3.7.12 \(\int \frac {x^{-n} (a+b n x^{-1+n})}{b+a x^{1-n}} \, dx\) [612]

Optimal. Leaf size=17 \[ n \log (x)+\log \left (b+a x^{1-n}\right ) \]

[Out]

n*ln(x)+ln(b+a*x^(1-n))

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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {528, 457, 78} \begin {gather*} \log \left (a x^{1-n}+b\right )+n \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*n*x^(-1 + n))/(x^n*(b + a*x^(1 - n))),x]

[Out]

n*Log[x] + Log[b + a*x^(1 - n)]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 528

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 0.59 \begin {gather*} \log \left (a x+b x^n\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a*x + b*x^n]

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Maple [A]
time = 0.33, size = 12, normalized size = 0.71

method	result	size
risch	\(\ln \left (x^{n}+\frac {a x}{b}\right )\)	\(12\)
norman	\(\ln \left (a x +b \,{\mathrm e}^{n \ln \left (x \right )}\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (17) = 34\).
time = 0.29, size = 86, normalized size = 5.06 \begin {gather*} b n {\left (\frac {\log \left (x\right )}{b} - \frac {n \log \left (x\right )}{b {\left (n - 1\right )}} + \frac {\log \left (\frac {a x + b x^{n}}{b}\right )}{b {\left (n - 1\right )}}\right )} + a {\left (\frac {n \log \left (x\right )}{a {\left (n - 1\right )}} - \frac {\log \left (\frac {a x + b x^{n}}{b}\right )}{a {\left (n - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 10, normalized size = 0.59 \begin {gather*} \log \left (a x + b x^{n}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b*x^n)

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Sympy [A]
time = 31.22, size = 8, normalized size = 0.47 \begin {gather*} \log {\left (a x + b x^{n} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b*x**n)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.37, size = 39, normalized size = 2.29 \begin {gather*} -\frac {\ln \left (b+a\,x^{1-n}\right )-2\,n\,\mathrm {atanh}\left (\frac {2\,a\,x^{1-n}}{b}+1\right )}{n-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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