3.75 \(\int \frac{\log (c (d+e x^n)^p)}{x^3} \, dx\)

Optimal. Leaf size=72 \[ -\frac{\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2}-\frac{e n p x^{n-2} \, _2F_1\left (1,-\frac{2-n}{n};2 \left (1-\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.0307312, antiderivative size = 72, 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} \[ -\frac{\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2}-\frac{e n p x^{n-2} \, _2F_1\left (1,-\frac{2-n}{n};2 \left (1-\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p]/x^3,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)) - Log[c*(d
+ e*x^n)^p]/(2*x^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 \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx &=-\frac{\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2}+\frac{1}{2} (e n p) \int \frac{x^{-3+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{\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0266078, size = 62, normalized size = 0.86 \[ \frac{\frac{e n p x^n \, _2F_1\left (1,\frac{n-2}{n};2-\frac{2}{n};-\frac{e x^n}{d}\right )}{d (n-2)}-\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^n)^p]/x^3,x]

[Out]

((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*x
^2)

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Maple [F]  time = 1.554, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e*x^n)^p)/x^3,x)

[Out]

int(ln(c*(d+e*x^n)^p)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -d n p \int \frac{1}{2 \,{\left (e x^{3} x^{n} + d x^{3}\right )}}\,{d x} - \frac{n p + 2 \, \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + 2 \, \log \left (c\right )}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)/x^3,x, algorithm="maxima")

[Out]

-d*n*p*integrate(1/2/(e*x^3*x^n + d*x^3), x) - 1/4*(n*p + 2*log((e*x^n + d)^p) + 2*log(c))/x^2

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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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)/x^3,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)/x^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(d+e*x**n)**p)/x**3,x)

[Out]

Timed out

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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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)/x^3,x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)/x^3, x)