3.5.95 \(\int \frac {x^{-1+2 n}}{b x^n+c x^{2 n}} \, dx\) [495]

Optimal. Leaf size=15 \[ \frac {\log \left (b+c x^n\right )}{c n} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1598, 266} \begin {gather*} \frac {\log \left (b+c x^n\right )}{c n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)/(b*x^n + c*x^(2*n)),x]

[Out]

Log[b + c*x^n]/(c*n)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.20 \begin {gather*} \frac {\log \left (b n+c n x^n\right )}{c n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[b*n + c*n*x^n]/(c*n)

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Maple [A]
time = 0.20, size = 18, normalized size = 1.20

method	result	size
norman	\(\frac {\ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{c n}\)	\(18\)
risch	\(\frac {\ln \left (x^{n}+\frac {b}{c}\right )}{c n}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c/n*ln(c*exp(n*ln(x))+b)

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Maxima [A]
time = 0.28, size = 19, normalized size = 1.27 \begin {gather*} \frac {\log \left (\frac {c x^{n} + b}{c}\right )}{c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((c*x^n + b)/c)/(c*n)

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Fricas [A]
time = 0.42, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (c x^{n} + b\right )}{c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^n + b)/(c*n)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (10) = 20\).
time = 3.02, size = 73, normalized size = 4.87 \begin {gather*} \begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: b = 0 \wedge c = 0 \wedge n = 0 \\\frac {x^{n}}{b n} & \text {for}\: c = 0 \\\frac {\log {\left (x \right )}}{b + c} & \text {for}\: n = 0 \\\frac {\frac {2 n \log {\left (x^{2 n} \right )}}{4 n^{2} - 2 n} - \frac {\log {\left (x^{2 n} \right )}}{4 n^{2} - 2 n}}{c} & \text {for}\: b = 0 \\- \frac {\log {\left (x \right )}}{c} + \frac {\log {\left (x^{n} + \frac {c x^{2 n}}{b} \right )}}{c n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*log(x), Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), (x**n/(b*n), Eq(c, 0)), (log(x)/(b + c), Eq(n, 0)), ((
2*n*log(x**(2*n))/(4*n**2 - 2*n) - log(x**(2*n))/(4*n**2 - 2*n))/c, Eq(b, 0)), (-log(x)/c + log(x**n + c*x**(2
*n)/b)/(c*n), True))

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

[Out]

integrate(x^(2*n - 1)/(c*x^(2*n) + b*x^n), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {x^{2\,n-1}}{b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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