\(\int \frac {x^{-1+\frac {n}{3}}}{(a+b x^n)^3} \, dx\) [570]

Optimal result

Integrand size = 19, antiderivative size = 200 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^3} \, dx=\frac {x^{n/3}}{2 a n \left (a+b x^n\right )^2}+\frac {5 x^{n/3}}{6 a^2 n \left (a+b x^n\right )}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b} n}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{9 a^{8/3} \sqrt [3]{b} n}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{18 a^{8/3} \sqrt [3]{b} n} \] Output:

1/2*x^(1/3*n)/a/n/(a+b*x^n)^2+5/6*x^(1/3*n)/a^2/n/(a+b*x^n)-5/9*arctan(1/3 
*(a^(1/3)-2*b^(1/3)*x^(1/3*n))*3^(1/2)/a^(1/3))*3^(1/2)/a^(8/3)/b^(1/3)/n+ 
5/9*ln(a^(1/3)+b^(1/3)*x^(1/3*n))/a^(8/3)/b^(1/3)/n-5/18*ln(a^(2/3)-a^(1/3 
)*b^(1/3)*x^(1/3*n)+b^(2/3)*x^(2/3*n))/a^(8/3)/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 = 0.16 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^3} \, dx=\frac {3 x^{n/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},3,\frac {4}{3},-\frac {b x^n}{a}\right )}{a^3 n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {868, 749, 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.

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

\(\Big \downarrow \) 868

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

\(\Big \downarrow \) 749

\(\displaystyle \frac {3 \left (\frac {5 \int \frac {1}{\left (b x^n+a\right )^2}dx^{n/3}}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 749

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \int \frac {1}{b x^n+a}dx^{n/3}}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x^{n/3}+\sqrt [3]{a}}dx^{n/3}}{3 a^{2/3}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}\right )}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}\right )}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}+\frac {3 \int \frac {1}{-x^{2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}+a^{2/3}}dx^{n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 \left (\frac {5 \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x^{n/3}}{3 a \left (a+b x^n\right )}\right )}{6 a}+\frac {x^{n/3}}{6 a \left (a+b x^n\right )^2}\right )}{n}\)

Input:

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

Output:

(3*(x^(n/3)/(6*a*(a + b*x^n)^2) + (5*(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)))/(6* 
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 = 0.91 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.38

method	result	size
risch	\(\frac {x^{\frac {n}{3}} \left (5 b \,x^{n}+8 a \right )}{6 a^{2} n \left (a +b \,x^{n}\right )^{2}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{8} b \,n^{3} \textit {\_Z}^{3}-125\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\frac {9 a^{3} n \textit {\_R}}{5}\right )\right )\)	\(76\)

Input:

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

Output:

1/6*x^(1/3*n)*(5*(x^(1/3*n))^3*b+8*a)/a^2/n/((x^(1/3*n))^3*b+a)^2+sum(_R*l 
n(x^(1/3*n)+9/5*a^3*n*_R),_R=RootOf(729*_Z^3*a^8*b*n^3-125))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (147) = 294\).

