3.60 \(\int x^4 \log (d (b x+c x^2)^n) \, dx\)

Optimal. Leaf size=99 \[ \frac{b^3 n x^2}{10 c^3}-\frac{b^2 n x^3}{15 c^2}-\frac{b^4 n x}{5 c^4}+\frac{b^5 n \log (b+c x)}{5 c^5}+\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^4}{20 c}-\frac{2 n x^5}{25} \]

[Out]

-(b^4*n*x)/(5*c^4) + (b^3*n*x^2)/(10*c^3) - (b^2*n*x^3)/(15*c^2) + (b*n*x^4)/(20*c) - (2*n*x^5)/25 + (b^5*n*Lo
g[b + c*x])/(5*c^5) + (x^5*Log[d*(b*x + c*x^2)^n])/5

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Rubi [A]  time = 0.0731365, antiderivative size = 99, 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} \[ \frac{b^3 n x^2}{10 c^3}-\frac{b^2 n x^3}{15 c^2}-\frac{b^4 n x}{5 c^4}+\frac{b^5 n \log (b+c x)}{5 c^5}+\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^4}{20 c}-\frac{2 n x^5}{25} \]

Antiderivative was successfully verified.

[In]

Int[x^4*Log[d*(b*x + c*x^2)^n],x]

[Out]

-(b^4*n*x)/(5*c^4) + (b^3*n*x^2)/(10*c^3) - (b^2*n*x^3)/(15*c^2) + (b*n*x^4)/(20*c) - (2*n*x^5)/25 + (b^5*n*Lo
g[b + c*x])/(5*c^5) + (x^5*Log[d*(b*x + c*x^2)^n])/5

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^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{5} n \int \frac{x^4 (b+2 c x)}{b+c x} \, dx\\ &=\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{5} n \int \left (\frac{b^4}{c^4}-\frac{b^3 x}{c^3}+\frac{b^2 x^2}{c^2}-\frac{b x^3}{c}+2 x^4-\frac{b^5}{c^4 (b+c x)}\right ) \, dx\\ &=-\frac{b^4 n x}{5 c^4}+\frac{b^3 n x^2}{10 c^3}-\frac{b^2 n x^3}{15 c^2}+\frac{b n x^4}{20 c}-\frac{2 n x^5}{25}+\frac{b^5 n \log (b+c x)}{5 c^5}+\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end{align*}

Mathematica [A]  time = 0.0484504, size = 85, normalized size = 0.86 \[ \frac{c n x \left (-20 b^2 c^2 x^2+30 b^3 c x-60 b^4+15 b c^3 x^3-24 c^4 x^4\right )+60 b^5 n \log (b+c x)+60 c^5 x^5 \log \left (d (x (b+c x))^n\right )}{300 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Log[d*(b*x + c*x^2)^n],x]

[Out]

(c*n*x*(-60*b^4 + 30*b^3*c*x - 20*b^2*c^2*x^2 + 15*b*c^3*x^3 - 24*c^4*x^4) + 60*b^5*n*Log[b + c*x] + 60*c^5*x^
5*Log[d*(x*(b + c*x))^n])/(300*c^5)

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\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^4*ln(d*(c*x^2+b*x)^n),x)

[Out]

int(x^4*ln(d*(c*x^2+b*x)^n),x)

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Maxima [A]  time = 1.04288, size = 117, normalized size = 1.18 \begin{align*} \frac{1}{5} \, x^{5} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + \frac{1}{300} \, n{\left (\frac{60 \, b^{5} \log \left (c x + b\right )}{c^{5}} - \frac{24 \, c^{4} x^{5} - 15 \, b c^{3} x^{4} + 20 \, b^{2} c^{2} x^{3} - 30 \, b^{3} c x^{2} + 60 \, b^{4} x}{c^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(d*(c*x^2+b*x)^n),x, algorithm="maxima")

[Out]

