Integrand size = 21, antiderivative size = 261 \[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=-\frac {4 (c x)^{-5 n/4}}{5 a c n}+\frac {4 b x^n (c x)^{-5 n/4}}{a^2 c n}+\frac {\sqrt {2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} c n}-\frac {\sqrt {2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} c n}-\frac {\sqrt {2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}}{\sqrt {b}+\sqrt {a} x^{-n/2}}\right )}{a^{9/4} c n} \] Output:
-4/5/a/c/n/((c*x)^(5/4*n))+4*b*x^n/a^2/c/n/((c*x)^(5/4*n))+2^(1/2)*b^(5/4) *x^(5/4*n)*arctan(1-2^(1/2)*a^(1/4)/b^(1/4)/(x^(1/4*n)))/a^(9/4)/c/n/((c*x )^(5/4*n))-2^(1/2)*b^(5/4)*x^(5/4*n)*arctan(1+2^(1/2)*a^(1/4)/b^(1/4)/(x^( 1/4*n)))/a^(9/4)/c/n/((c*x)^(5/4*n))-2^(1/2)*b^(5/4)*x^(5/4*n)*arctanh(2^( 1/2)*a^(1/4)*b^(1/4)/(x^(1/4*n))/(b^(1/2)+a^(1/2)/(x^(1/2*n))))/a^(9/4)/c/ n/((c*x)^(5/4*n))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.15 \[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=-\frac {4 x (c x)^{-1-\frac {5 n}{4}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\frac {b x^n}{a}\right )}{5 a n} \] Input:
Integrate[(c*x)^(-1 - (5*n)/4)/(a + b*x^n),x]
Output:
(-4*x*(c*x)^(-1 - (5*n)/4)*Hypergeometric2F1[-5/4, 1, -1/4, -((b*x^n)/a)]) /(5*a*n)
Time = 0.82 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {887, 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.
| \(\displaystyle \int \frac {(c x)^{-\frac {5 n}{4}-1}}{a+b x^n} \, dx\) |
| \(\Big \downarrow \) 887 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \int \frac {x^{-\frac {5 n}{4}-1}}{b x^n+a}dx}{c}\) |
| \(\Big \downarrow \) 886 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (-\frac {b \int \frac {x^{-\frac {n}{4}-1}}{b x^n+a}dx}{a}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 868 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \int \frac {1}{b x^n+a}dx^{-n/4}}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \int \frac {x^{-n}}{a x^{-n}+b}dx^{-n/4}}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \int \frac {1}{a x^{-n}+b}dx^{-n/4}}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{-n/2}}{a x^{-n}+b}dx^{-n/4}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {a} x^{-n/2}+\sqrt {b}}{a x^{-n}+b}dx^{-n/4}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{-n/2}}{a x^{-n}+b}dx^{-n/4}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx^{-n/4}}{2 \sqrt {a}}+\frac {\int \frac {1}{x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx^{-n/4}}{2 \sqrt {a}}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\frac {\int \frac {1}{-x^{-n/2}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{-n/2}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{-n/2}}{a x^{-n}+b}dx^{-n/4}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{-n/2}}{a x^{-n}+b}dx^{-n/4}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{a} \left (x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x^{-n/4}+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{a} \left (x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x^{-n/4}+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx^{-n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} x^{-n/4}}{x^{-n/2}-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx^{-n/4}}{2 \sqrt {2} \sqrt {a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}+\sqrt [4]{b}}{x^{-n/2}+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx^{-n/4}}{2 \sqrt {a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^{5 n/4} (c x)^{-5 n/4} \left (\frac {4 b \left (\frac {x^{-n/4}}{a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\right )}{a n}-\frac {4 x^{-5 n/4}}{5 a n}\right )}{c}\) |
Input:
Int[(c*x)^(-1 - (5*n)/4)/(a + b*x^n),x]
Output:
(x^((5*n)/4)*(-4/(5*a*n*x^((5*n)/4)) + (4*b*(1/(a*x^(n/4)) - (b*((-(ArcTan [1 - (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4))]/(Sqrt[2]*a^(1/4)*b^(1/4))) + Arc Tan[1 + (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4))]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2 *Sqrt[b]) + (-1/2*Log[Sqrt[b] + Sqrt[a]/x^(n/2) - (Sqrt[2]*a^(1/4)*b^(1/4) )/x^(n/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[b] + Sqrt[a]/x^(n/2) + (Sq rt[2]*a^(1/4)*b^(1/4))/x^(n/4)]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]))) /a))/(a*n)))/(c*(c*x)^((5*n)/4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]]))
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]
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]
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]
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]
Int[((c_)*(x_))^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[c^IntPart[ m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m/(a + b*x^n), x], x] /; FreeQ [{a, b, c, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && (SumSimplerQ[m, n ] || SumSimplerQ[m, -n])
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]
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]
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]
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]
\[\int \frac {\left (c x \right )^{-1-\frac {5 n}{4}}}{a +b \,x^{n}}d x\]
