3.2597 \(\int \frac{x^{-1+5 n}}{a+b x^n} \, dx\)

Optimal. Leaf size=82 \[ -\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}+\frac{a^4 \log \left (a+b x^n\right )}{b^5 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]

[Out]

-((a^3*x^n)/(b^4*n)) + (a^2*x^(2*n))/(2*b^3*n) - (a*x^(3*n))/(3*b^2*n) + x^(4*n)/(4*b*n) + (a^4*Log[a + b*x^n]
)/(b^5*n)

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Rubi [A]  time = 0.0451149, antiderivative size = 82, normalized size of antiderivative = 1., 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, 43} \[ -\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}+\frac{a^4 \log \left (a+b x^n\right )}{b^5 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 5*n)/(a + b*x^n),x]

[Out]

-((a^3*x^n)/(b^4*n)) + (a^2*x^(2*n))/(2*b^3*n) - (a*x^(3*n))/(3*b^2*n) + x^(4*n)/(4*b*n) + (a^4*Log[a + b*x^n]
)/(b^5*n)

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]]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0448766, size = 65, normalized size = 0.79 \[ \frac{b x^n \left (6 a^2 b x^n-12 a^3-4 a b^2 x^{2 n}+3 b^3 x^{3 n}\right )+12 a^4 \log \left (a+b x^n\right )}{12 b^5 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 5*n)/(a + b*x^n),x]

[Out]

(b*x^n*(-12*a^3 + 6*a^2*b*x^n - 4*a*b^2*x^(2*n) + 3*b^3*x^(3*n)) + 12*a^4*Log[a + b*x^n])/(12*b^5*n)

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Maple [A]  time = 0.025, size = 87, normalized size = 1.1 \begin{align*}{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,bn}}-{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,{b}^{2}n}}+{\frac{{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{3}n}}-{\frac{{a}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{4}n}}+{\frac{{a}^{4}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{5}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+5*n)/(a+b*x^n),x)

[Out]

1/4/b/n*exp(n*ln(x))^4-1/3*a/b^2/n*exp(n*ln(x))^3+1/2*a^2/b^3/n*exp(n*ln(x))^2-a^3/b^4/n*exp(n*ln(x))+a^4/b^5/
n*ln(a+b*exp(n*ln(x)))

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Maxima [A]  time = 0.962331, size = 97, normalized size = 1.18 \begin{align*} \frac{a^{4} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{5} n} + \frac{3 \, b^{3} x^{4 \, n} - 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} - 12 \, a^{3} x^{n}}{12 \, b^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+5*n)/(a+b*x^n),x, algorithm="maxima")

[Out]

a^4*log((b*x^n + a)/b)/(b^5*n) + 1/12*(3*b^3*x^(4*n) - 4*a*b^2*x^(3*n) + 6*a^2*b*x^(2*n) - 12*a^3*x^n)/(b^4*n)

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Fricas [A]  time = 1.00733, size = 144, normalized size = 1.76 \begin{align*} \frac{3 \, b^{4} x^{4 \, n} - 4 \, a b^{3} x^{3 \, n} + 6 \, a^{2} b^{2} x^{2 \, n} - 12 \, a^{3} b x^{n} + 12 \, a^{4} \log \left (b x^{n} + a\right )}{12 \, b^{5} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+5*n)/(a+b*x^n),x, algorithm="fricas")

[Out]

1/12*(3*b^4*x^(4*n) - 4*a*b^3*x^(3*n) + 6*a^2*b^2*x^(2*n) - 12*a^3*b*x^n + 12*a^4*log(b*x^n + a))/(b^5*n)

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Sympy [A]  time = 160.387, size = 87, normalized size = 1.06 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{5 n}}{5 a n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{a^{4} \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{5} n} - \frac{a^{3} x^{n}}{b^{4} n} + \frac{a^{2} x^{2 n}}{2 b^{3} n} - \frac{a x^{3 n}}{3 b^{2} n} + \frac{x^{4 n}}{4 b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+5*n)/(a+b*x**n),x)

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0)), (x**(5*n)/(5*a*n), Eq(b, 0)), (log(x)/(a + b), Eq(n, 0)), (a**4*log
(a/b + x**n)/(b**5*n) - a**3*x**n/(b**4*n) + a**2*x**(2*n)/(2*b**3*n) - a*x**(3*n)/(3*b**2*n) + x**(4*n)/(4*b*
n), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+5*n)/(a+b*x^n),x, algorithm="giac")

[Out]

integrate(x^(5*n - 1)/(b*x^n + a), x)