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

Optimal. Leaf size=57 \[ -\frac{b^2 n \log (b+c x)}{2 c^2}+\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x}{2 c}-\frac{n x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.040766, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2525, 77} \[ -\frac{b^2 n \log (b+c x)}{2 c^2}+\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x}{2 c}-\frac{n x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[x*Log[d*(b*x + c*x^2)^n],x]

[Out]

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

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

Mathematica [A]  time = 0.0202679, size = 49, normalized size = 0.86 \[ \frac{1}{2} x^2 \log \left (d (x (b+c x))^n\right )-\frac{1}{2} n \left (\frac{b^2 \log (b+c x)}{c^2}-\frac{b x}{c}+x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Log[d*(b*x + c*x^2)^n],x]

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int x\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*ln(d*(c*x^2+b*x)^n),x)

[Out]

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

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Maxima [A]  time = 1.01231, size = 69, normalized size = 1.21 \begin{align*} \frac{1}{2} \, x^{2} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) - \frac{1}{2} \, n{\left (\frac{b^{2} \log \left (c x + b\right )}{c^{2}} + \frac{c x^{2} - b x}{c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55161, size = 132, normalized size = 2.32 \begin{align*} \frac{c^{2} n x^{2} \log \left (c x^{2} + b x\right ) - c^{2} n x^{2} + c^{2} x^{2} \log \left (d\right ) + b c n x - b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.34266, size = 92, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{b^{2} n \log{\left (b + c x \right )}}{2 c^{2}} + \frac{b n x}{2 c} + \frac{n x^{2} \log{\left (b x + c x^{2} \right )}}{2} - \frac{n x^{2}}{2} + \frac{x^{2} \log{\left (d \right )}}{2} & \text{for}\: c \neq 0 \\\frac{n x^{2} \log{\left (b \right )}}{2} + \frac{n x^{2} \log{\left (x \right )}}{2} - \frac{n x^{2}}{4} + \frac{x^{2} \log{\left (d \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21367, size = 69, normalized size = 1.21 \begin{align*} \frac{1}{2} \, n x^{2} \log \left (c x^{2} + b x\right ) - \frac{1}{2} \,{\left (n - \log \left (d\right )\right )} x^{2} + \frac{b n x}{2 \, c} - \frac{b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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