Input:
int((c*x)^(-1-5/4*n)/(a+b*x^n),x)
Output:
int((c*x)^(-1-5/4*n)/(a+b*x^n),x)
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.65 \[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=-\frac {5 \, a^{2} n \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b c^{-n - \frac {4}{5}} x e^{\left (-\frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (c\right ) - \frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 5 \, a^{2} n \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} - b c^{-n - \frac {4}{5}} x e^{\left (-\frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (c\right ) - \frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + 5 i \, a^{2} n \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b c^{-n - \frac {4}{5}} x e^{\left (-\frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (c\right ) - \frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 5 i \, a^{2} n \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b c^{-n - \frac {4}{5}} x e^{\left (-\frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (c\right ) - \frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 20 \, b c^{-n - \frac {4}{5}} x^{\frac {1}{5}} e^{\left (-\frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (c\right ) - \frac {1}{20} \, {\left (5 \, n + 4\right )} \log \left (x\right )\right )} + 4 \, a x e^{\left (-\frac {1}{4} \, {\left (5 \, n + 4\right )} \log \left (c\right ) - \frac {1}{4} \, {\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{5 \, a^{2} n} \] Input:
integrate((c*x)^(-1-5/4*n)/(a+b*x^n),x, algorithm="fricas")
Output:
-1/5*(5*a^2*n*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4)*log((a^2*n*x^(4/5)*(-b^5 *c^(-5*n - 4)/(a^9*n^4))^(1/4) + b*c^(-n - 4/5)*x*e^(-1/20*(5*n + 4)*log(c ) - 1/20*(5*n + 4)*log(x)))/x) - 5*a^2*n*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/ 4)*log(-(a^2*n*x^(4/5)*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4) - b*c^(-n - 4/5 )*x*e^(-1/20*(5*n + 4)*log(c) - 1/20*(5*n + 4)*log(x)))/x) + 5*I*a^2*n*(-b ^5*c^(-5*n - 4)/(a^9*n^4))^(1/4)*log((I*a^2*n*x^(4/5)*(-b^5*c^(-5*n - 4)/( a^9*n^4))^(1/4) + b*c^(-n - 4/5)*x*e^(-1/20*(5*n + 4)*log(c) - 1/20*(5*n + 4)*log(x)))/x) - 5*I*a^2*n*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4)*log((-I*a^ 2*n*x^(4/5)*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4) + b*c^(-n - 4/5)*x*e^(-1/2 0*(5*n + 4)*log(c) - 1/20*(5*n + 4)*log(x)))/x) - 20*b*c^(-n - 4/5)*x^(1/5 )*e^(-1/20*(5*n + 4)*log(c) - 1/20*(5*n + 4)*log(x)) + 4*a*x*e^(-1/4*(5*n + 4)*log(c) - 1/4*(5*n + 4)*log(x)))/(a^2*n)
Result contains complex when optimal does not.
Time = 1.32 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.42 \[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\frac {c^{- \frac {5 n}{4} - 1} x^{- \frac {5 n}{4}} \Gamma \left (- \frac {5}{4}\right )}{a n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 b c^{- \frac {5 n}{4} - 1} x^{- \frac {n}{4}} \Gamma \left (- \frac {5}{4}\right )}{a^{2} n \Gamma \left (- \frac {1}{4}\right )} + \frac {5 b^{\frac {5}{4}} c^{- \frac {5 n}{4} - 1} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} + \frac {5 i b^{\frac {5}{4}} c^{- \frac {5 n}{4} - 1} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 b^{\frac {5}{4}} c^{- \frac {5 n}{4} - 1} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 i b^{\frac {5}{4}} c^{- \frac {5 n}{4} - 1} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} \] Input:
integrate((c*x)**(-1-5/4*n)/(a+b*x**n),x)
Output:
c**(-5*n/4 - 1)*gamma(-5/4)/(a*n*x**(5*n/4)*gamma(-1/4)) - 5*b*c**(-5*n/4 - 1)*gamma(-5/4)/(a**2*n*x**(n/4)*gamma(-1/4)) + 5*b**(5/4)*c**(-5*n/4 - 1 )*exp(-3*I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(I*pi/4)/a**(1/4))*gam ma(-5/4)/(4*a**(9/4)*n*gamma(-1/4)) + 5*I*b**(5/4)*c**(-5*n/4 - 1)*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)*c**(-5*n/4 - 1)*exp(-3*I*pi/4)*l og(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)*c**(-5*n/4 - 1)*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))
\[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {5}{4} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c*x)^(-1-5/4*n)/(a+b*x^n),x, algorithm="maxima")
Output:
b^2*integrate(x^(3/4*n)/(a^2*b*c^(5/4*n + 1)*x*x^n + a^3*c^(5/4*n + 1)*x), x) + 4/5*(5*b*x^n - a)*c^(-5/4*n - 1)/(a^2*n*x^(5/4*n))
\[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {5}{4} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c*x)^(-1-5/4*n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate((c*x)^(-5/4*n - 1)/(b*x^n + a), x)
Timed out. \[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\int \frac {1}{{\left (c\,x\right )}^{\frac {5\,n}{4}+1}\,\left (a+b\,x^n\right )} \,d x \] Input:
int(1/((c*x)^((5*n)/4 + 1)*(a + b*x^n)),x)
Output:
int(1/((c*x)^((5*n)/4 + 1)*(a + b*x^n)), x)
\[ \int \frac {(c x)^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx=\frac {\int \frac {1}{x^{\frac {9 n}{4}} b x +x^{\frac {5 n}{4}} a x}d x}{c^{\frac {5 n}{4}} c} \] Input:
int((c*x)^(-1-5/4*n)/(a+b*x^n),x)
Output:
int(1/(x**((9*n)/4)*b*x + x**((5*n)/4)*a*x),x)/(c**((5*n)/4)*c)