\(\int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx\) [503]

Optimal result

Integrand size = 19, antiderivative size = 176 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}+\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{7/3} n} \] Output:

-3/4/a/n/(x^(4/3*n))+3*b/a^2/n/(x^(1/3*n))+3^(1/2)*b^(4/3)*arctan(1/3*(b^( 
1/3)-2*a^(1/3)/(x^(1/3*n)))*3^(1/2)/b^(1/3))/a^(7/3)/n-b^(4/3)*ln(b^(1/3)+ 
a^(1/3)/(x^(1/3*n)))/a^(7/3)/n+1/2*b^(4/3)*ln(b^(2/3)+a^(2/3)/(x^(2/3*n))- 
a^(1/3)*b^(1/3)/(x^(1/3*n)))/a^(7/3)/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 = 34, normalized size of antiderivative = 0.19 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=-\frac {3 x^{-4 n/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},1,-\frac {1}{3},-\frac {b x^n}{a}\right )}{4 a n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {886, 868, 772, 843, 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.

\(\displaystyle \int \frac {x^{-\frac {4 n}{3}-1}}{a+b x^n} \, dx\)

\(\Big \downarrow \) 886

\(\displaystyle -\frac {b \int \frac {x^{-\frac {n}{3}-1}}{b x^n+a}dx}{a}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 868

\(\displaystyle \frac {3 b \int \frac {1}{b x^n+a}dx^{-n/3}}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 772

\(\displaystyle \frac {3 b \int \frac {x^{-n}}{a x^{-n}+b}dx^{-n/3}}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \int \frac {1}{a x^{-n}+b}dx^{-n/3}}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x^{-n/3}}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}}{3 b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}}dx^{-n/3}}{3 b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x^{-n/3}}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}-\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}\right )}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}+\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}\right )}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}}{2 \sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}+\frac {3 \int \frac {1}{-x^{-2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}}dx^{-n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 b \left (\frac {x^{-n/3}}{a}-\frac {b \left (\frac {-\frac {\log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}\right )}{a n}-\frac {3 x^{-4 n/3}}{4 a n}\)

Input:

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

Output:

-3/(4*a*n*x^((4*n)/3)) + (3*b*(1/(a*x^(n/3)) - (b*(Log[b^(1/3) + a^(1/3)/x 
^(n/3)]/(3*a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*a^(1/3))/(b^(1/3) 
*x^(n/3)))/Sqrt[3]])/a^(1/3)) - Log[b^(2/3) + a^(2/3)/x^((2*n)/3) - (a^(1/ 
3)*b^(1/3))/x^(n/3)]/(2*a^(1/3)))/(3*b^(2/3))))/a))/(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 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 772

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, 
x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
 

rule 843

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]
 

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 886

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[x^(m + 1)/(a*(m + 
 1)), x] - Simp[b/a   Int[x^Simplify[m + n]/(a + b*x^n), x], x] /; FreeQ[{a 
, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, 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 = 0.66 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.41

method	result	size
risch	\(\frac {3 b \,x^{-\frac {n}{3}}}{a^{2} n}-\frac {3 x^{-\frac {4 n}{3}}}{4 a n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} n^{3} \textit {\_Z}^{3}+b^{4}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\frac {a^{5} n^{2} \textit {\_R}^{2}}{b^{3}}\right )\right )\)	\(73\)

Input:

int(x^(-1-4/3*n)/(a+b*x^n),x,method=_RETURNVERBOSE)
 

Output:

3*b/a^2/n/(x^(1/3*n))-3/4/a/n/(x^(1/3*n))^4+sum(_R*ln(x^(1/3*n)+a^5*n^2/b^ 
3*_R^2),_R=RootOf(_Z^3*a^7*n^3+b^4))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=-\frac {3 \, a x x^{-\frac {4}{3} \, n - 1} - 4 \, \sqrt {3} b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}} \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x^{\frac {3}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + x x^{-\frac {2}{3} \, n - \frac {1}{2}} + \sqrt {x} \left (-\frac {b}{a}\right )^{\frac {2}{3}}}{x}\right ) - 4 \, b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x x^{-\frac {1}{3} \, n - \frac {1}{4}} - x^{\frac {3}{4}} \left (-\frac {b}{a}\right )^{\frac {1}{3}}}{x}\right ) - 12 \, b x^{\frac {1}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}}}{4 \, a^{2} n} \] Input:

integrate(x^(-1-4/3*n)/(a+b*x^n),x, algorithm="fricas")
 

Output:

-1/4*(3*a*x*x^(-4/3*n - 1) - 4*sqrt(3)*b*(-b/a)^(1/3)*arctan(1/3*(2*sqrt(3 
)*a*x^(1/4)*x^(-1/3*n - 1/4)*(-b/a)^(2/3) - sqrt(3)*b)/b) + 2*b*(-b/a)^(1/ 
3)*log((x^(3/4)*x^(-1/3*n - 1/4)*(-b/a)^(1/3) + x*x^(-2/3*n - 1/2) + sqrt( 
x)*(-b/a)^(2/3))/x) - 4*b*(-b/a)^(1/3)*log((x*x^(-1/3*n - 1/4) - x^(3/4)*( 
-b/a)^(1/3))/x) - 12*b*x^(1/4)*x^(-1/3*n - 1/4))/(a^2*n)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.19 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.31 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\frac {x^{- \frac {4 n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a n \Gamma \left (- \frac {1}{3}\right )} - \frac {4 b x^{- \frac {n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a^{2} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} \] Input:

integrate(x**(-1-4/3*n)/(a+b*x**n),x)
 

Output:

gamma(-4/3)/(a*n*x**(4*n/3)*gamma(-1/3)) - 4*b*gamma(-4/3)/(a**2*n*x**(n/3 
)*gamma(-1/3)) + 4*b**(4/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_p 
olar(I*pi/3)/a**(1/3))*gamma(-4/3)/(3*a**(7/3)*n*gamma(-1/3)) + 4*b**(4/3) 
*log(1 - b**(1/3)*x**(n/3)*exp_polar(I*pi)/a**(1/3))*gamma(-4/3)/(3*a**(7/ 
3)*n*gamma(-1/3)) + 4*b**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp 
_polar(5*I*pi/3)/a**(1/3))*gamma(-4/3)/(3*a**(7/3)*n*gamma(-1/3))
 

Maxima [F]

\[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {4}{3} \, n - 1}}{b x^{n} + a} \,d x } \] Input:

integrate(x^(-1-4/3*n)/(a+b*x^n),x, algorithm="maxima")
 

Output:

b^2*integrate(x^(2/3*n)/(a^2*b*x*x^n + a^3*x), x) + 3/4*(4*b*x^n - a)/(a^2 
*n*x^(4/3*n))
 

Giac [F]

\[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {4}{3} \, n - 1}}{b x^{n} + a} \,d x } \] Input:

integrate(x^(-1-4/3*n)/(a+b*x^n),x, algorithm="giac")
 

Output:

integrate(x^(-4/3*n - 1)/(b*x^n + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {4\,n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \] Input:

int(1/(x^((4*n)/3 + 1)*(a + b*x^n)),x)
 

Output:

int(1/(x^((4*n)/3 + 1)*(a + b*x^n)), x)
 

Reduce [F]

\[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {7 n}{3}} b x +x^{\frac {4 n}{3}} a x}d x \] Input:

int(x^(-1-4/3*n)/(a+b*x^n),x)
 

Output:

int(1/(x**((7*n)/3)*b*x + x**((4*n)/3)*a*x),x)