\(\int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx\) [2648]

Optimal result

Integrand size = 19, antiderivative size = 252 \[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {\sqrt {2} b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac {\sqrt {2} b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n} \]

[Out]

-4/5/a/n/(x^(5/4*n))+4*b/a^2/n/(x^(1/4*n))+1/2*b^(5/4)*ln(-a^(1/4)*b^(1/4)*2^(1/2)/(x^(1/4*n))+a^(1/2)/(x^(1/2
*n))+b^(1/2))/a^(9/4)/n*2^(1/2)-1/2*b^(5/4)*ln(a^(1/4)*b^(1/4)*2^(1/2)/(x^(1/4*n))+a^(1/2)/(x^(1/2*n))+b^(1/2)
)/a^(9/4)/n*2^(1/2)+b^(5/4)*arctan(1-a^(1/4)*2^(1/2)/b^(1/4)/(x^(1/4*n)))*2^(1/2)/a^(9/4)/n-b^(5/4)*arctan(1+a
^(1/4)*2^(1/2)/b^(1/4)/(x^(1/4*n)))*2^(1/2)/a^(9/4)/n

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {369, 352, 199, 327, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\frac {\sqrt {2} b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac {\sqrt {2} b^{5/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{9/4} n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {4 x^{-5 n/4}}{5 a n} \]

[In]

Int[x^(-1 - (5*n)/4)/(a + b*x^n),x]

[Out]

-4/(5*a*n*x^((5*n)/4)) + (4*b)/(a^2*n*x^(n/4)) + (Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4
))])/(a^(9/4)*n) - (Sqrt[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4))])/(a^(9/4)*n) + (b^(5/4)*Lo
g[Sqrt[b] + Sqrt[a]/x^(n/2) - (Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(9/4)*n) - (b^(5/4)*Log[Sqrt[b] +
Sqrt[a]/x^(n/2) + (Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(9/4)*n)

Rule 199

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

Rule 210

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 327

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] - Dist[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 352

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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 369

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[x^(m + 1)/(a*(m + 1)), x] - Dist[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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 x^{-5 n/4}}{5 a n}-\frac {b \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n} \, dx}{a} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{a+\frac {b}{x^4}} \, dx,x,x^{-n/4}\right )}{a n} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {(4 b) \text {Subst}\left (\int \frac {x^4}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a n} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {1}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a^2 n} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {\left (2 b^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a^2 n}-\frac {\left (2 b^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a^2 n} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {b^{5/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}+2 x}{-\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}+\frac {b^{5/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}-2 x}{-\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{3/2} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,x^{-n/4}\right )}{a^{5/2} n}-\frac {b^{3/2} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,x^{-n/4}\right )}{a^{5/2} n} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {\left (\sqrt {2} b^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}+\frac {\left (\sqrt {2} b^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n} \\ & = -\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=-\frac {4 x^{-5 n/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\frac {b x^n}{a}\right )}{5 a n} \]

[In]

Integrate[x^(-1 - (5*n)/4)/(a + b*x^n),x]

[Out]

(-4*Hypergeometric2F1[-5/4, 1, -1/4, -((b*x^n)/a)])/(5*a*n*x^((5*n)/4))

Maple [C] (verified)

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

Time = 3.87 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.29

method	result	size
risch	\(\frac {4 b \,x^{-\frac {n}{4}}}{a^{2} n}-\frac {4 x^{-\frac {5 n}{4}}}{5 a n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} n^{4} \textit {\_Z}^{4}+b^{5}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}+\frac {a^{7} n^{3} \textit {\_R}^{3}}{b^{4}}\right )\right )\)	\(73\)

