3.1.75 \(\int \frac {\log (c (d+e x^n)^p)}{x^3} \, dx\) [75]

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

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

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Rubi [A]
time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = {2505, 371} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p]/x^3,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)) - Log[c*(
d + e*x^n)^p]/(2*x^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*} \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*}

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Mathematica [A]
time = 0.02, size = 62, normalized size = 0.86 \begin {gather*} \frac {\frac {e n p x^n \, _2F_1\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 )}{2 x^2} \end {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}{x^{3}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^n*e + d)^p*c)/x^3, x)

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Sympy [C] Result contains complex when optimal does not.
time = 8.87, size = 51, normalized size = 0.71 \begin {gather*} - \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2 x^{2}} + \frac {p \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {2}{n}\right ) \Gamma \left (- \frac {2}{n}\right )}{n x^{2} \Gamma \left (1 - \frac {2}{n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^n*e + d)^p*c)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^n)^p)/x^3,x)

[Out]

int(log(c*(d + e*x^n)^p)/x^3, x)

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