3.2763 \(\int \frac{(c x)^{-1-\frac{2 n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=222 \[ -\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} c n}+\frac{\sqrt{3} b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} c n}-\frac{3 (c x)^{-2 n/3}}{2 a c n} \]

[Out]

-3/(2*a*c*n*(c*x)^((2*n)/3)) + (Sqrt[3]*b^(2/3)*x^((2*n)/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(n/3))/(Sqrt[3]*a^(1
/3))])/(a^(5/3)*c*n*(c*x)^((2*n)/3)) - (b^(2/3)*x^((2*n)/3)*Log[a^(1/3) + b^(1/3)*x^(n/3)])/(a^(5/3)*c*n*(c*x)
^((2*n)/3)) + (b^(2/3)*x^((2*n)/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(n/3) + b^(2/3)*x^((2*n)/3)])/(2*a^(5/3)*c*
n*(c*x)^((2*n)/3))

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Rubi [A]  time = 0.138106, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {363, 362, 345, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} c n}+\frac{\sqrt{3} b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} c n}-\frac{3 (c x)^{-2 n/3}}{2 a c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3/(2*a*c*n*(c*x)^((2*n)/3)) + (Sqrt[3]*b^(2/3)*x^((2*n)/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(n/3))/(Sqrt[3]*a^(1
/3))])/(a^(5/3)*c*n*(c*x)^((2*n)/3)) - (b^(2/3)*x^((2*n)/3)*Log[a^(1/3) + b^(1/3)*x^(n/3)])/(a^(5/3)*c*n*(c*x)
^((2*n)/3)) + (b^(2/3)*x^((2*n)/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(n/3) + b^(2/3)*x^((2*n)/3)])/(2*a^(5/3)*c*
n*(c*x)^((2*n)/3))

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 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{2 n}{3}}}{a+b x^n} \, dx &=\frac{\left (x^{2 n/3} (c x)^{-2 n/3}\right ) \int \frac{x^{-1-\frac{2 n}{3}}}{a+b x^n} \, dx}{c}\\ &=-\frac{3 (c x)^{-2 n/3}}{2 a c n}-\frac{\left (b x^{2 n/3} (c x)^{-2 n/3}\right ) \int \frac{x^{\frac{1}{3} (-3+n)}}{a+b x^n} \, dx}{a c}\\ &=-\frac{3 (c x)^{-2 n/3}}{2 a c n}-\frac{\left (3 b x^{2 n/3} (c x)^{-2 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,x^{1+\frac{1}{3} (-3+n)}\right )}{a c n}\\ &=-\frac{3 (c x)^{-2 n/3}}{2 a c n}-\frac{\left (b x^{2 n/3} (c x)^{-2 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^{1+\frac{1}{3} (-3+n)}\right )}{a^{5/3} c n}-\frac{\left (b x^{2 n/3} (c x)^{-2 n/3}\right ) \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^{1+\frac{1}{3} (-3+n)}\right )}{a^{5/3} c n}\\ &=-\frac{3 (c x)^{-2 n/3}}{2 a c n}-\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{\left (b^{2/3} x^{2 n/3} (c x)^{-2 n/3}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^{1+\frac{1}{3} (-3+n)}\right )}{2 a^{5/3} c n}-\frac{\left (3 b x^{2 n/3} (c x)^{-2 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^{1+\frac{1}{3} (-3+n)}\right )}{2 a^{4/3} c n}\\ &=-\frac{3 (c x)^{-2 n/3}}{2 a c n}-\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} c n}-\frac{\left (3 b^{2/3} x^{2 n/3} (c x)^{-2 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x^{1+\frac{1}{3} (-3+n)}}{\sqrt [3]{a}}\right )}{a^{5/3} c n}\\ &=-\frac{3 (c x)^{-2 n/3}}{2 a c n}+\frac{\sqrt{3} b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} c n}-\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} c n}\\ \end{align*}

Mathematica [C]  time = 0.011086, size = 39, normalized size = 0.18 \[ -\frac{3 x (c x)^{-\frac{2 n}{3}-1} \, _2F_1\left (-\frac{2}{3},1;\frac{1}{3};-\frac{b x^n}{a}\right )}{2 a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*(c*x)^(-1 - (2*n)/3)*Hypergeometric2F1[-2/3, 1, 1/3, -((b*x^n)/a)])/(2*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{2\,n}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.86318, size = 891, normalized size = 4.01 \begin{align*} -\frac{3 \, x e^{\left (-\frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - 2 \, \sqrt{3} \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \sqrt{x} e^{\left (-\frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} + \sqrt{3} b c^{-n - \frac{3}{2}}}{3 \, b c^{-n - \frac{3}{2}}}\right ) - 2 \, \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \log \left (\frac{b c^{-n - \frac{3}{2}} x e^{\left (-\frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} + a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{2}{3}} \sqrt{x}}{x}\right ) + \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \log \left (\frac{b c^{-n - \frac{3}{2}} x e^{\left (-\frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{2}{3}} \sqrt{x} e^{\left (-\frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - b c^{-n - \frac{3}{2}} \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}}}{x}\right )}{2 \, a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 5.01053, size = 226, normalized size = 1.02 \begin{align*} \frac{c^{- \frac{2 n}{3}} x^{- \frac{2 n}{3}} \Gamma \left (- \frac{2}{3}\right )}{a c n \Gamma \left (\frac{1}{3}\right )} - \frac{2 b^{\frac{2}{3}} c^{- \frac{2 n}{3}} e^{- \frac{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{2}{3}\right )}{3 a^{\frac{5}{3}} c n \Gamma \left (\frac{1}{3}\right )} + \frac{2 b^{\frac{2}{3}} c^{- \frac{2 n}{3}} \log{\left (1 - \frac{\sqrt [3]{b} x^{\frac{n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{2}{3}\right )}{3 a^{\frac{5}{3}} c n \Gamma \left (\frac{1}{3}\right )} - \frac{2 b^{\frac{2}{3}} c^{- \frac{2 n}{3}} e^{\frac{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{2}{3}\right )}{3 a^{\frac{5}{3}} c n \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**(-2*n/3)*x**(-2*n/3)*gamma(-2/3)/(a*c*n*gamma(1/3)) - 2*b**(2/3)*c**(-2*n/3)*exp(-I*pi/3)*log(1 - b**(1/3)*
x**(n/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-2/3)/(3*a**(5/3)*c*n*gamma(1/3)) + 2*b**(2/3)*c**(-2*n/3)*log(1 -
b**(1/3)*x**(n/3)*exp_polar(I*pi)/a**(1/3))*gamma(-2/3)/(3*a**(5/3)*c*n*gamma(1/3)) - 2*b**(2/3)*c**(-2*n/3)*e
xp(I*pi/3)*log(1 - b**(1/3)*x**(n/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-2/3)/(3*a**(5/3)*c*n*gamma(1/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{2}{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)^(-1-2/3*n)/(a+b*x^n),x, algorithm="giac")

[Out]

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