3.639 \(\int (a+b \log (c (d+\frac{e}{f+g x})^p)) \, dx\)

Optimal. Leaf size=50 \[ a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{b e p \log (d (f+g x)+e)}{d g} \]

[Out]

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

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Rubi [A]  time = 0.0376744, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2483, 2448, 263, 31} \[ a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{b e p \log (d (f+g x)+e)}{d g} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e/(f + g*x))^p],x]

[Out]

a*x + (b*(f + g*x)*Log[c*(d + e/(f + g*x))^p])/g + (b*e*p*Log[e + d*(f + g*x)])/(d*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 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right ) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x}\right )^p\right ) \, dx,x,f+g x\right )}{g}\\ &=a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{(b e p) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x}\right ) x} \, dx,x,f+g x\right )}{g}\\ &=a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{(b e p) \operatorname{Subst}\left (\int \frac{1}{e+d x} \, dx,x,f+g x\right )}{g}\\ &=a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{b e p \log (e+d (f+g x))}{d g}\\ \end{align*}

Mathematica [A]  time = 0.0408769, size = 70, normalized size = 1.4 \[ a x+b x \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b e g p \left (\frac{f \log (f+g x)}{e g^2}-\frac{(d f+e) \log (d f+d g x+e)}{d e g^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*x - b*e*g*p*((f*Log[f + g*x])/(e*g^2) - ((e + d*f)*Log[e + d*f + d*g*x])/(d*e*g^2)) + b*x*Log[c*(d + e/(f +
g*x))^p]

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Maple [A]  time = 0.106, size = 81, normalized size = 1.6 \begin{align*} ax+b\ln \left ( c \left ({\frac{dgx+df+e}{gx+f}} \right ) ^{p} \right ) x-{\frac{bpf\ln \left ( gx+f \right ) }{g}}+{\frac{bp\ln \left ( dgx+df+e \right ) f}{g}}+{\frac{ebp\ln \left ( dgx+df+e \right ) }{dg}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*ln(c*(d+e/(g*x+f))^p),x)

[Out]

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

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Maxima [A]  time = 1.02494, size = 95, normalized size = 1.9 \begin{align*} -b e g p{\left (\frac{f \log \left (g x + f\right )}{e g^{2}} - \frac{{\left (d f + e\right )} \log \left (d g x + d f + e\right )}{d e g^{2}}\right )} + b x \log \left (c{\left (d + \frac{e}{g x + f}\right )}^{p}\right ) + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.94278, size = 190, normalized size = 3.8 \begin{align*} \frac{b d g p x \log \left (\frac{d g x + d f + e}{g x + f}\right ) - b d f p \log \left (g x + f\right ) + b d g x \log \left (c\right ) + a d g x +{\left (b d f + b e\right )} p \log \left (d g x + d f + e\right )}{d g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.29712, size = 114, normalized size = 2.28 \begin{align*} a x + b \left (\begin{cases} x \log{\left (c \left (\frac{e}{f}\right )^{p} \right )} & \text{for}\: d = 0 \wedge g = 0 \\x \log{\left (c \left (d + \frac{e}{f}\right )^{p} \right )} & \text{for}\: g = 0 \\- \frac{f p \log{\left (f + g x \right )}}{g} + p x \log{\left (e \right )} - p x \log{\left (f + g x \right )} + p x + x \log{\left (c \right )} & \text{for}\: d = 0 \\\frac{f p \log{\left (d + \frac{e}{f + g x} \right )}}{g} + p x \log{\left (d + \frac{e}{f + g x} \right )} + x \log{\left (c \right )} + \frac{e p \log{\left (d f + d g x + e \right )}}{d g} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*ln(c*(d+e/(g*x+f))**p),x)

[Out]

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

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Giac [A]  time = 1.31974, size = 123, normalized size = 2.46 \begin{align*} a x + \frac{{\left (d g p x \log \left (d g x + d f + e\right ) - d g p x \log \left (g x + f\right ) + d f p \log \left (d g x + d f + e\right ) - d f p \log \left (-g x - f\right ) + d g x \log \left (c\right ) + p e \log \left (d g x + d f + e\right )\right )} b}{d g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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