3.69 \(\int \log ^3(a+b x+c x) \, dx\)

Optimal. Leaf size=73 \[ \frac{(a+x (b+c)) \log ^3(a+x (b+c))}{b+c}-\frac{3 (a+x (b+c)) \log ^2(a+x (b+c))}{b+c}+\frac{6 (a+x (b+c)) \log (a+x (b+c))}{b+c}-6 x \]

[Out]

-6*x + (6*(a + (b + c)*x)*Log[a + (b + c)*x])/(b + c) - (3*(a + (b + c)*x)*Log[a + (b + c)*x]^2)/(b + c) + ((a
+ (b + c)*x)*Log[a + (b + c)*x]^3)/(b + c)

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Rubi [A]  time = 0.0317987, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2444, 2389, 2296, 2295} \[ \frac{(a+x (b+c)) \log ^3(a+x (b+c))}{b+c}-\frac{3 (a+x (b+c)) \log ^2(a+x (b+c))}{b+c}+\frac{6 (a+x (b+c)) \log (a+x (b+c))}{b+c}-6 x \]

Antiderivative was successfully verified.

[In]

Int[Log[a + b*x + c*x]^3,x]

[Out]

-6*x + (6*(a + (b + c)*x)*Log[a + (b + c)*x])/(b + c) - (3*(a + (b + c)*x)*Log[a + (b + c)*x]^2)/(b + c) + ((a
+ (b + c)*x)*Log[a + (b + c)*x]^3)/(b + c)

Rule 2444

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]])

Rule 2389

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 2296

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 2295

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

Rubi steps

\begin{align*} \int \log ^3(a+b x+c x) \, dx &=\int \log ^3(a+(b+c) x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \log ^3(x) \, dx,x,a+(b+c) x\right )}{b+c}\\ &=\frac{(a+(b+c) x) \log ^3(a+(b+c) x)}{b+c}-\frac{3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,a+(b+c) x\right )}{b+c}\\ &=-\frac{3 (a+(b+c) x) \log ^2(a+(b+c) x)}{b+c}+\frac{(a+(b+c) x) \log ^3(a+(b+c) x)}{b+c}+\frac{6 \operatorname{Subst}(\int \log (x) \, dx,x,a+(b+c) x)}{b+c}\\ &=-6 x+\frac{6 (a+(b+c) x) \log (a+(b+c) x)}{b+c}-\frac{3 (a+(b+c) x) \log ^2(a+(b+c) x)}{b+c}+\frac{(a+(b+c) x) \log ^3(a+(b+c) x)}{b+c}\\ \end{align*}

Mathematica [A]  time = 0.0084843, size = 67, normalized size = 0.92 \[ \frac{(a+x (b+c)) \log ^3(a+x (b+c))-3 (a+x (b+c)) \log ^2(a+x (b+c))+6 (a+x (b+c)) \log (a+x (b+c))-6 x (b+c)}{b+c} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*x + c*x]^3,x]

[Out]

(-6*(b + c)*x + 6*(a + (b + c)*x)*Log[a + (b + c)*x] - 3*(a + (b + c)*x)*Log[a + (b + c)*x]^2 + (a + (b + c)*x
)*Log[a + (b + c)*x]^3)/(b + c)

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Maple [B]  time = 0.059, size = 187, normalized size = 2.6 \begin{align*}{\frac{ \left ( \ln \left ( a+ \left ( b+c \right ) x \right ) \right ) ^{3}xb}{b+c}}+{\frac{ \left ( \ln \left ( a+ \left ( b+c \right ) x \right ) \right ) ^{3}xc}{b+c}}+{\frac{ \left ( \ln \left ( a+ \left ( b+c \right ) x \right ) \right ) ^{3}a}{b+c}}-3\,{\frac{ \left ( \ln \left ( a+ \left ( b+c \right ) x \right ) \right ) ^{2}xb}{b+c}}-3\,{\frac{ \left ( \ln \left ( a+ \left ( b+c \right ) x \right ) \right ) ^{2}xc}{b+c}}-3\,{\frac{ \left ( \ln \left ( a+ \left ( b+c \right ) x \right ) \right ) ^{2}a}{b+c}}+6\,{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) xb}{b+c}}+6\,{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) xc}{b+c}}+6\,{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) a}{b+c}}-6\,{\frac{bx}{b+c}}-6\,{\frac{cx}{b+c}}-6\,{\frac{a}{b+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(b*x+c*x+a)^3,x)

