3.16 \(\int x^2 \log (c (a+b x^3)^p) \, dx\)

Optimal. Leaf size=35 \[ \frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b}-\frac{p x^3}{3} \]

[Out]

-(p*x^3)/3 + ((a + b*x^3)*Log[c*(a + b*x^3)^p])/(3*b)

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Rubi [A]  time = 0.0291231, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2389, 2295} \[ \frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b}-\frac{p x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[c*(a + b*x^3)^p],x]

[Out]

-(p*x^3)/3 + ((a + b*x^3)*Log[c*(a + b*x^3)^p])/(3*b)

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int x^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^3\right )}{3 b}\\ &=-\frac{p x^3}{3}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0095017, size = 34, normalized size = 0.97 \[ \frac{1}{3} \left (\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{b}-p x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[c*(a + b*x^3)^p],x]

[Out]

(-(p*x^3) + ((a + b*x^3)*Log[c*(a + b*x^3)^p])/b)/3

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Maple [A]  time = 0.062, size = 50, normalized size = 1.4 \begin{align*}{\frac{{x}^{3}\ln \left ( c \left ( b{x}^{3}+a \right ) ^{p} \right ) }{3}}-{\frac{p{x}^{3}}{3}}+{\frac{\ln \left ( c \left ( b{x}^{3}+a \right ) ^{p} \right ) a}{3\,b}}-{\frac{ap}{3\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(c*(b*x^3+a)^p),x)

[Out]

1/3*x^3*ln(c*(b*x^3+a)^p)-1/3*p*x^3+1/3/b*ln(c*(b*x^3+a)^p)*a-1/3/b*a*p

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Maxima [A]  time = 1.15659, size = 59, normalized size = 1.69 \begin{align*} \frac{1}{3} \, x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) - \frac{1}{3} \,{\left (\frac{x^{3}}{b} - \frac{a \log \left (b x^{3} + a\right )}{b^{2}}\right )} b p \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(c*(b*x^3+a)^p),x, algorithm="maxima")

[Out]

1/3*x^3*log((b*x^3 + a)^p*c) - 1/3*(x^3/b - a*log(b*x^3 + a)/b^2)*b*p

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Fricas [A]  time = 1.92167, size = 89, normalized size = 2.54 \begin{align*} -\frac{b p x^{3} - b x^{3} \log \left (c\right ) -{\left (b p x^{3} + a p\right )} \log \left (b x^{3} + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(c*(b*x^3+a)^p),x, algorithm="fricas")

[Out]

-1/3*(b*p*x^3 - b*x^3*log(c) - (b*p*x^3 + a*p)*log(b*x^3 + a))/b

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Sympy [A]  time = 7.52976, size = 56, normalized size = 1.6 \begin{align*} \begin{cases} \frac{a p \log{\left (a + b x^{3} \right )}}{3 b} + \frac{p x^{3} \log{\left (a + b x^{3} \right )}}{3} - \frac{p x^{3}}{3} + \frac{x^{3} \log{\left (c \right )}}{3} & \text{for}\: b \neq 0 \\\frac{x^{3} \log{\left (a^{p} c \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(c*(b*x**3+a)**p),x)

[Out]

Piecewise((a*p*log(a + b*x**3)/(3*b) + p*x**3*log(a + b*x**3)/3 - p*x**3/3 + x**3*log(c)/3, Ne(b, 0)), (x**3*l
og(a**p*c)/3, True))

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Giac [A]  time = 1.27323, size = 58, normalized size = 1.66 \begin{align*} -\frac{{\left (b x^{3} -{\left (b x^{3} + a\right )} \log \left (b x^{3} + a\right ) + a\right )} p -{\left (b x^{3} + a\right )} \log \left (c\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(c*(b*x^3+a)^p),x, algorithm="giac")

[Out]

-1/3*((b*x^3 - (b*x^3 + a)*log(b*x^3 + a) + a)*p - (b*x^3 + a)*log(c))/b