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

Optimal. Leaf size=76 \[ \frac{(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac{e p q (f+g x)^{p+1} \, _2F_1\left (1,1+\frac{1}{p};2+\frac{1}{p};-\frac{e (f+g x)^p}{d}\right )}{d g (p+1)} \]

[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

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Rubi [A]  time = 0.0443628, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2483, 2448, 364} \[ \frac{(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac{e p q (f+g x)^{p+1} \, _2F_1\left (1,1+\frac{1}{p};2+\frac{1}{p};-\frac{e (f+g x)^p}{d}\right )}{d g (p+1)} \]

Antiderivative was successfully verified.

[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 2483

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])

Rule 2448

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx &=\frac{\operatorname{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) \operatorname{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]  time = 0.024157, size = 65, normalized size = 0.86 \[ \frac{(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}+\frac{p q (f+g x) \, _2F_1\left (1,\frac{1}{p};1+\frac{1}{p};-\frac{e (f+g x)^p}{d}\right )}{g}-p q x \]

Antiderivative was successfully verified.

[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

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Maple [F]  time = 2.597, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( c \left ( d+e \left ( gx+f \right ) ^{p} \right ) ^{q} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d g p q \int \frac{x}{d g x +{\left (e g x + e f\right )}{\left (g x + f\right )}^{p} + d f}\,{d x} + \frac{f p q \log \left (g x + f\right ) + g x \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q}\right ) -{\left (g p q - g \log \left (c\right )\right )} x}{g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (c \left (d + e \left (f + g x\right )^{p}\right )^{q} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)