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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 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]))))

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]

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*}

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Mathematica [A]  time = 0.03, size = 63, normalized size = 0.89 \[ \frac {6 b^3 n \log (b+c x)+c n x \left (-6 b^2+3 b c x-4 c^2 x^2\right )+6 c^3 x^3 \log \left (d (x (b+c x))^n\right )}{18 c^3} \]

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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fricas [A]  time = 0.46, size = 74, normalized size = 1.04 \[ \frac {6 \, c^{3} n x^{3} \log \left (c x^{2} + b x\right ) - 4 \, c^{3} n x^{3} + 6 \, c^{3} x^{3} \log \relax (d) + 3 \, b c^{2} n x^{2} - 6 \, b^{2} c n x + 6 \, b^{3} n \log \left (c x + b\right )}{18 \, c^{3}} \]

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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giac [A]  time = 0.17, size = 65, normalized size = 0.92 \[ \frac {1}{3} \, n x^{3} \log \left (c x^{2} + b x\right ) - \frac {1}{9} \, {\left (2 \, n - 3 \, \log \relax (d)\right )} x^{3} + \frac {b n x^{2}}{6 \, c} - \frac {b^{2} n x}{3 \, c^{2}} + \frac {b^{3} n \log \left (c x + b\right )}{3 \, c^{3}} \]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int x^{2} \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )\, dx \]

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 = 0.44, size = 65, normalized size = 0.92 \[ \frac {1}{3} \, x^{3} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + \frac {1}{18} \, n {\left (\frac {6 \, b^{3} \log \left (c x + b\right )}{c^{3}} - \frac {4 \, c^{2} x^{3} - 3 \, b c x^{2} + 6 \, b^{2} x}{c^{2}}\right )} \]

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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mupad [B]  time = 0.32, size = 61, normalized size = 0.86 \[ \frac {x^3\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{3}-\frac {2\,n\,x^3}{9}+\frac {b^3\,n\,\ln \left (b+c\,x\right )}{3\,c^3}+\frac {b\,n\,x^2}{6\,c}-\frac {b^2\,n\,x}{3\,c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*log(d*(b*x + c*x^2)^n),x)

[Out]

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

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sympy [A]  time = 5.35, size = 107, normalized size = 1.51 \[ \begin {cases} \frac {b^{3} n \log {\left (b + c x \right )}}{3 c^{3}} - \frac {b^{2} n x}{3 c^{2}} + \frac {b n x^{2}}{6 c} + \frac {n x^{3} \log {\left (b x + c x^{2} \right )}}{3} - \frac {2 n x^{3}}{9} + \frac {x^{3} \log {\relax (d )}}{3} & \text {for}\: c \neq 0 \\\frac {n x^{3} \log {\relax (b )}}{3} + \frac {n x^{3} \log {\relax (x )}}{3} - \frac {n x^{3}}{9} + \frac {x^{3} \log {\relax (d )}}{3} & \text {otherwise} \end {cases} \]

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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