### 3.64 $$\int \log (d (b x+c x^2)^n) \, dx$$

Optimal. Leaf size=33 $x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n \log (b+c x)}{c}-2 n x$

[Out]

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

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Rubi [A]  time = 0.0163603, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {2523, 43} $x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n \log (b+c x)}{c}-2 n x$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[d*(b*x + c*x^2)^n],x]

[Out]

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

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.006888, size = 31, normalized size = 0.94 $x \log \left (d (x (b+c x))^n\right )+\frac{b n \log (b+c x)}{c}-2 n x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[d*(b*x + c*x^2)^n],x]

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 1. \begin{align*} -2\,nx+{\frac{bn\ln \left ( cx+b \right ) }{c}}+x\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*n*x+b*n*ln(c*x+b)/c+x*ln(d*(c*x^2+b*x)^n)

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Maxima [A]  time = 1.15723, size = 49, normalized size = 1.48 \begin{align*} -n{\left (2 \, x - \frac{b \log \left (c x + b\right )}{c}\right )} + x \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52262, size = 95, normalized size = 2.88 \begin{align*} \frac{c n x \log \left (c x^{2} + b x\right ) - 2 \, c n x + b n \log \left (c x + b\right ) + c x \log \left (d\right )}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.21605, size = 56, normalized size = 1.7 \begin{align*} \begin{cases} \frac{b n \log{\left (b + c x \right )}}{c} + n x \log{\left (b x + c x^{2} \right )} - 2 n x + x \log{\left (d \right )} & \text{for}\: c \neq 0 \\n x \log{\left (b \right )} + n x \log{\left (x \right )} - n x + x \log{\left (d \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22886, size = 50, normalized size = 1.52 \begin{align*} n x \log \left (c x^{2} + b x\right ) -{\left (2 \, n - \log \left (d\right )\right )} x + \frac{b n \log \left (c x + b\right )}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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