Integrand size = 21, antiderivative size = 237 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=-\frac {4 (c x)^{-3 n/4}}{3 a c n}+\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}}{\sqrt {a}+\sqrt {b} x^{n/2}}\right )}{a^{7/4} c n} \] Output:
-4/3/a/c/n/((c*x)^(3/4*n))-2^(1/2)*b^(3/4)*x^(3/4*n)*arctan(-1+2^(1/2)*b^( 1/4)*x^(1/4*n)/a^(1/4))/a^(7/4)/c/n/((c*x)^(3/4*n))-2^(1/2)*b^(3/4)*x^(3/4 *n)*arctan(1+2^(1/2)*b^(1/4)*x^(1/4*n)/a^(1/4))/a^(7/4)/c/n/((c*x)^(3/4*n) )-2^(1/2)*b^(3/4)*x^(3/4*n)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/4*n)/(a^( 1/2)+b^(1/2)*x^(1/2*n)))/a^(7/4)/c/n/((c*x)^(3/4*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 = 0.16 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=-\frac {4 x (c x)^{-1-\frac {3 n}{4}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {b x^n}{a}\right )}{3 a n} \] Input:
Integrate[(c*x)^(-1 - (3*n)/4)/(a + b*x^n),x]
Output:
(-4*x*(c*x)^(-1 - (3*n)/4)*Hypergeometric2F1[-3/4, 1, 1/4, -((b*x^n)/a)])/ (3*a*n)
Time = 0.80 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {887, 886, 868, 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.
| \(\displaystyle \int \frac {(c x)^{-\frac {3 n}{4}-1}}{a+b x^n} \, dx\) |
| \(\Big \downarrow \) 887 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \int \frac {x^{-\frac {3 n}{4}-1}}{b x^n+a}dx}{c}\) |
| \(\Big \downarrow \) 886 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {b \int \frac {x^{\frac {n-4}{4}}}{b x^n+a}dx}{a}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 868 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \int \frac {1}{b x^n+a}dx^{n/4}}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{n/2}}{b x^n+a}dx^{n/4}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x^{n/2}+\sqrt {a}}{b x^n+a}dx^{n/4}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\frac {\int \frac {1}{-\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}}dx^{n/4}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}}dx^{n/4}}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{n/2}}{b x^n+a}dx^{n/4}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\frac {\int \frac {1}{-x^{n/2}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{n/2}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{n/2}}{b x^n+a}dx^{n/4}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{n/2}}{b x^n+a}dx^{n/4}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x^{n/4}}{\sqrt [4]{b} \left (-\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^{n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x^{n/4}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^{n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x^{n/4}}{\sqrt [4]{b} \left (-\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^{n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x^{n/4}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^{n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x^{n/4}}{-\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}}dx^{n/4}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x^{n/4}+\sqrt [4]{a}}{\frac {\sqrt {2} \sqrt [4]{a} x^{n/4}}{\sqrt [4]{b}}+x^{n/2}+\frac {\sqrt {a}}{\sqrt {b}}}dx^{n/4}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^{3 n/4} (c x)^{-3 n/4} \left (-\frac {4 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a n}-\frac {4 x^{-3 n/4}}{3 a n}\right )}{c}\) |
Input:
Int[(c*x)^(-1 - (3*n)/4)/(a + b*x^n),x]
Output:
(x^((3*n)/4)*(-4/(3*a*n*x^((3*n)/4)) - (4*b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4 )*x^(n/4))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/ 4)*x^(n/4))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sq rt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^(n/4) + Sqrt[b]*x^(n/2)]/(Sqrt[2]*a^(1/4 )*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^(n/4) + Sqrt[b]*x^(n/ 2)]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(a*n)))/(c*(c*x)^((3*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[(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]
