\(\int (a+b \log (c (d (e+f x)^m)^n)) \, dx\) [407]

Optimal result

Integrand size = 18, antiderivative size = 34 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=a x-b m n x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f} \]

[Out]

a*x-b*m*n*x+b*(f*x+e)*ln(c*(d*(f*x+e)^m)^n)/f

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2436, 2332, 2495} \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=a x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]

[In]

Int[a + b*Log[c*(d*(e + f*x)^m)^n],x]

[Out]

a*x - b*m*n*x + (b*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f

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 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 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps \begin{align*} \text {integral}& = a x+b \int \log \left (c \left (d (e+f x)^m\right )^n\right ) \, dx \\ & = a x+b \text {Subst}\left (\int \log \left (c d^n (e+f x)^{m n}\right ) \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = a x+b \text {Subst}\left (\frac {\text {Subst}\left (\int \log \left (c d^n x^{m n}\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = a x-b m n x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=a x-b m n x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f} \]

[In]

Integrate[a + b*Log[c*(d*(e + f*x)^m)^n],x]

[Out]

a*x - b*m*n*x + (b*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24

method	result	size
default	\(a x +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) x -b m n x +\frac {b n m e \ln \left (f x +e \right )}{f}\)	\(42\)
parts	\(a x +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) x -b m n x +\frac {b n m e \ln \left (f x +e \right )}{f}\)	\(42\)
parallelrisch	\(\frac {b \left (2 \ln \left (f x +e \right ) e^{2} m n -x e f m n +x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) e f -\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) e^{2}\right )}{e f}+a x\)	\(71\)

[In]

int(a+b*ln(c*(d*(f*x+e)^m)^n),x,method=_RETURNVERBOSE)

[Out]

a*x+b*ln(c*(d*(f*x+e)^m)^n)*x-b*m*n*x+b*n*m/f*e*ln(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=\frac {b f n x \log \left (d\right ) + b f x \log \left (c\right ) - {\left (b f m n - a f\right )} x + {\left (b f m n x + b e m n\right )} \log \left (f x + e\right )}{f} \]

[In]

integrate(a+b*log(c*(d*(f*x+e)^m)^n),x, algorithm="fricas")

[Out]

(b*f*n*x*log(d) + b*f*x*log(c) - (b*f*m*n - a*f)*x + (b*f*m*n*x + b*e*m*n)*log(f*x + e))/f

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=a x + b \left (\begin {cases} \frac {e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - m n x + x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{m}\right )^{n} \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate(a+b*ln(c*(d*(f*x+e)**m)**n),x)

[Out]

a*x + b*Piecewise((e*log(c*(d*(e + f*x)**m)**n)/f - m*n*x + x*log(c*(d*(e + f*x)**m)**n), Ne(f, 0)), (x*log(c*
(d*e**m)**n), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=-b f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a x \]

[In]

integrate(a+b*log(c*(d*(f*x+e)^m)^n),x, algorithm="maxima")

[Out]

-b*f*m*n*(x/f - e*log(f*x + e)/f^2) + b*x*log(((f*x + e)^m*d)^n*c) + a*x

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx={\left (\frac {{\left (f x + e\right )} m n \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} m n}{f} + \frac {{\left (f x + e\right )} n \log \left (d\right )}{f} + \frac {{\left (f x + e\right )} \log \left (c\right )}{f}\right )} b + a x \]

[In]

integrate(a+b*log(c*(d*(f*x+e)^m)^n),x, algorithm="giac")

[Out]

((f*x + e)*m*n*log(f*x + e)/f - (f*x + e)*m*n/f + (f*x + e)*n*log(d)/f + (f*x + e)*log(c)/f)*b + a*x

Mupad [B] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx=x\,\left (a-b\,m\,n\right )+b\,x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )+\frac {b\,e\,m\,n\,\ln \left (e+f\,x\right )}{f} \]

[In]

int(a + b*log(c*(d*(e + f*x)^m)^n),x)

[Out]

x*(a - b*m*n) + b*x*log(c*(d*(e + f*x)^m)^n) + (b*e*m*n*log(e + f*x))/f