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

Optimal. Leaf size=65 \[ \frac{1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.0241193, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2455, 364} \[ \frac{1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[x*Log[c*(d + e*x^n)^p],x]

[Out]

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

Rule 2455

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]

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 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{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]  time = 0.0320371, size = 61, normalized size = 0.94 \[ \frac{1}{2} x^2 \left (\log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^n \, _2F_1\left (1,\frac{n+2}{n};2+\frac{2}{n};-\frac{e x^n}{d}\right )}{d (n+2)}\right ) \]

Antiderivative was successfully verified.

[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

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Maple [F]  time = 1.892, size = 0, normalized size = 0. \begin{align*} \int x\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(x*ln(c*(d+e*x^n)^p),x)

[Out]

int(x*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{x}{2 \,{\left (e x^{n} + d\right )}}\,{d x} - \frac{1}{4} \,{\left (n p - 2 \, \log \left (c\right )\right )} x^{2} + \frac{1}{2} \, x^{2} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

integral(x*log((e*x^n + d)^p*c), x)

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Sympy [C]  time = 38.3548, size = 104, normalized size = 1.6 \begin{align*} \frac{x^{2} \log{\left (c \left (d + e x^{n}\right )^{p} \right )}}{2} - \frac{e p x^{2} x^{n} \Phi \left (\frac{e x^{n} e^{i \pi }}{d}, 1, 1 + \frac{2}{n}\right ) \Gamma \left (1 + \frac{2}{n}\right )}{2 d \Gamma \left (2 + \frac{2}{n}\right )} - \frac{e p x^{2} x^{n} \Phi \left (\frac{e x^{n} e^{i \pi }}{d}, 1, 1 + \frac{2}{n}\right ) \Gamma \left (1 + \frac{2}{n}\right )}{d n \Gamma \left (2 + \frac{2}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(x*log((e*x^n + d)^p*c), x)