\[\int \frac {\left (c x \right )^{-1-\frac {3 n}{4}}}{a +b \,x^{n}}d x\]
Input:
int((c*x)^(-1-3/4*n)/(a+b*x^n),x)
Output:
int((c*x)^(-1-3/4*n)/(a+b*x^n),x)
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.70 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\frac {3 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 3 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} - b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 3 i \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + 3 i \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 4 \, x e^{\left (-\frac {1}{4} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{4} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{3 \, a n} \] Input:
integrate((c*x)^(-1-3/4*n)/(a+b*x^n),x, algorithm="fricas")
Output:
1/3*(3*a*n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4)*log((a^5*n^3*x^(2/3)*(-b^3* c^(-3*n - 4)/(a^7*n^4))^(3/4) + b^2*c^(-2*n - 8/3)*x*e^(-1/12*(3*n + 4)*lo g(c) - 1/12*(3*n + 4)*log(x)))/x) - 3*a*n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1 /4)*log(-(a^5*n^3*x^(2/3)*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(3/4) - b^2*c^(-2* n - 8/3)*x*e^(-1/12*(3*n + 4)*log(c) - 1/12*(3*n + 4)*log(x)))/x) - 3*I*a* n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4)*log((I*a^5*n^3*x^(2/3)*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(3/4) + b^2*c^(-2*n - 8/3)*x*e^(-1/12*(3*n + 4)*log(c) - 1/12*(3*n + 4)*log(x)))/x) + 3*I*a*n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4)*l og((-I*a^5*n^3*x^(2/3)*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(3/4) + b^2*c^(-2*n - 8/3)*x*e^(-1/12*(3*n + 4)*log(c) - 1/12*(3*n + 4)*log(x)))/x) - 4*x*e^(-1 /4*(3*n + 4)*log(c) - 1/4*(3*n + 4)*log(x)))/(a*n)
Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.34 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\frac {c^{- \frac {3 n}{4} - 1} x^{- \frac {3 n}{4}} \Gamma \left (- \frac {3}{4}\right )}{a n \Gamma \left (\frac {1}{4}\right )} - \frac {3 b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} + \frac {3 i b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} + \frac {3 b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} - \frac {3 i b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate((c*x)**(-1-3/4*n)/(a+b*x**n),x)
Output:
c**(-3*n/4 - 1)*gamma(-3/4)/(a*n*x**(3*n/4)*gamma(1/4)) - 3*b**(3/4)*c**(- 3*n/4 - 1)*exp(-I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(I*pi/4)/a**(1/ 4))*gamma(-3/4)/(4*a**(7/4)*n*gamma(1/4)) + 3*I*b**(3/4)*c**(-3*n/4 - 1)*e xp(-I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(3*I*pi/4)/a**(1/4))*gamma( -3/4)/(4*a**(7/4)*n*gamma(1/4)) + 3*b**(3/4)*c**(-3*n/4 - 1)*exp(-I*pi/4)* log(1 - b**(1/4)*x**(n/4)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(-3/4)/(4*a** (7/4)*n*gamma(1/4)) - 3*I*b**(3/4)*c**(-3*n/4 - 1)*exp(-I*pi/4)*log(1 - b* *(1/4)*x**(n/4)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(-3/4)/(4*a**(7/4)*n*ga mma(1/4))
\[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {3}{4} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c*x)^(-1-3/4*n)/(a+b*x^n),x, algorithm="maxima")
Output:
-b*integrate(x^(1/4*n)/(a*b*c^(3/4*n + 1)*x*x^n + a^2*c^(3/4*n + 1)*x), x) - 4/3*c^(-3/4*n - 1)/(a*n*x^(3/4*n))
\[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {3}{4} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c*x)^(-1-3/4*n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate((c*x)^(-3/4*n - 1)/(b*x^n + a), x)
Timed out. \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\int \frac {1}{{\left (c\,x\right )}^{\frac {3\,n}{4}+1}\,\left (a+b\,x^n\right )} \,d x \] Input:
int(1/((c*x)^((3*n)/4 + 1)*(a + b*x^n)),x)
Output:
int(1/((c*x)^((3*n)/4 + 1)*(a + b*x^n)), x)
\[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\frac {\int \frac {1}{x^{\frac {7 n}{4}} b x +x^{\frac {3 n}{4}} a x}d x}{c^{\frac {3 n}{4}} c} \] Input:
int((c*x)^(-1-3/4*n)/(a+b*x^n),x)
Output:
int(1/(x**((7*n)/4)*b*x + x**((3*n)/4)*a*x),x)/(c**((3*n)/4)*c)