}{3}}}{x}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (\frac {a b x^{2} x^{\frac {2}{3} \, n - 2} - \left (a^{2} b\right )^{\frac {2}{3}} x x^{\frac {1}{3} \, n - 1} + \left (a^{2} b\right )^{\frac {1}{3}} a}{x^{2}}\right )}{2 \, a^{2} b n}, \frac {2 \, \sqrt {3} a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x x^{\frac {1}{3} \, n - 1} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{3 \, a^{2}}\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (\frac {a b x x^{\frac {1}{3} \, n - 1} + \left (a^{2} b\right )^{\frac {2}{3}}}{x}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (\frac {a b x^{2} x^{\frac {2}{3} \, n - 2} - \left (a^{2} b\right )^{\frac {2}{3}} x x^{\frac {1}{3} \, n - 1} + \left (a^{2} b\right )^{\frac {1}{3}} a}{x^{2}}\right )}{2 \, a^{2} b n}\right ] \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.01 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=- \frac {e^{- \frac {i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {2}{3}} \sqrt [3]{b} n \Gamma \left (\frac {4}{3}\right )} + \frac {\log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {2}{3}} \sqrt [3]{b} n \Gamma \left (\frac {4}{3}\right )} - \frac {e^{\frac {i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {2}{3}} \sqrt [3]{b} n \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

-exp(-I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(I*pi/3)/a**(1/3))*gamma( 
1/3)/(3*a**(2/3)*b**(1/3)*n*gamma(4/3)) + log(1 - b**(1/3)*x**(n/3)*exp_po 
lar(I*pi)/a**(1/3))*gamma(1/3)/(3*a**(2/3)*b**(1/3)*n*gamma(4/3)) - exp(I* 
pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(1/3)/( 
3*a**(2/3)*b**(1/3)*n*gamma(4/3))
 

Maxima [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{b x^{n} + a} \,d x } \] Input:

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

Output:

integrate(x^(1/3*n - 1)/(b*x^n + a), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.92 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=-\frac {\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3} \, n} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3} \, n} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3} \, n} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + {\left (x^{n}\right )}^{\frac {2}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b}}{2 \, n} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=\int \frac {x^{\frac {n}{3}-1}}{a+b\,x^n} \,d x \] Input:

int(x^(n/3 - 1)/(a + b*x^n),x)
 

Output:

int(x^(n/3 - 1)/(a + b*x^n), x)
 

Reduce [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n} \, dx=\int \frac {x^{\frac {n}{3}}}{x^{n} b x +a x}d x \] Input:

int(x^(-1+1/3*n)/(a+b*x^n),x)
 

Output:

int(x**(n/3)/(x**n*b*x + a*x),x)