3.2533 \(\int x^{-1-n} (a+b x^n)^2 \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2 x^{-n}}{n}+2 a b \log (x)+\frac{b^2 x^n}{n} \]

[Out]

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

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Rubi [A]  time = 0.0152437, antiderivative size = 30, 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} \[ -\frac{a^2 x^{-n}}{n}+2 a b \log (x)+\frac{b^2 x^n}{n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*(a + b*x^n)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0164571, size = 29, normalized size = 0.97 \[ \frac{-a^2 x^{-n}+2 a b n \log (x)+b^2 x^n}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)*(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.011, size = 43, normalized size = 1.4 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}+2\,ab\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{2}}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04035, size = 68, normalized size = 2.27 \begin{align*} \frac{2 \, a b n x^{n} \log \left (x\right ) + b^{2} x^{2 \, n} - a^{2}}{n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 39.3753, size = 175, normalized size = 5.83 \begin{align*} \begin{cases} a^{2} x + 2 a b \log{\left (x \right )} - \frac{b^{2}}{x} & \text{for}\: n = -1 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{a^{2} n}{n^{2} x^{n} + n x^{n}} - \frac{a^{2}}{n^{2} x^{n} + n x^{n}} + \frac{2 a b n^{2} x^{n} \log{\left (x \right )}}{n^{2} x^{n} + n x^{n}} + \frac{2 a b n x^{n} \log{\left (x \right )}}{n^{2} x^{n} + n x^{n}} + \frac{2 a b n x^{n}}{n^{2} x^{n} + n x^{n}} + \frac{b^{2} n x^{2 n}}{n^{2} x^{n} + n x^{n}} + \frac{b^{2} x^{2 n}}{n^{2} x^{n} + n x^{n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x + 2*a*b*log(x) - b**2/x, Eq(n, -1)), ((a + b)**2*log(x), Eq(n, 0)), (-a**2*n/(n**2*x**n + n*
x**n) - a**2/(n**2*x**n + n*x**n) + 2*a*b*n**2*x**n*log(x)/(n**2*x**n + n*x**n) + 2*a*b*n*x**n*log(x)/(n**2*x*
*n + n*x**n) + 2*a*b*n*x**n/(n**2*x**n + n*x**n) + b**2*n*x**(2*n)/(n**2*x**n + n*x**n) + b**2*x**(2*n)/(n**2*
x**n + n*x**n), True))

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Giac [A]  time = 1.18877, size = 46, normalized size = 1.53 \begin{align*} \frac{2 \, a b n x^{n} \log \left (x\right ) + b^{2} x^{2 \, n} - a^{2}}{n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-n)*(a+b*x^n)^2,x, algorithm="giac")

[Out]

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