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

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Rubi [A]  time = 0.14, antiderivative size = 220, normalized size of antiderivative = 1.00, 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 31

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

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 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 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 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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 0.01, 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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fricas [A]  time = 0.56, size = 255, normalized size = 1.16 \[ -\frac {6 \, x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \relax (c) - \frac {1}{3} \, {\left (n + 3\right )} \log \relax (x)\right )} - 2 \, \sqrt {3} \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {2}{3}} x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \relax (c) - \frac {1}{3} \, {\left (n + 3\right )} \log \relax (x)\right )} - \sqrt {3} b c^{-n - 3}}{3 \, b c^{-n - 3}}\right ) - 2 \, \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} \log \left (\frac {x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \relax (c) - \frac {1}{3} \, {\left (n + 3\right )} \log \relax (x)\right )} + \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}}}{x}\right ) + \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} e^{\left (-\frac {2}{3} \, {\left (n + 3\right )} \log \relax (c) - \frac {2}{3} \, {\left (n + 3\right )} \log \relax (x)\right )} - \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \relax (c) - \frac {1}{3} \, {\left (n + 3\right )} \log \relax (x)\right )} + \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{-\frac {1}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]

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)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {n}{3}-1}}{b \,x^{n}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -b \int \frac {x^{\frac {2}{3} \, n}}{a b c^{\frac {1}{3} \, n + 1} x x^{n} + a^{2} c^{\frac {1}{3} \, n + 1} x}\,{d x} - \frac {3 \, c^{-\frac {1}{3} \, n - 1}}{a n x^{\frac {1}{3} \, n}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,x\right )}^{\frac {n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 4.23, size = 216, normalized size = 0.98 \[ \frac {c^{- \frac {n}{3}} x^{- \frac {n}{3}} \Gamma \left (- \frac {1}{3}\right )}{a c n \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{b} c^{- \frac {n}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} c n \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{b} c^{- \frac {n}{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} c n \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{b} c^{- \frac {n}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} c n \Gamma \left (\frac {2}{3}\right )} \]

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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