3.2605 \(\int \frac {x^{-1-3 n}}{a+b x^n} \, dx\)

Optimal. Leaf size=76 \[ \frac {b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac {b^3 \log (x)}{a^4}-\frac {b^2 x^{-n}}{a^3 n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {x^{-3 n}}{3 a n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 44} \[ -\frac {b^2 x^{-n}}{a^3 n}+\frac {b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac {b^3 \log (x)}{a^4}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {x^{-3 n}}{3 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)/(a + b*x^n),x]

[Out]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 62, normalized size = 0.82 \[ -\frac {a x^{-3 n} \left (2 a^2-3 a b x^n+6 b^2 x^{2 n}\right )-6 b^3 \log \left (a+b x^n\right )+6 b^3 n \log (x)}{6 a^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 3*n)/(a + b*x^n),x]

[Out]

-1/6*((a*(2*a^2 - 3*a*b*x^n + 6*b^2*x^(2*n)))/x^(3*n) + 6*b^3*n*Log[x] - 6*b^3*Log[a + b*x^n])/(a^4*n)

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 72, normalized size = 0.95 \[ -\frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) - 6 \, b^{3} x^{3 \, n} \log \left (b x^{n} + a\right ) + 6 \, a b^{2} x^{2 \, n} - 3 \, a^{2} b x^{n} + 2 \, a^{3}}{6 \, a^{4} n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*b^3*n*x^(3*n)*log(x) - 6*b^3*x^(3*n)*log(b*x^n + a) + 6*a*b^2*x^(2*n) - 3*a^2*b*x^n + 2*a^3)/(a^4*n*x^
(3*n))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 88, normalized size = 1.16 \[ \left (-\frac {b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{a^{4}}+\frac {b \,{\mathrm e}^{n \ln \relax (x )}}{2 a^{2} n}-\frac {b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{a^{3} n}-\frac {1}{3 a n}\right ) {\mathrm e}^{-3 n \ln \relax (x )}+\frac {b^{3} \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 71, normalized size = 0.93 \[ -\frac {b^{3} \log \relax (x)}{a^{4}} + \frac {b^{3} \log \left (\frac {b x^{n} + a}{b}\right )}{a^{4} n} - \frac {6 \, b^{2} x^{2 \, n} - 3 \, a b x^{n} + 2 \, a^{2}}{6 \, a^{3} n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3*log(x)/a^4 + b^3*log((b*x^n + a)/b)/(a^4*n) - 1/6*(6*b^2*x^(2*n) - 3*a*b*x^n + 2*a^2)/(a^3*n*x^(3*n))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{3\,n+1}\,\left (a+b\,x^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3*n + 1)*(a + b*x^n)),x)

[Out]

int(1/(x^(3*n + 1)*(a + b*x^n)), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________