3.72 \(\int \log (c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=54 \[ x \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)} \]

[Out]

-((e*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((e*x^n)/d)])/(d*(1 + n))) + x*Log[c*(d + e*x
^n)^p]

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Rubi [A]  time = 0.0202858, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2448, 364} \[ x \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p],x]

[Out]

-((e*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((e*x^n)/d)])/(d*(1 + n))) + x*Log[c*(d + e*x
^n)^p]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=x \log \left (c \left (d+e x^n\right )^p\right )-(e n p) \int \frac{x^n}{d+e x^n} \, dx\\ &=-\frac{e n p x^{1+n} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (1+n)}+x \log \left (c \left (d+e x^n\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.029246, size = 52, normalized size = 0.96 \[ x \left (\log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^n \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^n)^p],x]

[Out]

x*(-((e*n*p*x^n*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((e*x^n)/d)])/(d*(1 + n))) + Log[c*(d + e*x^n)^p
])

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Maple [F]  time = 1.6, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e*x^n)^p),x)

[Out]

int(ln(c*(d+e*x^n)^p),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d n p \int \frac{1}{e x^{n} + d}\,{d x} -{\left (n p - \log \left (c\right )\right )} x + x \log \left ({\left (e x^{n} + d\right )}^{p}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p),x, algorithm="maxima")

[Out]

d*n*p*integrate(1/(e*x^n + d), x) - (n*p - log(c))*x + x*log((e*x^n + d)^p)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c), x)

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Sympy [C]  time = 4.67734, size = 48, normalized size = 0.89 \begin{align*} x \log{\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac{p x \Phi \left (\frac{d x^{- n} e^{i \pi }}{e}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{n \Gamma \left (1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(d+e*x**n)**p),x)

[Out]

x*log(c*(d + e*x**n)**p) + p*x*lerchphi(d*x**(-n)*exp_polar(I*pi)/e, 1, exp_polar(I*pi)/n)*gamma(1/n)/(n*gamma
(1 + 1/n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c), x)