3.70 \(\int x^2 \log (c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=65 \[ \frac{1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{3 d (n+3)} \]

[Out]

-(e*n*p*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(3*d*(3 + n)) + (x^3*Log[c*(d + e*x^
n)^p])/3

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Rubi [A]  time = 0.0280675, antiderivative size = 65, 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} \[ \frac{1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{3 d (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[c*(d + e*x^n)^p],x]

[Out]

-(e*n*p*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(3*d*(3 + n)) + (x^3*Log[c*(d + e*x^
n)^p])/3

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^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac{1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac{1}{3} (e n p) \int \frac{x^{2+n}}{d+e x^n} \, dx\\ &=-\frac{e n p x^{3+n} \, _2F_1\left (1,\frac{3+n}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{3 d (3+n)}+\frac{1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.0314956, size = 61, normalized size = 0.94 \[ \frac{1}{3} x^3 \left (\log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^n \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{d (n+3)}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[c*(d + e*x^n)^p],x]

[Out]

(x^3*(-((e*n*p*x^n*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(d*(3 + n))) + Log[c*(d + e*x^n)^p]
))/3

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Maple [F]  time = 1.69, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\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^2*ln(c*(d+e*x^n)^p),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{9} \,{\left (n p - 3 \, \log \left (c\right )\right )} x^{3} + d n p \int \frac{x^{2}}{3 \,{\left (e x^{n} + d\right )}}\,{d x} + \frac{1}{3} \, x^{3} \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^2*log(c*(d+e*x^n)^p),x, algorithm="maxima")

[Out]

-1/9*(n*p - 3*log(c))*x^3 + d*n*p*integrate(1/3*x^2/(e*x^n + d), x) + 1/3*x^3*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^{2} \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^2*log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

integral(x^2*log((e*x^n + d)^p*c), 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(x**2*ln(c*(d+e*x**n)**p),x)

[Out]

Timed out

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

[Out]

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