3.2608 \(\int \frac{x^{2+3 (-1+n)}}{a+b x^n} \, dx\)

Optimal. Leaf size=46 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]

[Out]

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

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Rubi [A]  time = 0.0242214, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(2 + 3*(-1 + n))/(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0123571, size = 38, normalized size = 0.83 \[ \frac{2 a^2 \log \left (a+b x^n\right )+b x^n \left (b x^n-2 a\right )}{2 b^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 51, normalized size = 1.1 \begin{align*}{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,bn}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}+{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.997813, size = 61, normalized size = 1.33 \begin{align*} \frac{a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} + \frac{b x^{2 \, n} - 2 \, a x^{n}}{2 \, b^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.02448, size = 84, normalized size = 1.83 \begin{align*} \frac{b^{2} x^{2 \, n} - 2 \, a b x^{n} + 2 \, a^{2} \log \left (b x^{n} + a\right )}{2 \, b^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 31.0822, size = 56, normalized size = 1.22 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{3 n}}{3 a n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{a^{2} \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{3} n} - \frac{a x^{n}}{b^{2} n} + \frac{x^{2 n}}{2 b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)