3.423 \(\int (a+b \log (c (d (e+f x)^p)^q)) \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0294347, 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} \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d*(e + f*x)^p)^q],x]

[Out]

a*x - b*p*q*x + (b*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/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)^p\right )^q\right )\right ) \, dx &=a x+b \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx\\ &=a x+b \operatorname{Subst}\left (\int \log \left (c d^q (e+f x)^{p q}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=a x+b \operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=a x-b p q x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}\\ \end{align*}

Mathematica [A]  time = 0.0084686, size = 34, normalized size = 1. \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.066, size = 42, normalized size = 1.2 \begin{align*} ax+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) x-bpqx+{\frac{bqpe\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)^p)^q),x)

[Out]

a*x+b*ln(c*(d*(f*x+e)^p)^q)*x-b*p*q*x+b*q*p/f*e*ln(f*x+e)

________________________________________________________________________________________

Maxima [A]  time = 1.06097, size = 61, normalized size = 1.79 \begin{align*} -b f p q{\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 )}^{p} d\right )^{q} 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)^p)^q),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.23357, size = 124, normalized size = 3.65 \begin{align*} \frac{b f q x \log \left (d\right ) + b f x \log \left (c\right ) -{\left (b f p q - a f\right )} x +{\left (b f p q x + b e p q\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)^p)^q),x, algorithm="fricas")

[Out]

(b*f*q*x*log(d) + b*f*x*log(c) - (b*f*p*q - a*f)*x + (b*f*p*q*x + b*e*p*q)*log(f*x + e))/f

________________________________________________________________________________________

Sympy [A]  time = 0.928201, size = 58, normalized size = 1.71 \begin{align*} a x + b \left (\begin{cases} \frac{e p q \log{\left (e + f x \right )}}{f} + p q x \log{\left (e + f x \right )} - p q x + q x \log{\left (d \right )} + x \log{\left (c \right )} & \text{for}\: f \neq 0 \\x \log{\left (c \left (d e^{p}\right )^{q} \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)**p)**q),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.29779, size = 86, normalized size = 2.53 \begin{align*}{\left (\frac{{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac{{\left (f x + e\right )} p q}{f} + \frac{{\left (f x + e\right )} q \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)^p)^q),x, algorithm="giac")

[Out]

((f*x + e)*p*q*log(f*x + e)/f - (f*x + e)*p*q/f + (f*x + e)*q*log(d)/f + (f*x + e)*log(c)/f)*b + a*x