3.5.44 \(\int \log (c (d (e+f x)^p)^q) \, dx\) [444]

Optimal. Leaf size=29 \[ -p q x+\frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2436, 2332, 2495} \begin {gather*} \frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(d*(e + f*x)^p)^q],x]

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} -p q x+\frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d*(e + f*x)^p)^q],x]

[Out]

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

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Maple [A]
time = 0.07, size = 41, normalized size = 1.41

method	result	size
default	\(\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) x -q p f \left (\frac {x}{f}-\frac {e \ln \left (f x +e \right )}{f^{2}}\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d*(f*x+e)^p)^q),x,method=_RETURNVERBOSE)

[Out]

ln(c*(d*(f*x+e)^p)^q)*x-q*p*f*(1/f*x-e/f^2*ln(f*x+e))

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Maxima [A]
time = 0.27, size = 43, normalized size = 1.48 \begin {gather*} -f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 44, normalized size = 1.52 \begin {gather*} -\frac {f p q x - f q x \log \left (d\right ) - f x \log \left (c\right ) - {\left (f p q x + p q e\right )} \log \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.22, size = 48, normalized size = 1.66 \begin {gather*} \begin {cases} \frac {e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - p q x + x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{p}\right )^{q} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((e*log(c*(d*(e + f*x)**p)**q)/f - p*q*x + x*log(c*(d*(e + f*x)**p)**q), Ne(f, 0)), (x*log(c*(d*e**p)
**q), True))

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Giac [A]
time = 4.06, size = 58, normalized size = 2.00 \begin {gather*} \frac {{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} p q}{f} + \frac {{\left (f x + e\right )} q \log \left (d\right )}{f} + \frac {{\left (f x + e\right )} \log \left (c\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 36, normalized size = 1.24 \begin {gather*} x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )+\frac {p\,q\,\left (e\,\ln \left (e+f\,x\right )-f\,x\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d*(e + f*x)^p)^q),x)

[Out]

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

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