\(\int \log (c (d+e (f+g x)^p)^q) \, dx\) [628]

Optimal result

Integrand size = 16, antiderivative size = 76 \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=-\frac {e p q (f+g x)^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{p},2+\frac {1}{p},-\frac {e (f+g x)^p}{d}\right )}{d g (1+p)}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g} \]

[Out]

-e*p*q*(g*x+f)^(p+1)*hypergeom([1, 1+1/p],[2+1/p],-e*(g*x+f)^p/d)/d/g/(p+1)+(g*x+f)*ln(c*(d+e*(g*x+f)^p)^q)/g

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2533, 2498, 371} \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac {e p q (f+g x)^{p+1} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{p},2+\frac {1}{p},-\frac {e (f+g x)^p}{d}\right )}{d g (p+1)} \]

[In]

Int[Log[c*(d + e*(f + g*x)^p)^q],x]

[Out]

-((e*p*q*(f + g*x)^(1 + p)*Hypergeometric2F1[1, 1 + p^(-1), 2 + p^(-1), -((e*(f + g*x)^p)/d)])/(d*g*(1 + p)))
+ ((f + g*x)*Log[c*(d + e*(f + g*x)^p)^q])/g

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2498

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

Rule 2533

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_.))^(q_.), x_Symbol] :> Dist[1/g, Su
bst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q
, 0] && (EqQ[q, 1] || IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \log \left (c \left (d+e x^p\right )^q\right ) \, dx,x,f+g x\right )}{g} \\ & = \frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac {(e p q) \text {Subst}\left (\int \frac {x^p}{d+e x^p} \, dx,x,f+g x\right )}{g} \\ & = -\frac {e p q (f+g x)^{1+p} \, _2F_1\left (1,1+\frac {1}{p};2+\frac {1}{p};-\frac {e (f+g x)^p}{d}\right )}{d g (1+p)}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=-p q x+\frac {p q (f+g x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{p},1+\frac {1}{p},-\frac {e (f+g x)^p}{d}\right )}{g}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g} \]

[In]

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

[Out]

-(p*q*x) + (p*q*(f + g*x)*Hypergeometric2F1[1, p^(-1), 1 + p^(-1), -((e*(f + g*x)^p)/d)])/g + ((f + g*x)*Log[c
*(d + e*(f + g*x)^p)^q])/g

Maple [F]

\[\int \ln \left (c \left (d +e \left (g x +f \right )^{p}\right )^{q}\right )d x\]

[In]

int(ln(c*(d+e*(g*x+f)^p)^q),x)

[Out]

int(ln(c*(d+e*(g*x+f)^p)^q),x)

Fricas [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ) \,d x } \]

[In]

integrate(log(c*(d+e*(g*x+f)^p)^q),x, algorithm="fricas")

[Out]

integral(log(((g*x + f)^p*e + d)^q*c), x)

Sympy [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int \log {\left (c \left (d + e \left (f + g x\right )^{p}\right )^{q} \right )}\, dx \]

[In]

integrate(ln(c*(d+e*(g*x+f)**p)**q),x)

[Out]

Integral(log(c*(d + e*(f + g*x)**p)**q), x)

Maxima [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ) \,d x } \]

[In]

integrate(log(c*(d+e*(g*x+f)^p)^q),x, algorithm="maxima")

[Out]

d*g*p*q*integrate(x/(d*g*x + (e*g*x + e*f)*(g*x + f)^p + d*f), x) + (f*p*q*log(g*x + f) + g*x*log(((g*x + f)^p
*e + d)^q) - (g*p*q - g*log(c))*x)/g

Giac [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ) \,d x } \]

[In]

integrate(log(c*(d+e*(g*x+f)^p)^q),x, algorithm="giac")

[Out]

integrate(log(((g*x + f)^p*e + d)^q*c), x)

Mupad [F(-1)]

Timed out. \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int \ln \left (c\,{\left (d+e\,{\left (f+g\,x\right )}^p\right )}^q\right ) \,d x \]

[In]

int(log(c*(d + e*(f + g*x)^p)^q),x)

[Out]

int(log(c*(d + e*(f + g*x)^p)^q), x)