\(\int \frac {x^{-1+\frac {n}{3}}}{(a+b x^n)^2} \, dx\) [569]

Optimal result

Integrand size = 19, antiderivative size = 169 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx=\frac {x^{n/3}}{a n \left (a+b x^n\right )}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b} n}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{5/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{3 a^{5/3} \sqrt [3]{b} n} \] Output:

x^(1/3*n)/a/n/(a+b*x^n)-2/3*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3*n))*3^(1/ 
2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(1/3)/n+2/3*ln(a^(1/3)+b^(1/3)*x^(1/3*n))/a^ 
(5/3)/b^(1/3)/n-1/3*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3*n)+b^(2/3)*x^(2/3*n) 
)/a^(5/3)/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 = 0.19 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx=\frac {3 x^{n/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},2,\frac {4}{3},-\frac {b x^n}{a}\right )}{a^2 n} \] Input:

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

Output:

(3*x^(n/3)*Hypergeometric2F1[1/3, 2, 4/3, -((b*x^n)/a)])/(a^2*n)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {868, 749, 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.

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

\(\Big \downarrow \) 868

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

\(\Big \downarrow \) 749

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

\(\Big \downarrow \) 750

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(3*(x^(n/3)/(3*a*(a + b*x^n)) + (2*(Log[a^(1/3) + b^(1/3)*x^(n/3)]/(3*a^(2 
/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(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))))/(3*a)))/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 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

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 = 0.70 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.36

method	result	size
risch	\(\frac {x^{\frac {n}{3}}}{a n \left (a +b \,x^{n}\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 a^{5} b \,n^{3} \textit {\_Z}^{3}-8\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\frac {3 a^{2} n \textit {\_R}}{2}\right )\right )\)	\(61\)

Input:

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

Output:

1/a/n*x^(1/3*n)/((x^(1/3*n))^3*b+a)+sum(_R*ln(x^(1/3*n)+3/2*a^2*n*_R),_R=R 
ootOf(27*_Z^3*a^5*b*n^3-8))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (124) = 248\).

Time = 0.09 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.40 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/3*(3*a^2*b*x*x^(1/3*n - 1) + 3*sqrt(1/3)*(a*b^2*x^3*x^(n - 3) + a^2*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 + 3*sqrt(1/3)*(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)*b*x^3*x^(n - 3) + (a^2*b)^(2/3)*a)*log((a*b*x*x^(1 
/3*n - 1) + (a^2*b)^(2/3))/x) - ((a^2*b)^(2/3)*b*x^3*x^(n - 3) + (a^2*b)^( 
2/3)*a)*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^3*b^2*n*x^3*x^(n - 3) + a^4*b*n), 1/3*(3*a^2*b*x*x^(1 
/3*n - 1) + 6*sqrt(1/3)*(a*b^2*x^3*x^(n - 3) + a^2*b)*sqrt((a^2*b)^(1/3)/b 
)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x*x^(1/3*n - 1) - (a^2*b)^(1/3)*a)*sqr 
t((a^2*b)^(1/3)/b)/a^2) + 2*((a^2*b)^(2/3)*b*x^3*x^(n - 3) + (a^2*b)^(2/3) 
*a)*log((a*b*x*x^(1/3*n - 1) + (a^2*b)^(2/3))/x) - ((a^2*b)^(2/3)*b*x^3*x^ 
(n - 3) + (a^2*b)^(2/3)*a)*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^3*b^2*n*x^3*x^(n - 3) + a^4*b*n)]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.49 (sec) , antiderivative size = 751, normalized size of antiderivative = 4.44 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-2*a**(4/3)*x**(2*n/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(I*pi/3)/a**(1/3 
))*gamma(1/3)/(9*a**3*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) + 9*a** 
2*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3)) + 2*a**(4/3)*x**(2*n/3)*ex 
p(I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(I*pi)/a**(1/3))*gamma(1/3)/( 
9*a**3*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) + 9*a**2*b**(4/3)*n*x* 
*(5*n/3)*exp(I*pi/3)*gamma(4/3)) - 2*a**(4/3)*x**(2*n/3)*exp(2*I*pi/3)*log 
(1 - b**(1/3)*x**(n/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(1/3)/(9*a**3*b* 
*(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) + 9*a**2*b**(4/3)*n*x**(5*n/3)* 
exp(I*pi/3)*gamma(4/3)) - 2*a**(1/3)*b*x**(5*n/3)*log(1 - b**(1/3)*x**(n/3 
)*exp_polar(I*pi/3)/a**(1/3))*gamma(1/3)/(9*a**3*b**(1/3)*n*x**(2*n/3)*exp 
(I*pi/3)*gamma(4/3) + 9*a**2*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3)) 
 + 2*a**(1/3)*b*x**(5*n/3)*exp(I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar 
(I*pi)/a**(1/3))*gamma(1/3)/(9*a**3*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamm 
a(4/3) + 9*a**2*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3)) - 2*a**(1/3) 
*b*x**(5*n/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(5*I*pi/3)/ 
a**(1/3))*gamma(1/3)/(9*a**3*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) 
+ 9*a**2*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3)) + 3*a*b**(1/3)*x**n 
*exp(I*pi/3)*gamma(1/3)/(9*a**3*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/ 
3) + 9*a**2*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3))
 

Maxima [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx=-\frac {\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3} \, n} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} - \frac {3 \, x^{\frac {1}{3} \, n}}{{\left (b x^{n} + a\right )} a} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3} \, n} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3} \, n} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + {\left (x^{n}\right )}^{\frac {2}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} b}}{3 \, n} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx=\int \frac {x^{\frac {n}{3}-1}}{{\left (a+b\,x^n\right )}^2} \,d x \] Input:

int(x^(n/3 - 1)/(a + b*x^n)^2,x)
 

Output:

int(x^(n/3 - 1)/(a + b*x^n)^2, x)
 

Reduce [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^2} \, dx=\int \frac {x^{\frac {n}{3}}}{x^{2 n} b^{2} x +2 x^{n} a b x +a^{2} x}d x \] Input:

int(x^(-1+1/3*n)/(a+b*x^n)^2,x)
 

Output:

int(x**(n/3)/(x**(2*n)*b**2*x + 2*x**n*a*b*x + a**2*x),x)