\(\int (a+b \log (c (d (e+f x)^m)^n))^3 \, dx\) [405]

Optimal result

Integrand size = 20, antiderivative size = 121 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=6 a b^2 m^2 n^2 x-6 b^3 m^3 n^3 x+\frac {6 b^3 m^2 n^2 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f} \]

[Out]

6*a*b^2*m^2*n^2*x-6*b^3*m^3*n^3*x+6*b^3*m^2*n^2*(f*x+e)*ln(c*(d*(f*x+e)^m)^n)/f-3*b*m*n*(f*x+e)*(a+b*ln(c*(d*(
f*x+e)^m)^n))^2/f+(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n))^3/f

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2333, 2332, 2495} \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=6 a b^2 m^2 n^2 x-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {6 b^3 m^2 n^2 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-6 b^3 m^3 n^3 x \]

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^3,x]

[Out]

6*a*b^2*m^2*n^2*x - 6*b^3*m^3*n^3*x + (6*b^3*m^2*n^2*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f - (3*b*m*n*(e + f*x
)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^3)/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 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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*} \text {integral}& = \text {Subst}\left (\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^3 \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^3 \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}-\text {Subst}\left (\frac {(3 b m n) \text {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^2 \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\text {Subst}\left (\frac {\left (6 b^2 m^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = 6 a b^2 m^2 n^2 x-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\text {Subst}\left (\frac {\left (6 b^3 m^2 n^2\right ) \text {Subst}\left (\int \log \left (c d^n x^{m n}\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = 6 a b^2 m^2 n^2 x-6 b^3 m^3 n^3 x+\frac {6 b^3 m^2 n^2 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3-3 b m n \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2-2 b m n \left (f (a-b m n) x+b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{f} \]

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^3,x]

[Out]

((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^3 - 3*b*m*n*((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2 - 2*b*m*
n*(f*(a - b*m*n)*x + b*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])))/f

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(121)=242\).

Time = 1.51 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.19

method	result	size
parallelrisch	\(\frac {6 \ln \left (f x +e \right ) b^{3} e^{2} m^{3} n^{3}-6 x \,b^{3} e f \,m^{3} n^{3}+6 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) b^{3} e f \,m^{2} n^{2}+6 b^{3} e^{2} m^{3} n^{3}-6 \ln \left (f x +e \right ) a \,b^{2} e^{2} m^{2} n^{2}-3 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} b^{3} e f m n +6 x a \,b^{2} e f \,m^{2} n^{2}+x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} b^{3} e f -6 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a \,b^{2} e f m n -3 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} b^{3} e^{2} m n -6 a \,b^{2} e^{2} m^{2} n^{2}+3 \ln \left (f x +e \right ) a^{2} b \,e^{2} m n +3 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a \,b^{2} e f -3 x \,a^{2} b e f m n +{\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} b^{3} e^{2}+3 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a^{2} b e f +3 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a \,b^{2} e^{2}+3 a^{2} b \,e^{2} m n +x \,a^{3} e f -a^{3} e^{2}}{e f}\)	\(386\)

[In]

int((a+b*ln(c*(d*(f*x+e)^m)^n))^3,x,method=_RETURNVERBOSE)

[Out]

(6*ln(f*x+e)*b^3*e^2*m^3*n^3-6*x*b^3*e*f*m^3*n^3+6*x*ln(c*(d*(f*x+e)^m)^n)*b^3*e*f*m^2*n^2+6*b^3*e^2*m^3*n^3-6
*ln(f*x+e)*a*b^2*e^2*m^2*n^2-3*x*ln(c*(d*(f*x+e)^m)^n)^2*b^3*e*f*m*n+6*x*a*b^2*e*f*m^2*n^2+x*ln(c*(d*(f*x+e)^m
)^n)^3*b^3*e*f-6*x*ln(c*(d*(f*x+e)^m)^n)*a*b^2*e*f*m*n-3*ln(c*(d*(f*x+e)^m)^n)^2*b^3*e^2*m*n-6*a*b^2*e^2*m^2*n
^2+3*ln(f*x+e)*a^2*b*e^2*m*n+3*x*ln(c*(d*(f*x+e)^m)^n)^2*a*b^2*e*f-3*x*a^2*b*e*f*m*n+ln(c*(d*(f*x+e)^m)^n)^3*b
^3*e^2+3*x*ln(c*(d*(f*x+e)^m)^n)*a^2*b*e*f+3*ln(c*(d*(f*x+e)^m)^n)^2*a*b^2*e^2+3*a^2*b*e^2*m*n+x*a^3*e*f-a^3*e
^2)/e/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (121) = 242\).

