3.74 \(\int \frac{\log (c (d+e x^n)^p)}{x^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{\log \left (c \left (d+e x^n\right )^p\right )}{x}-\frac{e n p x^{n-1} \, _2F_1\left (1,-\frac{1-n}{n};2-\frac{1}{n};-\frac{e x^n}{d}\right )}{d (1-n)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0315251, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2455, 364} \[ -\frac{\log \left (c \left (d+e x^n\right )^p\right )}{x}-\frac{e n p x^{n-1} \, _2F_1\left (1,-\frac{1-n}{n};2-\frac{1}{n};-\frac{e x^n}{d}\right )}{d (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p]/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0321867, size = 59, normalized size = 0.89 \[ \frac{\frac{e n p x^n \, _2F_1\left (1,\frac{n-1}{n};2-\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n-1)}-\log \left (c \left (d+e x^n\right )^p\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^n)^p]/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 1.523, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e*x^n)^p)/x^2,x)

[Out]

int(ln(c*(d+e*x^n)^p)/x^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -d n p \int \frac{1}{e x^{2} x^{n} + d x^{2}}\,{d x} - \frac{n p + \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + \log \left (c\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)/x^2,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)/x^2, x)

________________________________________________________________________________________

Sympy [C]  time = 14.7649, size = 46, normalized size = 0.7 \begin{align*} - \frac{\log{\left (c \left (d + e x^{n}\right )^{p} \right )}}{x} + \frac{p \Phi \left (\frac{d x^{- n} e^{i \pi }}{e}, 1, \frac{1}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{n x \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**2,x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)/x^2,x, algorithm="giac")

[Out]

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