1/5*x^5*log((c*x^2 + b*x)^n*d) + 1/300*n*(60*b^5*log(c*x + b)/c^5 - (24*c^4*x^5 - 15*b*c^3*x^4 + 20*b^2*c^2*x^
3 - 30*b^3*c*x^2 + 60*b^4*x)/c^4)

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Fricas [A]  time = 1.6118, size = 232, normalized size = 2.34 \begin{align*} \frac{60 \, c^{5} n x^{5} \log \left (c x^{2} + b x\right ) - 24 \, c^{5} n x^{5} + 60 \, c^{5} x^{5} \log \left (d\right ) + 15 \, b c^{4} n x^{4} - 20 \, b^{2} c^{3} n x^{3} + 30 \, b^{3} c^{2} n x^{2} - 60 \, b^{4} c n x + 60 \, b^{5} n \log \left (c x + b\right )}{300 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(d*(c*x^2+b*x)^n),x, algorithm="fricas")

[Out]

1/300*(60*c^5*n*x^5*log(c*x^2 + b*x) - 24*c^5*n*x^5 + 60*c^5*x^5*log(d) + 15*b*c^4*n*x^4 - 20*b^2*c^3*n*x^3 +
30*b^3*c^2*n*x^2 - 60*b^4*c*n*x + 60*b^5*n*log(c*x + b))/c^5

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Sympy [A]  time = 14.2238, size = 134, normalized size = 1.35 \begin{align*} \begin{cases} \frac{b^{5} n \log{\left (b + c x \right )}}{5 c^{5}} - \frac{b^{4} n x}{5 c^{4}} + \frac{b^{3} n x^{2}}{10 c^{3}} - \frac{b^{2} n x^{3}}{15 c^{2}} + \frac{b n x^{4}}{20 c} + \frac{n x^{5} \log{\left (b x + c x^{2} \right )}}{5} - \frac{2 n x^{5}}{25} + \frac{x^{5} \log{\left (d \right )}}{5} & \text{for}\: c \neq 0 \\\frac{n x^{5} \log{\left (b \right )}}{5} + \frac{n x^{5} \log{\left (x \right )}}{5} - \frac{n x^{5}}{25} + \frac{x^{5} \log{\left (d \right )}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*ln(d*(c*x**2+b*x)**n),x)

[Out]

Piecewise((b**5*n*log(b + c*x)/(5*c**5) - b**4*n*x/(5*c**4) + b**3*n*x**2/(10*c**3) - b**2*n*x**3/(15*c**2) +
b*n*x**4/(20*c) + n*x**5*log(b*x + c*x**2)/5 - 2*n*x**5/25 + x**5*log(d)/5, Ne(c, 0)), (n*x**5*log(b)/5 + n*x*
*5*log(x)/5 - n*x**5/25 + x**5*log(d)/5, True))

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Giac [A]  time = 1.30019, size = 120, normalized size = 1.21 \begin{align*} \frac{1}{5} \, n x^{5} \log \left (c x^{2} + b x\right ) - \frac{1}{25} \,{\left (2 \, n - 5 \, \log \left (d\right )\right )} x^{5} + \frac{b n x^{4}}{20 \, c} - \frac{b^{2} n x^{3}}{15 \, c^{2}} + \frac{b^{3} n x^{2}}{10 \, c^{3}} - \frac{b^{4} n x}{5 \, c^{4}} + \frac{b^{5} n \log \left (c x + b\right )}{5 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(d*(c*x^2+b*x)^n),x, algorithm="giac")

[Out]

1/5*n*x^5*log(c*x^2 + b*x) - 1/25*(2*n - 5*log(d))*x^5 + 1/20*b*n*x^4/c - 1/15*b^2*n*x^3/c^2 + 1/10*b^3*n*x^2/
c^3 - 1/5*b^4*n*x/c^4 + 1/5*b^5*n*log(c*x + b)/c^5