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

Optimal. Leaf size=34 \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]

[Out]

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

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Rubi [A]  time = 0.0315224, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2389, 2295, 2445} \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]

Antiderivative was successfully verified.

[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 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 2295

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

Rule 2445

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*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d (e+f x)^m\right )^n\right ) \, dx\\ &=a x+b \operatorname{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 \operatorname{Subst}\left (\frac{\operatorname{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]  time = 0.006675, size = 34, normalized size = 1. \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.066, size = 42, normalized size = 1.2 \begin{align*} ax+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) x-bmnx+{\frac{bemn\ln \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12163, size = 61, normalized size = 1.79 \begin{align*} -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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.27636, size = 124, normalized size = 3.65 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.920717, size = 58, normalized size = 1.71 \begin{align*} a x + b \left (\begin{cases} \frac{e m n \log{\left (e + f x \right )}}{f} + m n x \log{\left (e + f x \right )} - m n x + n x \log{\left (d \right )} + x \log{\left (c \right )} & \text{for}\: f \neq 0 \\x \log{\left (c \left (d e^{m}\right )^{n} \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23734, size = 86, normalized size = 2.53 \begin{align*}{\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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