\(\int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx\) [2764]

Optimal result

Integrand size = 21, antiderivative size = 317 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=-\frac {4 (c x)^{-3 n/4}}{3 a c n}+\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}+\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n} \]

[Out]

-4/3/a/c/n/((c*x)^(3/4*n))+1/2*b^(3/4)*x^(3/4*n)*ln(-a^(1/4)*b^(1/4)*x^(1/4*n)*2^(1/2)+a^(1/2)+x^(1/2*n)*b^(1/
2))/a^(7/4)/c/n/((c*x)^(3/4*n))*2^(1/2)-1/2*b^(3/4)*x^(3/4*n)*ln(a^(1/4)*b^(1/4)*x^(1/4*n)*2^(1/2)+a^(1/2)+x^(
1/2*n)*b^(1/2))/a^(7/4)/c/n/((c*x)^(3/4*n))*2^(1/2)-b^(3/4)*x^(3/4*n)*arctan(-1+b^(1/4)*x^(1/4*n)*2^(1/2)/a^(1
/4))*2^(1/2)/a^(7/4)/c/n/((c*x)^(3/4*n))-b^(3/4)*x^(3/4*n)*arctan(1+b^(1/4)*x^(1/4*n)*2^(1/2)/a^(1/4))*2^(1/2)
/a^(7/4)/c/n/((c*x)^(3/4*n))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {370, 369, 352, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac {\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}+\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {a}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {4 (c x)^{-3 n/4}}{3 a c n} \]

[In]

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

[Out]

-4/(3*a*c*n*(c*x)^((3*n)/4)) + (Sqrt[2]*b^(3/4)*x^((3*n)/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^(n/4))/a^(1/4)])/(a^
(7/4)*c*n*(c*x)^((3*n)/4)) - (Sqrt[2]*b^(3/4)*x^((3*n)/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^(n/4))/a^(1/4)])/(a^(7
/4)*c*n*(c*x)^((3*n)/4)) + (b^(3/4)*x^((3*n)/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^(n/4) + Sqrt[b]*x^(n/2
)])/(Sqrt[2]*a^(7/4)*c*n*(c*x)^((3*n)/4)) - (b^(3/4)*x^((3*n)/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^(n/4)
+ Sqrt[b]*x^(n/2)])/(Sqrt[2]*a^(7/4)*c*n*(c*x)^((3*n)/4))

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 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 370

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

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 {\left (x^{3 n/4} (c x)^{-3 n/4}\right ) \int \frac {x^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx}{c} \\ & = -\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (b x^{3 n/4} (c x)^{-3 n/4}\right ) \int \frac {x^{\frac {1}{4} (-4+n)}}{a+b x^n} \, dx}{a c} \\ & = -\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (4 b x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a c n} \\ & = -\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (2 b x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}-\frac {\left (2 b x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n} \\ & = -\frac {4 (c x)^{-3 n/4}}{3 a c n}-\frac {\left (\sqrt {b} x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}-\frac {\left (\sqrt {b} x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{a^{3/2} c n}+\frac {\left (b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} a^{7/4} c n}+\frac {\left (b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} a^{7/4} c n} \\ & = -\frac {4 (c x)^{-3 n/4}}{3 a c n}+\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {\left (\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}+\frac {\left (\sqrt {2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{a}}\right )}{a^{7/4} c n} \\ & = -\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} \tan ^{-1}\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} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}+\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n}-\frac {b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt {b} x^{n/2}\right )}{\sqrt {2} a^{7/4} c n} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.12 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=-\frac {4 x (c x)^{-1-\frac {3 n}{4}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {b x^n}{a}\right )}{3 a n} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (c x \right )^{-1-\frac {3 n}{4}}}{a +b \,x^{n}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.27 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\frac {3 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 3 \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} - b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 3 i \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + 3 i \, a n \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{5} n^{3} x^{\frac {2}{3}} \left (-\frac {b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac {3}{4}} + b^{2} c^{-2 \, n - \frac {8}{3}} x e^{\left (-\frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{12} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 4 \, x e^{\left (-\frac {1}{4} \, {\left (3 \, n + 4\right )} \log \left (c\right ) - \frac {1}{4} \, {\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{3 \, a n} \]

[In]

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

[Out]

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)*log(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)*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) - 4*x*e^(-1/4*(3*n + 4)*log(c) - 1/4*(3*n
+ 4)*log(x)))/(a*n)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00 \[ \int \frac {(c x)^{-1-\frac {3 n}{4}}}{a+b x^n} \, dx=\frac {c^{- \frac {3 n}{4} - 1} x^{- \frac {3 n}{4}} \Gamma \left (- \frac {3}{4}\right )}{a n \Gamma \left (\frac {1}{4}\right )} - \frac {3 b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} + \frac {3 i b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} + \frac {3 b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} - \frac {3 i b^{\frac {3}{4}} c^{- \frac {3 n}{4} - 1} e^{- \frac {i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {3}{4}\right )}{4 a^{\frac {7}{4}} n \Gamma \left (\frac {1}{4}\right )} \]

[In]

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

[Out]

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)
*exp(-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*gamma(1/4))

Maxima [F]

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

[In]

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

[Out]

-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)
)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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