3.2764 \(\int \frac{(c x)^{-1-\frac{3 n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=317 \[ \frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}-\frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}+\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}-\frac{4 (c x)^{-3 n/4}}{3 a c n} \]

[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))

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Rubi [A]  time = 0.242825, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {363, 362, 345, 211, 1165, 628, 1162, 617, 204} \[ \frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}-\frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}+\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}-\frac{4 (c x)^{-3 n/4}}{3 a c n} \]

Antiderivative was successfully verified.

[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 363

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 362

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 345

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 211

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 1165

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]

Rule 628

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 1162

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 617

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(c x)^{-1-\frac{3 n}{4}}}{a+b x^n} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.0105467, size = 39, normalized size = 0.12 \[ -\frac{4 x (c x)^{-\frac{3 n}{4}-1} \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{b x^n}{a}\right )}{3 a n} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{3\,n}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{x^{\frac{1}{4} \, n}}{a b c^{\frac{3}{4} \, n + 1} x x^{n} + a^{2} c^{\frac{3}{4} \, n + 1} x}\,{d x} - \frac{4 \, c^{-\frac{3}{4} \, n - 1}}{3 \, a n x^{\frac{3}{4} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)
)

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Fricas [A]  time = 4.49438, size = 1112, normalized size = 3.51 \begin{align*} -\frac{12 \, a n \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{2} b^{2} c^{-2 \, n - \frac{8}{3}} n x^{\frac{1}{3}} \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{1}{4}} 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 )} - a^{2} n x^{\frac{1}{3}} \sqrt{-\frac{a^{3} b^{3} c^{-3 \, n - 4} n^{2} x^{\frac{1}{3}} \sqrt{-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}} - b^{4} c^{-4 \, n - \frac{16}{3}} x e^{\left (-\frac{1}{6} \,{\left (3 \, n + 4\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x}} \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{1}{4}}}{b^{3} c^{-3 \, n - 4}}\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 \, 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 ) + 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*a*n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4)*arctan(-(a^2*b^2*c^(-2*n - 8/3)*n*x^(1/3)*(-b^3*c^(-3*n - 4)/
(a^7*n^4))^(1/4)*e^(-1/12*(3*n + 4)*log(c) - 1/12*(3*n + 4)*log(x)) - a^2*n*x^(1/3)*sqrt(-(a^3*b^3*c^(-3*n - 4
)*n^2*x^(1/3)*sqrt(-b^3*c^(-3*n - 4)/(a^7*n^4)) - b^4*c^(-4*n - 16/3)*x*e^(-1/6*(3*n + 4)*log(c) - 1/6*(3*n +
4)*log(x)))/x)*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4))/(b^3*c^(-3*n - 4))) - 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) + 4*x*e^(
-1/4*(3*n + 4)*log(c) - 1/4*(3*n + 4)*log(x)))/(a*n)

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Sympy [C]  time = 5.26776, size = 309, normalized size = 0.97 \begin{align*} \frac{c^{- \frac{3 n}{4}} x^{- \frac{3 n}{4}} \Gamma \left (- \frac{3}{4}\right )}{a c n \Gamma \left (\frac{1}{4}\right )} - \frac{3 b^{\frac{3}{4}} c^{- \frac{3 n}{4}} 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}} c n \Gamma \left (\frac{1}{4}\right )} + \frac{3 i b^{\frac{3}{4}} c^{- \frac{3 n}{4}} 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}} c n \Gamma \left (\frac{1}{4}\right )} + \frac{3 b^{\frac{3}{4}} c^{- \frac{3 n}{4}} 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}} c n \Gamma \left (\frac{1}{4}\right )} - \frac{3 i b^{\frac{3}{4}} c^{- \frac{3 n}{4}} 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}} c n \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**(-3*n/4)*x**(-3*n/4)*gamma(-3/4)/(a*c*n*gamma(1/4)) - 3*b**(3/4)*c**(-3*n/4)*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)*c*n*gamma(1/4)) + 3*I*b**(3/4)*c**(-3*n/4)*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)*c*n*gamma(1/4)) + 3*b**
(3/4)*c**(-3*n/4)*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)
*c*n*gamma(1/4)) - 3*I*b**(3/4)*c**(-3*n/4)*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)*c*n*gamma(1/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{3}{4} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)