3.1.64 \(\int \log (d (b x+c x^2)^n) \, dx\) [64]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {2603, 45} \begin {gather*} x \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n \log (b+c x)}{c}-2 n x \end {gather*}

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 45

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])

Rule 2603

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]

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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.94 \begin {gather*} -2 n x+\frac {b n \log (b+c x)}{c}+x \log \left (d (x (b+c x))^n\right ) \end {gather*}

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 = 0.02, size = 37, normalized size = 1.12

method	result	size
default	\(x \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )-n \left (2 x -\frac {b \ln \left (c x +b \right )}{c}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 36, normalized size = 1.09 \begin {gather*} -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 {gather*}

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 = 0.36, size = 38, normalized size = 1.15 \begin {gather*} \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 {gather*}

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 = 0.39, size = 44, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {b n \log {\left (b + c x \right )}}{c} - 2 n x + x \log {\left (d \left (b x + c x^{2}\right )^{n} \right )} & \text {for}\: c \neq 0 \\- n x + x \log {\left (d \left (b x\right )^{n} \right )} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 5.51, size = 37, normalized size = 1.12 \begin {gather*} 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 {gather*}

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

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Mupad [B]
time = 0.36, size = 33, normalized size = 1.00 \begin {gather*} x\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )-2\,n\,x+\frac {b\,n\,\ln \left (b+c\,x\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(d*(b*x + c*x^2)^n),x)

[Out]

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

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