3.2534 \(\int x^{-1-2 n} (a+b x^n)^2 \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2 x^{-2 n}}{2 n}-\frac{2 a b x^{-n}}{n}+b^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0159883, antiderivative size = 34, 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} \[ -\frac{a^2 x^{-2 n}}{2 n}-\frac{2 a b x^{-n}}{n}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*(a + b*x^n)^2,x]

[Out]

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

Mathematica [A]  time = 0.0272676, size = 28, normalized size = 0.82 \[ b^2 \log (x)-\frac{a x^{-2 n} \left (a+4 b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 43, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{2}}{2\,n}}-2\,{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.20478, size = 81, normalized size = 2.38 \begin{align*} \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 4 \, a b x^{n} - a^{2}}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 74.7313, size = 235, normalized size = 6.91 \begin{align*} \begin{cases} a^{2} x + 4 a b \sqrt{x} + b^{2} \log{\left (x \right )} & \text{for}\: n = - \frac{1}{2} \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{2 a^{2} n}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac{a^{2}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac{8 a b n x^{n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac{4 a b x^{n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac{4 b^{2} n^{2} x^{2 n} \log{\left (x \right )}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac{2 b^{2} n x^{2 n} \log{\left (x \right )}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac{2 b^{2} n x^{2 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x + 4*a*b*sqrt(x) + b**2*log(x), Eq(n, -1/2)), ((a + b)**2*log(x), Eq(n, 0)), (-2*a**2*n/(4*n*
*2*x**(2*n) + 2*n*x**(2*n)) - a**2/(4*n**2*x**(2*n) + 2*n*x**(2*n)) - 8*a*b*n*x**n/(4*n**2*x**(2*n) + 2*n*x**(
2*n)) - 4*a*b*x**n/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 4*b**2*n**2*x**(2*n)*log(x)/(4*n**2*x**(2*n) + 2*n*x**(2
*n)) + 2*b**2*n*x**(2*n)*log(x)/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 2*b**2*n*x**(2*n)/(4*n**2*x**(2*n) + 2*n*x*
*(2*n)), True))

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Giac [A]  time = 1.22043, size = 51, normalized size = 1.5 \begin{align*} \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 4 \, a b x^{n} - a^{2}}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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