3.1.68 \(\int \log ^2(a+b x+c x) \, dx\) [68]

Optimal. Leaf size=49 \[ 2 x-\frac {2 (a+(b+c) x) \log (a+(b+c) x)}{b+c}+\frac {(a+(b+c) x) \log ^2(a+(b+c) x)}{b+c} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2494, 2436, 2333, 2332} \begin {gather*} \frac {(a+x (b+c)) \log ^2(a+x (b+c))}{b+c}-\frac {2 (a+x (b+c)) \log (a+x (b+c))}{b+c}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[a + b*x + c*x]^2,x]

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2494

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 48, normalized size = 0.98 \begin {gather*} \frac {2 (b+c) x-2 (a+(b+c) x) \log (a+(b+c) x)+(a+(b+c) x) \log ^2(a+(b+c) x)}{b+c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*x + c*x]^2,x]

[Out]

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

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Maple [A]
time = 0.14, size = 52, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {\ln \left (a +\left (b +c \right ) x \right )^{2} \left (a +\left (b +c \right ) x \right )-2 \left (a +\left (b +c \right ) x \right ) \ln \left (a +\left (b +c \right ) x \right )+2 a +2 \left (b +c \right ) x}{b +c}\)	\(52\)
default	\(\frac {\ln \left (a +\left (b +c \right ) x \right )^{2} \left (a +\left (b +c \right ) x \right )-2 \left (a +\left (b +c \right ) x \right ) \ln \left (a +\left (b +c \right ) x \right )+2 a +2 \left (b +c \right ) x}{b +c}\)	\(52\)
norman	\(x \ln \left (b x +c x +a \right )^{2}+\frac {a \ln \left (b x +c x +a \right )^{2}}{b +c}+2 x -2 x \ln \left (b x +c x +a \right )-\frac {2 a \ln \left (b x +c x +a \right )}{b +c}\)	\(65\)
risch	\(\frac {\ln \left (b x +c x +a \right )^{2} \left (b x +c x +a \right )}{b +c}-2 x \ln \left (b x +c x +a \right )-\frac {2 a \ln \left (a +\left (b +c \right ) x \right )}{b +c}+\frac {2 b x}{b +c}+\frac {2 c x}{b +c}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(b+c)*(ln(a+(b+c)*x)^2*(a+(b+c)*x)-2*(a+(b+c)*x)*ln(a+(b+c)*x)+2*a+2*(b+c)*x)

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Maxima [A]
time = 0.29, size = 38, normalized size = 0.78 \begin {gather*} \frac {{\left (b x + c x + a\right )} {\left (\log \left (b x + c x + a\right )^{2} - 2 \, \log \left (b x + c x + a\right ) + 2\right )}}{b + c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 48, normalized size = 0.98 \begin {gather*} \frac {{\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )^{2} + 2 \, {\left (b + c\right )} x - 2 \, {\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )}{b + c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.12, size = 63, normalized size = 1.29 \begin {gather*} - 2 x \log {\left (a + b x + c x \right )} + \left (2 b + 2 c\right ) \left (- \frac {a \log {\left (a + x \left (b + c\right ) \right )}}{\left (b + c\right )^{2}} + \frac {x}{b + c}\right ) + \frac {\left (a + b x + c x\right ) \log {\left (a + b x + c x \right )}^{2}}{b + c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(b*x+c*x+a)**2,x)

[Out]

-2*x*log(a + b*x + c*x) + (2*b + 2*c)*(-a*log(a + x*(b + c))/(b + c)**2 + x/(b + c)) + (a + b*x + c*x)*log(a +
b*x + c*x)**2/(b + c)

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Giac [A]
time = 4.56, size = 65, normalized size = 1.33 \begin {gather*} \frac {{\left (b x + c x + a\right )} \log \left (b x + c x + a\right )^{2}}{b + c} - \frac {2 \, {\left (b x + c x + a\right )} \log \left (b x + c x + a\right )}{b + c} + \frac {2 \, {\left (b x + c x + a\right )}}{b + c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.36, size = 94, normalized size = 1.92 \begin {gather*} \frac {2\,b\,x+2\,c\,x-2\,a\,\ln \left (a+b\,x+c\,x\right )+a\,{\ln \left (a+b\,x+c\,x\right )}^2+b\,x\,{\ln \left (a+b\,x+c\,x\right )}^2+c\,x\,{\ln \left (a+b\,x+c\,x\right )}^2-2\,b\,x\,\ln \left (a+b\,x+c\,x\right )-2\,c\,x\,\ln \left (a+b\,x+c\,x\right )}{b+c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a + b*x + c*x)^2,x)

[Out]

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

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