\(\int \log (c (d+e x^n)^p) \, dx\) [215]

Optimal result

Integrand size = 12, antiderivative size = 54 \[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e n p x^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1+n)}+x \log \left (c \left (d+e x^n\right )^p\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2498, 371} \[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=x \log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^{n+1} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (n+1)} \]

[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2498

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]

Rubi steps \begin{align*} \text {integral}& = 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] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=x \left (-\frac {e n p x^n \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1+n)}+\log \left (c \left (d+e x^n\right )^p\right )\right ) \]

[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
])

Maple [F]

\[\int \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

integral(log((e*x^n + d)^p*c), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=x \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {d^{- \frac {1}{n}} d^{1 + \frac {1}{n}} e e^{\frac {1}{n}} e^{-1 - \frac {1}{n}} p x \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{d n \Gamma \left (1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]

[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)

Giac [F]

\[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]

[In]

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

[Out]

integrate(log((e*x^n + d)^p*c), x)

Mupad [F(-1)]

Timed out. \[ \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right ) \,d x \]

[In]

int(log(c*(d + e*x^n)^p),x)

[Out]

int(log(c*(d + e*x^n)^p), x)