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

Optimal. Leaf size=61 \[ \frac{(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{q r (b c-a d) \log (c+d x)}{b d}+r x (-(p+q)) \]

[Out]

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

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Rubi [A]  time = 0.0148794, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2487, 31, 8} \[ \frac{(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{q r (b c-a d) \log (c+d x)}{b d}+r x (-(p+q)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2487

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + (Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1
), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] &&
LtQ[s, 4]

Rule 31

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0628468, size = 57, normalized size = 0.93 \[ x \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-r (p+q)\right )+\frac{a p r \log (a+b x)}{b}+\frac{c q r \log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.071, size = 61, normalized size = 1. \begin{align*} \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) x-rpx-rqx+{\frac{rap\ln \left ( bx+a \right ) }{b}}+{\frac{rqc\ln \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15706, size = 101, normalized size = 1.66 \begin{align*} x \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) - \frac{{\left (b f p{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} + d f q{\left (\frac{x}{d} - \frac{c \log \left (d x + c\right )}{d^{2}}\right )}\right )} r}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.08855, size = 182, normalized size = 2.98 \begin{align*} \frac{b d r x \log \left (f\right ) + b d x \log \left (e\right ) -{\left (b d p + b d q\right )} r x +{\left (b d p r x + a d p r\right )} \log \left (b x + a\right ) +{\left (b d q r x + b c q r\right )} \log \left (d x + c\right )}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)

[Out]

Timed out

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Giac [B]  time = 1.23847, size = 242, normalized size = 3.97 \begin{align*} p r x \log \left (b x + a\right ) + q r x \log \left (d x + c\right ) -{\left (p r + q r - r \log \left (f\right ) - 1\right )} x + \frac{{\left (a d p r + b c q r\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{2 \, b d} + \frac{{\left (a b c d p r - a^{2} d^{2} p r - b^{2} c^{2} q r + a b c d q r\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | -b c + a d \right |}}{2 \, b d x + b c + a d +{\left | -b c + a d \right |}} \right |}\right )}{2 \, b d{\left | -b c + a d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

p*r*x*log(b*x + a) + q*r*x*log(d*x + c) - (p*r + q*r - r*log(f) - 1)*x + 1/2*(a*d*p*r + b*c*q*r)*log(abs(b*d*x
^2 + b*c*x + a*d*x + a*c))/(b*d) + 1/2*(a*b*c*d*p*r - a^2*d^2*p*r - b^2*c^2*q*r + a*b*c*d*q*r)*log(abs((2*b*d*
x + b*c + a*d - abs(-b*c + a*d))/(2*b*d*x + b*c + a*d + abs(-b*c + a*d))))/(b*d*abs(-b*c + a*d))