3.62 \(\int x^2 \log (d (b x+c x^2)^n) \, dx\)

Optimal. Leaf size=71 \[ -\frac{b^2 n x}{3 c^2}+\frac{b^3 n \log (b+c x)}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9} \]

[Out]

-(b^2*n*x)/(3*c^2) + (b*n*x^2)/(6*c) - (2*n*x^3)/9 + (b^3*n*Log[b + c*x])/(3*c^3) + (x^3*Log[d*(b*x + c*x^2)^n
])/3

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Rubi [A]  time = 0.0526, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2525, 77} \[ -\frac{b^2 n x}{3 c^2}+\frac{b^3 n \log (b+c x)}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[d*(b*x + c*x^2)^n],x]

[Out]

-(b^2*n*x)/(3*c^2) + (b*n*x^2)/(6*c) - (2*n*x^3)/9 + (b^3*n*Log[b + c*x])/(3*c^3) + (x^3*Log[d*(b*x + c*x^2)^n
])/3

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
 1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^2 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{3} n \int \frac{x^2 (b+2 c x)}{b+c x} \, dx\\ &=\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{3} n \int \left (\frac{b^2}{c^2}-\frac{b x}{c}+2 x^2-\frac{b^3}{c^2 (b+c x)}\right ) \, dx\\ &=-\frac{b^2 n x}{3 c^2}+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9}+\frac{b^3 n \log (b+c x)}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end{align*}

Mathematica [A]  time = 0.0288719, size = 63, normalized size = 0.89 \[ \frac{c n x \left (-6 b^2+3 b c x-4 c^2 x^2\right )+6 b^3 n \log (b+c x)+6 c^3 x^3 \log \left (d (x (b+c x))^n\right )}{18 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[d*(b*x + c*x^2)^n],x]

[Out]

(c*n*x*(-6*b^2 + 3*b*c*x - 4*c^2*x^2) + 6*b^3*n*Log[b + c*x] + 6*c^3*x^3*Log[d*(x*(b + c*x))^n])/(18*c^3)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08318, size = 88, normalized size = 1.24 \begin{align*} \frac{1}{3} \, x^{3} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + \frac{1}{18} \, n{\left (\frac{6 \, b^{3} \log \left (c x + b\right )}{c^{3}} - \frac{4 \, c^{2} x^{3} - 3 \, b c x^{2} + 6 \, b^{2} x}{c^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(d*(c*x^2+b*x)^n),x, algorithm="maxima")

[Out]

1/3*x^3*log((c*x^2 + b*x)^n*d) + 1/18*n*(6*b^3*log(c*x + b)/c^3 - (4*c^2*x^3 - 3*b*c*x^2 + 6*b^2*x)/c^2)

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Fricas [A]  time = 1.50534, size = 171, normalized size = 2.41 \begin{align*} \frac{6 \, c^{3} n x^{3} \log \left (c x^{2} + b x\right ) - 4 \, c^{3} n x^{3} + 6 \, c^{3} x^{3} \log \left (d\right ) + 3 \, b c^{2} n x^{2} - 6 \, b^{2} c n x + 6 \, b^{3} n \log \left (c x + b\right )}{18 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(d*(c*x^2+b*x)^n),x, algorithm="fricas")

[Out]

1/18*(6*c^3*n*x^3*log(c*x^2 + b*x) - 4*c^3*n*x^3 + 6*c^3*x^3*log(d) + 3*b*c^2*n*x^2 - 6*b^2*c*n*x + 6*b^3*n*lo
g(c*x + b))/c^3

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Sympy [A]  time = 4.21884, size = 107, normalized size = 1.51 \begin{align*} \begin{cases} \frac{b^{3} n \log{\left (b + c x \right )}}{3 c^{3}} - \frac{b^{2} n x}{3 c^{2}} + \frac{b n x^{2}}{6 c} + \frac{n x^{3} \log{\left (b x + c x^{2} \right )}}{3} - \frac{2 n x^{3}}{9} + \frac{x^{3} \log{\left (d \right )}}{3} & \text{for}\: c \neq 0 \\\frac{n x^{3} \log{\left (b \right )}}{3} + \frac{n x^{3} \log{\left (x \right )}}{3} - \frac{n x^{3}}{9} + \frac{x^{3} \log{\left (d \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(d*(c*x**2+b*x)**n),x)

[Out]

Piecewise((b**3*n*log(b + c*x)/(3*c**3) - b**2*n*x/(3*c**2) + b*n*x**2/(6*c) + n*x**3*log(b*x + c*x**2)/3 - 2*
n*x**3/9 + x**3*log(d)/3, Ne(c, 0)), (n*x**3*log(b)/3 + n*x**3*log(x)/3 - n*x**3/9 + x**3*log(d)/3, True))

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Giac [A]  time = 1.1419, size = 88, normalized size = 1.24 \begin{align*} \frac{1}{3} \, n x^{3} \log \left (c x^{2} + b x\right ) - \frac{1}{9} \,{\left (2 \, n - 3 \, \log \left (d\right )\right )} x^{3} + \frac{b n x^{2}}{6 \, c} - \frac{b^{2} n x}{3 \, c^{2}} + \frac{b^{3} n \log \left (c x + b\right )}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(d*(c*x^2+b*x)^n),x, algorithm="giac")

[Out]

1/3*n*x^3*log(c*x^2 + b*x) - 1/9*(2*n - 3*log(d))*x^3 + 1/6*b*n*x^2/c - 1/3*b^2*n*x/c^2 + 1/3*b^3*n*log(c*x +
b)/c^3