3.61 \(\int x^3 \log (d (b x+c x^2)^n) \, dx\)

Optimal. Leaf size=85 \[ -\frac{b^2 n x^2}{8 c^2}+\frac{b^3 n x}{4 c^3}-\frac{b^4 n \log (b+c x)}{4 c^4}+\frac{1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^3}{12 c}-\frac{n x^4}{8} \]

[Out]

(b^3*n*x)/(4*c^3) - (b^2*n*x^2)/(8*c^2) + (b*n*x^3)/(12*c) - (n*x^4)/8 - (b^4*n*Log[b + c*x])/(4*c^4) + (x^4*L
og[d*(b*x + c*x^2)^n])/4

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Rubi [A]  time = 0.0616151, antiderivative size = 85, 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} \[ -\frac{b^2 n x^2}{8 c^2}+\frac{b^3 n x}{4 c^3}-\frac{b^4 n \log (b+c x)}{4 c^4}+\frac{1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^3}{12 c}-\frac{n x^4}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[d*(b*x + c*x^2)^n],x]

[Out]

(b^3*n*x)/(4*c^3) - (b^2*n*x^2)/(8*c^2) + (b*n*x^3)/(12*c) - (n*x^4)/8 - (b^4*n*Log[b + c*x])/(4*c^4) + (x^4*L
og[d*(b*x + c*x^2)^n])/4

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^3 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{4} n \int \frac{x^3 (b+2 c x)}{b+c x} \, dx\\ &=\frac{1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{4} n \int \left (-\frac{b^3}{c^3}+\frac{b^2 x}{c^2}-\frac{b x^2}{c}+2 x^3+\frac{b^4}{c^3 (b+c x)}\right ) \, dx\\ &=\frac{b^3 n x}{4 c^3}-\frac{b^2 n x^2}{8 c^2}+\frac{b n x^3}{12 c}-\frac{n x^4}{8}-\frac{b^4 n \log (b+c x)}{4 c^4}+\frac{1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end{align*}

Mathematica [A]  time = 0.0357714, size = 74, normalized size = 0.87 \[ \frac{c n x \left (-3 b^2 c x+6 b^3+2 b c^2 x^2-3 c^3 x^3\right )-6 b^4 n \log (b+c x)+6 c^4 x^4 \log \left (d (x (b+c x))^n\right )}{24 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Log[d*(b*x + c*x^2)^n],x]

[Out]

(c*n*x*(6*b^3 - 3*b^2*c*x + 2*b*c^2*x^2 - 3*c^3*x^3) - 6*b^4*n*Log[b + c*x] + 6*c^4*x^4*Log[d*(x*(b + c*x))^n]
)/(24*c^4)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\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^3*ln(d*(c*x^2+b*x)^n),x)

[Out]

int(x^3*ln(d*(c*x^2+b*x)^n),x)

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Maxima [A]  time = 1.09299, size = 101, normalized size = 1.19 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) - \frac{1}{24} \, n{\left (\frac{6 \, b^{4} \log \left (c x + b\right )}{c^{4}} + \frac{3 \, c^{3} x^{4} - 2 \, b c^{2} x^{3} + 3 \, b^{2} c x^{2} - 6 \, b^{3} x}{c^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*log((c*x^2 + b*x)^n*d) - 1/24*n*(6*b^4*log(c*x + b)/c^4 + (3*c^3*x^4 - 2*b*c^2*x^3 + 3*b^2*c*x^2 - 6*b
^3*x)/c^3)

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Fricas [A]  time = 1.56903, size = 196, normalized size = 2.31 \begin{align*} \frac{6 \, c^{4} n x^{4} \log \left (c x^{2} + b x\right ) - 3 \, c^{4} n x^{4} + 6 \, c^{4} x^{4} \log \left (d\right ) + 2 \, b c^{3} n x^{3} - 3 \, b^{2} c^{2} n x^{2} + 6 \, b^{3} c n x - 6 \, b^{4} n \log \left (c x + b\right )}{24 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*c^4*n*x^4*log(c*x^2 + b*x) - 3*c^4*n*x^4 + 6*c^4*x^4*log(d) + 2*b*c^3*n*x^3 - 3*b^2*c^2*n*x^2 + 6*b^3*
c*n*x - 6*b^4*n*log(c*x + b))/c^4

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Sympy [A]  time = 7.43731, size = 119, normalized size = 1.4 \begin{align*} \begin{cases} - \frac{b^{4} n \log{\left (b + c x \right )}}{4 c^{4}} + \frac{b^{3} n x}{4 c^{3}} - \frac{b^{2} n x^{2}}{8 c^{2}} + \frac{b n x^{3}}{12 c} + \frac{n x^{4} \log{\left (b x + c x^{2} \right )}}{4} - \frac{n x^{4}}{8} + \frac{x^{4} \log{\left (d \right )}}{4} & \text{for}\: c \neq 0 \\\frac{n x^{4} \log{\left (b \right )}}{4} + \frac{n x^{4} \log{\left (x \right )}}{4} - \frac{n x^{4}}{16} + \frac{x^{4} \log{\left (d \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*ln(d*(c*x**2+b*x)**n),x)

[Out]

Piecewise((-b**4*n*log(b + c*x)/(4*c**4) + b**3*n*x/(4*c**3) - b**2*n*x**2/(8*c**2) + b*n*x**3/(12*c) + n*x**4
*log(b*x + c*x**2)/4 - n*x**4/8 + x**4*log(d)/4, Ne(c, 0)), (n*x**4*log(b)/4 + n*x**4*log(x)/4 - n*x**4/16 + x
**4*log(d)/4, True))

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Giac [A]  time = 1.22759, size = 101, normalized size = 1.19 \begin{align*} \frac{1}{4} \, n x^{4} \log \left (c x^{2} + b x\right ) - \frac{1}{8} \,{\left (n - 2 \, \log \left (d\right )\right )} x^{4} + \frac{b n x^{3}}{12 \, c} - \frac{b^{2} n x^{2}}{8 \, c^{2}} + \frac{b^{3} n x}{4 \, c^{3}} - \frac{b^{4} n \log \left (c x + b\right )}{4 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*n*x^4*log(c*x^2 + b*x) - 1/8*(n - 2*log(d))*x^4 + 1/12*b*n*x^3/c - 1/8*b^2*n*x^2/c^2 + 1/4*b^3*n*x/c^3 - 1
/4*b^4*n*log(c*x + b)/c^4