[In]

int(x^(-1-5/4*n)/(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

4*b/a^2/n/(x^(1/4*n))-4/5/a/n/(x^(1/4*n))^5+sum(_R*ln(x^(1/4*n)+a^7*n^3/b^4*_R^3),_R=RootOf(_Z^4*a^9*n^4+b^5))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.04 \[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=-\frac {5 \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b x x^{-\frac {1}{4} \, n - \frac {1}{5}}}{x}\right ) - 5 \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} - b x x^{-\frac {1}{4} \, n - \frac {1}{5}}}{x}\right ) + 5 i \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b x x^{-\frac {1}{4} \, n - \frac {1}{5}}}{x}\right ) - 5 i \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b x x^{-\frac {1}{4} \, n - \frac {1}{5}}}{x}\right ) + 4 \, a x x^{-\frac {5}{4} \, n - 1} - 20 \, b x^{\frac {1}{5}} x^{-\frac {1}{4} \, n - \frac {1}{5}}}{5 \, a^{2} n} \]

[In]

integrate(x^(-1-5/4*n)/(a+b*x^n),x, algorithm="fricas")

[Out]

-1/5*(5*a^2*n*(-b^5/(a^9*n^4))^(1/4)*log((a^2*n*x^(4/5)*(-b^5/(a^9*n^4))^(1/4) + b*x*x^(-1/4*n - 1/5))/x) - 5*
a^2*n*(-b^5/(a^9*n^4))^(1/4)*log(-(a^2*n*x^(4/5)*(-b^5/(a^9*n^4))^(1/4) - b*x*x^(-1/4*n - 1/5))/x) + 5*I*a^2*n
*(-b^5/(a^9*n^4))^(1/4)*log((I*a^2*n*x^(4/5)*(-b^5/(a^9*n^4))^(1/4) + b*x*x^(-1/4*n - 1/5))/x) - 5*I*a^2*n*(-b
^5/(a^9*n^4))^(1/4)*log((-I*a^2*n*x^(4/5)*(-b^5/(a^9*n^4))^(1/4) + b*x*x^(-1/4*n - 1/5))/x) + 4*a*x*x^(-5/4*n
- 1) - 20*b*x^(1/5)*x^(-1/4*n - 1/5))/(a^2*n)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.23 \[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\frac {x^{- \frac {5 n}{4}} \Gamma \left (- \frac {5}{4}\right )}{a n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 b x^{- \frac {n}{4}} \Gamma \left (- \frac {5}{4}\right )}{a^{2} n \Gamma \left (- \frac {1}{4}\right )} + \frac {5 b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} + \frac {5 i b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 i b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} \]

[In]

integrate(x**(-1-5/4*n)/(a+b*x**n),x)

[Out]

gamma(-5/4)/(a*n*x**(5*n/4)*gamma(-1/4)) - 5*b*gamma(-5/4)/(a**2*n*x**(n/4)*gamma(-1/4)) + 5*b**(5/4)*exp(-3*I
*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(I*pi/4)/a**(1/4))*gamma(-5/4)/(4*a**(9/4)*n*gamma(-1/4)) + 5*I*b**(
5/4)*exp(-3*I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(-5/4)/(4*a**(9/4)*n*gamma(-1
/4)) - 5*b**(5/4)*exp(-3*I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(-5/4)/(4*a**(9/
4)*n*gamma(-1/4)) - 5*I*b**(5/4)*exp(-3*I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(
-5/4)/(4*a**(9/4)*n*gamma(-1/4))

Maxima [F]

\[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {5}{4} \, n - 1}}{b x^{n} + a} \,d x } \]

[In]

integrate(x^(-1-5/4*n)/(a+b*x^n),x, algorithm="maxima")

[Out]

b^2*integrate(x^(3/4*n)/(a^2*b*x*x^n + a^3*x), x) + 4/5*(5*b*x^n - a)/(a^2*n*x^(5/4*n))

Giac [F]

\[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {5}{4} \, n - 1}}{b x^{n} + a} \,d x } \]

[In]

integrate(x^(-1-5/4*n)/(a+b*x^n),x, algorithm="giac")

[Out]

integrate(x^(-5/4*n - 1)/(b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {5\,n}{4}+1}\,\left (a+b\,x^n\right )} \,d x \]

[In]

int(1/(x^((5*n)/4 + 1)*(a + b*x^n)),x)

[Out]

int(1/(x^((5*n)/4 + 1)*(a + b*x^n)), x)