Time = 0.09 (sec) , antiderivative size = 785, normalized size of antiderivative = 3.92 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/18*(15*a^2*b^2*x^4*x^(4/3*n - 4) + 24*a^3*b*x*x^(1/3*n - 1) + 15*sqrt(1 
/3)*(a*b^3*x^6*x^(2*n - 6) + 2*a^2*b^2*x^3*x^(n - 3) + a^3*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)) + 10*(( 
a^2*b)^(2/3)*b^2*x^6*x^(2*n - 6) + 2*(a^2*b)^(2/3)*a*b*x^3*x^(n - 3) + (a^ 
2*b)^(2/3)*a^2)*log((a*b*x*x^(1/3*n - 1) + (a^2*b)^(2/3))/x) - 5*((a^2*b)^ 
(2/3)*b^2*x^6*x^(2*n - 6) + 2*(a^2*b)^(2/3)*a*b*x^3*x^(n - 3) + (a^2*b)^(2 
/3)*a^2)*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^4*b^3*n*x^6*x^(2*n - 6) + 2*a^5*b^2*n*x^3*x^(n - 3) 
+ a^6*b*n), 1/18*(15*a^2*b^2*x^4*x^(4/3*n - 4) + 24*a^3*b*x*x^(1/3*n - 1) 
+ 30*sqrt(1/3)*(a*b^3*x^6*x^(2*n - 6) + 2*a^2*b^2*x^3*x^(n - 3) + a^3*b)*s 
qrt((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)*sqrt((a^2*b)^(1/3)/b)/a^2) + 10*((a^2*b)^(2/3)*b^2*x^6*x^( 
2*n - 6) + 2*(a^2*b)^(2/3)*a*b*x^3*x^(n - 3) + (a^2*b)^(2/3)*a^2)*log((a*b 
*x*x^(1/3*n - 1) + (a^2*b)^(2/3))/x) - 5*((a^2*b)^(2/3)*b^2*x^6*x^(2*n - 6 
) + 2*(a^2*b)^(2/3)*a*b*x^3*x^(n - 3) + (a^2*b)^(2/3)*a^2)*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^4* 
b^3*n*x^6*x^(2*n - 6) + 2*a^5*b^2*n*x^3*x^(n - 3) + a^6*b*n)]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.24 (sec) , antiderivative size = 2531, normalized size of antiderivative = 12.66 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-10*a**(13/3)*x**(2*n/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(I*pi/3)/a**(1 
/3))*gamma(1/3)/(54*a**7*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) + 16 
2*a**6*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3) + 162*a**5*b**(7/3)*n* 
x**(8*n/3)*exp(I*pi/3)*gamma(4/3) + 54*a**4*b**(10/3)*n*x**(11*n/3)*exp(I* 
pi/3)*gamma(4/3)) + 10*a**(13/3)*x**(2*n/3)*exp(I*pi/3)*log(1 - b**(1/3)*x 
**(n/3)*exp_polar(I*pi)/a**(1/3))*gamma(1/3)/(54*a**7*b**(1/3)*n*x**(2*n/3 
)*exp(I*pi/3)*gamma(4/3) + 162*a**6*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamm 
a(4/3) + 162*a**5*b**(7/3)*n*x**(8*n/3)*exp(I*pi/3)*gamma(4/3) + 54*a**4*b 
**(10/3)*n*x**(11*n/3)*exp(I*pi/3)*gamma(4/3)) - 10*a**(13/3)*x**(2*n/3)*e 
xp(2*I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma 
(1/3)/(54*a**7*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) + 162*a**6*b** 
(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3) + 162*a**5*b**(7/3)*n*x**(8*n/3) 
*exp(I*pi/3)*gamma(4/3) + 54*a**4*b**(10/3)*n*x**(11*n/3)*exp(I*pi/3)*gamm 
a(4/3)) - 30*a**(10/3)*b*x**(5*n/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(I* 
pi/3)/a**(1/3))*gamma(1/3)/(54*a**7*b**(1/3)*n*x**(2*n/3)*exp(I*pi/3)*gamm 
a(4/3) + 162*a**6*b**(4/3)*n*x**(5*n/3)*exp(I*pi/3)*gamma(4/3) + 162*a**5* 
b**(7/3)*n*x**(8*n/3)*exp(I*pi/3)*gamma(4/3) + 54*a**4*b**(10/3)*n*x**(11* 
n/3)*exp(I*pi/3)*gamma(4/3)) + 30*a**(10/3)*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)/(54*a**7*b**(1/3 
)*n*x**(2*n/3)*exp(I*pi/3)*gamma(4/3) + 162*a**6*b**(4/3)*n*x**(5*n/3)*...
 

Maxima [F]

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

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

Output:

1/6*(5*b*x^(4/3*n) + 8*a*x^(1/3*n))/(a^2*b^2*n*x^(2*n) + 2*a^3*b*n*x^n + a 
^4*n) + 5*integrate(1/9*x^(1/3*n)/(a^2*b*x*x^n + a^3*x), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.81 \[ \int \frac {x^{-1+\frac {n}{3}}}{\left (a+b x^n\right )^3} \, dx=-\frac {\frac {10 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3} \, n} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{3}} - \frac {10 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3} \, n} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3} b} - \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3} \, n} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + {\left (x^{n}\right )}^{\frac {2}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{3} b} - \frac {3 \, {\left (5 \, b {\left (x^{n}\right )}^{\frac {4}{3}} + 8 \, a x^{\frac {1}{3} \, n}\right )}}{{\left (b x^{n} + a\right )}^{2} a^{2}}}{18 \, n} \] Input:

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

Output:

-1/18*(10*(-a/b)^(1/3)*log(abs(x^(1/3*n) - (-a/b)^(1/3)))/a^3 - 10*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^3*b) - 5*(-a*b^2)^(1/3)*log(x^(1/3*n)*(-a/b)^(1/3) + (x^n)^(2/3) + 
(-a/b)^(2/3))/(a^3*b) - 3*(5*b*(x^n)^(4/3) + 8*a*x^(1/3*n))/((b*x^n + a)^2 
*a^2))/n
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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