3.26.33 \(\int x^{-1-n} (a+b x^n)^2 \, dx\) [2533]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, 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 = {272, 45} \begin {gather*} -\frac {a^2 x^{-n}}{n}+2 a b \log (x)+\frac {b^2 x^n}{n} \end {gather*}

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 45

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])

Rule 272

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

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Mathematica [A]
time = 0.03, size = 31, normalized size = 1.03 \begin {gather*} -\frac {a^2 x^{-n}-b^2 x^n-2 a b \log \left (x^n\right )}{n} \end {gather*}

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*Log[x^n])/n)

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Maple [A]
time = 0.22, size = 31, normalized size = 1.03

method	result	size
risch	\(-\frac {a^{2} x^{-n}}{n}+\frac {b^{2} x^{n}}{n}+2 a b \ln \left (x \right )\)	\(31\)
norman	\(\left (\frac {b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{n}+2 a b \ln \left (x \right ) {\mathrm e}^{n \ln \left (x \right )}-\frac {a^{2}}{n}\right ) {\mathrm e}^{-n \ln \left (x \right )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 30, normalized size = 1.00 \begin {gather*} 2 \, a b \log \left (x\right ) + \frac {b^{2} x^{n}}{n} - \frac {a^{2}}{n x^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.42, size = 34, normalized size = 1.13 \begin {gather*} \frac {2 \, a b n x^{n} \log \left (x\right ) + b^{2} x^{2 \, n} - a^{2}}{n x^{n}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (24) = 48\).
time = 2.58, size = 178, normalized size = 5.93 \begin {gather*} \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 x^{n} \log {\left (x^{- n} \right )}}{n^{2} x^{n} + n x^{n}} + \frac {2 a b n x^{n}}{n^{2} x^{n} + n x^{n}} - \frac {2 a b x^{n} \log {\left (x^{- n} \right )}}{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 {gather*}

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*x**n*log(x**(-n))/(n**2*x**n + n*x**n) + 2*a*b*n*x**n/(n**2*x**n +
n*x**n) - 2*a*b*x**n*log(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 = 2.31, size = 34, normalized size = 1.13 \begin {gather*} \frac {2 \, a b n x^{n} \log \left (x\right ) + b^{2} x^{2 \, n} - a^{2}}{n x^{n}} \end {gather*}

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)

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Mupad [B]
time = 1.32, size = 30, normalized size = 1.00 \begin {gather*} 2\,a\,b\,\ln \left (x\right )+\frac {b^2\,x^n}{n}-\frac {a^2}{n\,x^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^2/x^(n + 1),x)

[Out]

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

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