3.1.29 \(\int \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [29]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {2579, 31, 8} \begin {gather*} \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)) \end {gather*}

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 8

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

Rule 31

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

Rule 2579

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]

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*}

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Mathematica [A]
time = 0.04, size = 57, normalized size = 0.93 \begin {gather*} \frac {a p r \log (a+b x)}{b}+\frac {c q r \log (c+d x)}{d}+x \left (-((p+q) r)+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \end {gather*}

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 = 0.08, size = 61, normalized size = 1.00

method	result	size
default	\(\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) x -r \left (p x +q x -\frac {a p \ln \left (b x +a \right )}{b}-\frac {c q \ln \left (d x +c \right )}{d}\right )\)	\(61\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 76, normalized size = 1.25 \begin {gather*} 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 {gather*}

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 = 0.47, size = 71, normalized size = 1.16 \begin {gather*} \frac {b d r x \log \left (f\right ) + {\left (b d - {\left (b d p + b d q\right )} r\right )} 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 {gather*}

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 - (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 [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (54) = 108\).
time = 6.67, size = 184, normalized size = 3.02 \begin {gather*} \begin {cases} x \log {\left (e \left (a^{p} c^{q} f\right )^{r} \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {a \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{b} - p r x + x \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )} & \text {for}\: d = 0 \\\frac {c \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}}{d} - q r x + x \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )} & \text {for}\: b = 0 \\- \frac {a q r \log {\left (c + d x \right )}}{b} + \frac {a \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{b} + \frac {c q r \log {\left (c + d x \right )}}{d} - p r x - q r x + x \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )} & \text {otherwise} \end {cases} \end {gather*}

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]

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

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Giac [A]
time = 4.23, size = 66, normalized size = 1.08 \begin {gather*} p r x \log \left (b x + a\right ) + q r x \log \left (d x + c\right ) + \frac {a p r \log \left (b x + a\right )}{b} + \frac {c q r \log \left (-d x - c\right )}{d} - {\left (p r + q r - r \log \left (f\right ) - 1\right )} x \end {gather*}

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

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Mupad [B]
time = 0.22, size = 60, normalized size = 0.98 \begin {gather*} x\,\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )-p\,r\,x-q\,r\,x+\frac {a\,p\,r\,\ln \left (a+b\,x\right )}{b}+\frac {c\,q\,r\,\ln \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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