3.6.5 \(\int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx\) [505]

3.6.5.1 Optimal result

Integrand size = 25, antiderivative size = 252 \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {\sqrt {2} c^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n} \]

output

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

3.6.5.2 Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-5 n/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\frac {c x^n}{b}\right )}{5 b n} \]

input

Integrate[x^(-1 - n/4)/(b*x^n + c*x^(2*n)),x]
 

output

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

3.6.5.3 Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {10, 886, 868, 772, 843, 755, 1476, 1082, 217, 1479, 25, 27, 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.

\(\displaystyle \int \frac {x^{-\frac {n}{4}-1}}{b x^n+c x^{2 n}} \, dx\)

\(\Big \downarrow \) 10

\(\displaystyle \int \frac {x^{-\frac {5 n}{4}-1}}{b+c x^n}dx\)

\(\Big \downarrow \) 886

\(\displaystyle -\frac {c \int \frac {x^{-\frac {n}{4}-1}}{c x^n+b}dx}{b}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 868

\(\displaystyle \frac {4 c \int \frac {1}{c x^n+b}dx^{-n/4}}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 772

\(\displaystyle \frac {4 c \int \frac {x^{-n}}{b x^{-n}+c}dx^{-n/4}}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \int \frac {1}{b x^{-n}+c}dx^{-n/4}}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\int \frac {\sqrt {c}-\sqrt {b} x^{-n/2}}{b x^{-n}+c}dx^{-n/4}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {b} x^{-n/2}+\sqrt {c}}{b x^{-n}+c}dx^{-n/4}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\int \frac {\sqrt {c}-\sqrt {b} x^{-n/2}}{b x^{-n}+c}dx^{-n/4}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}}dx^{-n/4}}{2 \sqrt {b}}+\frac {\int \frac {1}{x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}}dx^{-n/4}}{2 \sqrt {b}}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\frac {\int \frac {1}{-x^{-n/2}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x^{-n/2}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {c}-\sqrt {b} x^{-n/2}}{b x^{-n}+c}dx^{-n/4}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\int \frac {\sqrt {c}-\sqrt {b} x^{-n/2}}{b x^{-n}+c}dx^{-n/4}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{b} \left (x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x^{-n/4}+\sqrt [4]{c}\right )}{\sqrt [4]{b} \left (x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{b} \left (x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x^{-n/4}+\sqrt [4]{c}\right )}{\sqrt [4]{b} \left (x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{b} x^{-n/4}}{x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}}dx^{-n/4}}{2 \sqrt {2} \sqrt {b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}+\sqrt [4]{c}}{x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{c} x^{-n/4}}{\sqrt [4]{b}}+\frac {\sqrt {c}}{\sqrt {b}}}dx^{-n/4}}{2 \sqrt {b} \sqrt [4]{c}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {4 c \left (\frac {x^{-n/4}}{b}-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}\right )}{b n}-\frac {4 x^{-5 n/4}}{5 b n}\)

input

Int[x^(-1 - n/4)/(b*x^n + c*x^(2*n)),x]
 

output

-4/(5*b*n*x^((5*n)/4)) + (4*c*(1/(b*x^(n/4)) - (c*((-(ArcTan[1 - (Sqrt[2]* 
b^(1/4))/(c^(1/4)*x^(n/4))]/(Sqrt[2]*b^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[ 
2]*b^(1/4))/(c^(1/4)*x^(n/4))]/(Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[c]) + (- 
1/2*Log[Sqrt[c] + Sqrt[b]/x^(n/2) - (Sqrt[2]*b^(1/4)*c^(1/4))/x^(n/4)]/(Sq 
rt[2]*b^(1/4)*c^(1/4)) + Log[Sqrt[c] + Sqrt[b]/x^(n/2) + (Sqrt[2]*b^(1/4)* 
c^(1/4))/x^(n/4)]/(2*Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[c])))/b))/(b*n)
 

3.6.5.3.1 Defintions of rubi rules used

rule 10

Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x 
_Symbol] :> Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], 
 x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ 
[e, 0]) && PosQ[s - r]
 

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 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[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 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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
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 1479

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

3.6.5.4 Maple [C] (verified)

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

Time = 0.69 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.29

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

input

int(x^(-1-1/4*n)/(b*x^n+c*x^(2*n)),x,method=_RETURNVERBOSE)
 

output

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

3.6.5.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.99 \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 \, b x^{5} x^{-\frac {5}{4} \, n - 5} + 5 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 5 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} - c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) + 5 i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 5 i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 20 \, c x x^{-\frac {1}{4} \, n - 1}}{5 \, b^{2} n} \]

input

integrate(x^(-1-1/4*n)/(b*x^n+c*x^(2*n)),x, algorithm="fricas")
 

output

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

3.6.5.6 Sympy [F]

\[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{- n} x^{- \frac {n}{4} - 1}}{b + c x^{n}}\, dx \]

input

integrate(x**(-1-1/4*n)/(b*x**n+c*x**(2*n)),x)
 

output

Integral(x**(-n/4 - 1)/(x**n*(b + c*x**n)), x)
 

3.6.5.7 Maxima [F]

\[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]

input

integrate(x^(-1-1/4*n)/(b*x^n+c*x^(2*n)),x, algorithm="maxima")
 

output

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

3.6.5.8 Giac [F]

\[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]

input

integrate(x^(-1-1/4*n)/(b*x^n+c*x^(2*n)),x, algorithm="giac")
 

output

integrate(x^(-1/4*n - 1)/(c*x^(2*n) + b*x^n), x)
 

3.6.5.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{\frac {n}{4}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]

input

int(1/(x^(n/4 + 1)*(b*x^n + c*x^(2*n))),x)
 

output

int(1/(x^(n/4 + 1)*(b*x^n + c*x^(2*n))), x)