3.2767 \(\int \frac{(c x)^{-1-\frac{n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=220 \[ -\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} c n}+\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} c n}-\frac{\sqrt{3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} c n}-\frac{3 (c x)^{-n/3}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.137471, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {347, 345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} c n}+\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} c n}-\frac{\sqrt{3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} c n}-\frac{3 (c x)^{-n/3}}{a c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - n/3)/(a + b*x^n),x]

[Out]

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

Rule 347

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracP
art[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !In
tegerQ[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 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

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] - Dist[(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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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])

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]

Rubi steps

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

Mathematica [C]  time = 0.009353, size = 37, normalized size = 0.17 \[ -\frac{3 x (c x)^{-\frac{n}{3}-1} \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};-\frac{b x^n}{a}\right )}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*(c*x)^(-1 - n/3)*Hypergeometric2F1[-1/3, 1, 2/3, -((b*x^n)/a)])/(a*n)

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{n}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*x)^(-1-1/3*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{2}{3} \, n}}{a b c^{\frac{1}{3} \, n + 1} x x^{n} + a^{2} c^{\frac{1}{3} \, n + 1} x}\,{d x} - \frac{3 \, c^{-\frac{1}{3} \, n - 1}}{a n x^{\frac{1}{3} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*integrate(x^(2/3*n)/(a*b*c^(1/3*n + 1)*x*x^n + a^2*c^(1/3*n + 1)*x), x) - 3*c^(-1/3*n - 1)/(a*n*x^(1/3*n))

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Fricas [A]  time = 1.53029, size = 705, normalized size = 3.2 \begin{align*} -\frac{6 \, x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} - 2 \, \sqrt{3} \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{2}{3}} x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} - \sqrt{3} b c^{-n - 3}}{3 \, b c^{-n - 3}}\right ) - 2 \, \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} \log \left (\frac{x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} + \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}}}{x}\right ) + \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} e^{\left (-\frac{2}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{2}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} - \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} + \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{2}{3}}}{x^{2}}\right )}{2 \, a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 5.06201, size = 216, normalized size = 0.98 \begin{align*} \frac{c^{- \frac{n}{3}} x^{- \frac{n}{3}} \Gamma \left (- \frac{1}{3}\right )}{a c n \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} c^{- \frac{n}{3}} e^{- \frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} x^{\frac{n}{3}} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} c n \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} c^{- \frac{n}{3}} \log{\left (1 - \frac{\sqrt [3]{b} x^{\frac{n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} c n \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} c^{- \frac{n}{3}} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} x^{\frac{n}{3}} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} c n \Gamma \left (\frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**(-n/3)*x**(-n/3)*gamma(-1/3)/(a*c*n*gamma(2/3)) - b**(1/3)*c**(-n/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*x**(n/
3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*c*n*gamma(2/3)) - b**(1/3)*c**(-n/3)*log(1 - b**(1/3)*x
**(n/3)*exp_polar(I*pi)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*c*n*gamma(2/3)) - b**(1/3)*c**(-n/3)*exp(2*I*pi/3)*l
og(1 - b**(1/3)*x**(n/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*c*n*gamma(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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