[Out]

1/(b+c)*ln(a+(b+c)*x)^3*x*b+1/(b+c)*ln(a+(b+c)*x)^3*x*c+1/(b+c)*ln(a+(b+c)*x)^3*a-3/(b+c)*ln(a+(b+c)*x)^2*x*b-
3/(b+c)*ln(a+(b+c)*x)^2*x*c-3/(b+c)*ln(a+(b+c)*x)^2*a+6/(b+c)*ln(a+(b+c)*x)*x*b+6/(b+c)*ln(a+(b+c)*x)*x*c+6/(b
+c)*ln(a+(b+c)*x)*a-6/(b+c)*b*x-6/(b+c)*c*x-6/(b+c)*a

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Maxima [A]  time = 1.23922, size = 69, normalized size = 0.95 \begin{align*} \frac{{\left (\log \left (b x + c x + a\right )^{3} - 3 \, \log \left (b x + c x + a\right )^{2} + 6 \, \log \left (b x + c x + a\right ) - 6\right )}{\left (b x + c x + a\right )}}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(b*x+c*x+a)^3,x, algorithm="maxima")

[Out]

(log(b*x + c*x + a)^3 - 3*log(b*x + c*x + a)^2 + 6*log(b*x + c*x + a) - 6)*(b*x + c*x + a)/(b + c)

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Fricas [A]  time = 2.04405, size = 192, normalized size = 2.63 \begin{align*} \frac{{\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )^{3} - 3 \,{\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )^{2} - 6 \,{\left (b + c\right )} x + 6 \,{\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(b*x+c*x+a)^3,x, algorithm="fricas")

[Out]

(((b + c)*x + a)*log((b + c)*x + a)^3 - 3*((b + c)*x + a)*log((b + c)*x + a)^2 - 6*(b + c)*x + 6*((b + c)*x +
a)*log((b + c)*x + a))/(b + c)

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Sympy [A]  time = 0.623432, size = 95, normalized size = 1.3 \begin{align*} 6 x \log{\left (a + b x + c x \right )} + \left (- 6 b - 6 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 (- 3 a - 3 b x - 3 c x\right ) \log{\left (a + b x + c x \right )}^{2}}{b + c} + \frac{\left (a + b x + c x\right ) \log{\left (a + b x + c x \right )}^{3}}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(b*x+c*x+a)**3,x)

[Out]

6*x*log(a + b*x + c*x) + (-6*b - 6*c)*(-a*log(a + x*(b + c))/(b + c)**2 + x/(b + c)) + (-3*a - 3*b*x - 3*c*x)*
log(a + b*x + c*x)**2/(b + c) + (a + b*x + c*x)*log(a + b*x + c*x)**3/(b + c)

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Giac [A]  time = 1.24107, size = 123, normalized size = 1.68 \begin{align*} \frac{{\left (b x + c x + a\right )} \log \left (b x + c x + a\right )^{3}}{b + c} - \frac{3 \,{\left (b x + c x + a\right )} \log \left (b x + c x + a\right )^{2}}{b + c} + \frac{6 \,{\left (b x + c x + a\right )} \log \left (b x + c x + a\right )}{b + c} - \frac{6 \,{\left (b x + c x + a\right )}}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(b*x+c*x+a)^3,x, algorithm="giac")

[Out]

(b*x + c*x + a)*log(b*x + c*x + a)^3/(b + c) - 3*(b*x + c*x + a)*log(b*x + c*x + a)^2/(b + c) + 6*(b*x + c*x +
a)*log(b*x + c*x + a)/(b + c) - 6*(b*x + c*x + a)/(b + c)