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

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

[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)

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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {2505, 371} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \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*}

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Mathematica [A]
time = 0.02, size = 61, normalized size = 0.94 \begin {gather*} \frac {1}{2} x^2 \left (-\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 )\right ) \end {gather*}

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

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 = 0.00, 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(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 = 0.00, 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(x*log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

integral(x*log((x^n*e + d)^p*c), x)

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Sympy [C] Result contains complex when optimal does not.
time = 3.68, size = 104, normalized size = 1.60 \begin {gather*} \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 {gather*}

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 = 0.00, 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(x*log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

integrate(x*log((x^n*e + d)^p*c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\ln \left (c\,{\left (d+e\,x^n\right )}^p\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(c*(d + e*x^n)^p),x)

[Out]

int(x*log(c*(d + e*x^n)^p), x)

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