\(\int x \log (c (d+e x^n)^p) \, dx\) [71]

Optimal result

Integrand size = 14, antiderivative size = 65 \[ \int x \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e n p x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{n},2 \left (1+\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{2 d (2+n)}+\frac {1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2505, 371} \[ \int x \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{n},2 \left (1+\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{2 d (n+2)} \]

[In]

Int[x*Log[c*(d + e*x^n)^p],x]

[Out]

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

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 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int x \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {1}{2} x^2 \left (-\frac {e n p x^n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{n},2+\frac {2}{n},-\frac {e x^n}{d}\right )}{d (2+n)}+\log \left (c \left (d+e x^n\right )^p\right )\right ) \]

[In]

Integrate[x*Log[c*(d + e*x^n)^p],x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*log((e*x^n + d)^p*c), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.88 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.97 \[ \int x \log \left (c \left (d+e x^n\right )^p\right ) \, dx=- \frac {d^{-2 - \frac {2}{n}} d^{1 + \frac {2}{n}} e p x^{n + 2} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{2 \Gamma \left (2 + \frac {2}{n}\right )} - \frac {d^{-2 - \frac {2}{n}} d^{1 + \frac {2}{n}} e p x^{n + 2} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{n \Gamma \left (2 + \frac {2}{n}\right )} + \frac {x^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2} \]

[In]

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

[Out]

-d**(-2 - 2/n)*d**(1 + 2/n)*e*p*x**(n + 2)*lerchphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 2/n)*gamma(1 + 2/n)/(2*ga
mma(2 + 2/n)) - d**(-2 - 2/n)*d**(1 + 2/n)*e*p*x**(n + 2)*lerchphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 2/n)*gamma
(1 + 2/n)/(n*gamma(2 + 2/n)) + x**2*log(c*(d + e*x**n)**p)/2

Maxima [F]

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

[In]

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

[Out]

d*n*p*integrate(1/2*x/(e*x^n + d), x) - 1/4*(n*p - 2*log(c))*x^2 + 1/2*x^2*log((e*x^n + d)^p)

Giac [F]

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

[In]

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

[Out]

integrate(x*log((e*x^n + d)^p*c), x)

Mupad [F(-1)]

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

[In]

int(x*log(c*(d + e*x^n)^p),x)

[Out]

int(x*log(c*(d + e*x^n)^p), x)