Time = 0.32 (sec) , antiderivative size = 639, normalized size of antiderivative = 5.28 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=\frac {b^{3} f n^{3} x \log \left (d\right )^{3} + b^{3} f x \log \left (c\right )^{3} + {\left (b^{3} f m^{3} n^{3} x + b^{3} e m^{3} n^{3}\right )} \log \left (f x + e\right )^{3} - 3 \, {\left (b^{3} f m n - a b^{2} f\right )} x \log \left (c\right )^{2} - 3 \, {\left (b^{3} e m^{3} n^{3} - a b^{2} e m^{2} n^{2} + {\left (b^{3} f m^{3} n^{3} - a b^{2} f m^{2} n^{2}\right )} x - {\left (b^{3} f m^{2} n^{2} x + b^{3} e m^{2} n^{2}\right )} \log \left (c\right ) - {\left (b^{3} f m^{2} n^{3} x + b^{3} e m^{2} n^{3}\right )} \log \left (d\right )\right )} \log \left (f x + e\right )^{2} + 3 \, {\left (2 \, b^{3} f m^{2} n^{2} - 2 \, a b^{2} f m n + a^{2} b f\right )} x \log \left (c\right ) + 3 \, {\left (b^{3} f n^{2} x \log \left (c\right ) - {\left (b^{3} f m n^{3} - a b^{2} f n^{2}\right )} x\right )} \log \left (d\right )^{2} - {\left (6 \, b^{3} f m^{3} n^{3} - 6 \, a b^{2} f m^{2} n^{2} + 3 \, a^{2} b f m n - a^{3} f\right )} x + 3 \, {\left (2 \, b^{3} e m^{3} n^{3} - 2 \, a b^{2} e m^{2} n^{2} + a^{2} b e m n + {\left (b^{3} f m n x + b^{3} e m n\right )} \log \left (c\right )^{2} + {\left (b^{3} f m n^{3} x + b^{3} e m n^{3}\right )} \log \left (d\right )^{2} + {\left (2 \, b^{3} f m^{3} n^{3} - 2 \, a b^{2} f m^{2} n^{2} + a^{2} b f m n\right )} x - 2 \, {\left (b^{3} e m^{2} n^{2} - a b^{2} e m n + {\left (b^{3} f m^{2} n^{2} - a b^{2} f m n\right )} x\right )} \log \left (c\right ) - 2 \, {\left (b^{3} e m^{2} n^{3} - a b^{2} e m n^{2} + {\left (b^{3} f m^{2} n^{3} - a b^{2} f m n^{2}\right )} x - {\left (b^{3} f m n^{2} x + b^{3} e m n^{2}\right )} \log \left (c\right )\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 3 \, {\left (b^{3} f n x \log \left (c\right )^{2} - 2 \, {\left (b^{3} f m n^{2} - a b^{2} f n\right )} x \log \left (c\right ) + {\left (2 \, b^{3} f m^{2} n^{3} - 2 \, a b^{2} f m n^{2} + a^{2} b f n\right )} x\right )} \log \left (d\right )}{f} \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="fricas")

[Out]

(b^3*f*n^3*x*log(d)^3 + b^3*f*x*log(c)^3 + (b^3*f*m^3*n^3*x + b^3*e*m^3*n^3)*log(f*x + e)^3 - 3*(b^3*f*m*n - a
*b^2*f)*x*log(c)^2 - 3*(b^3*e*m^3*n^3 - a*b^2*e*m^2*n^2 + (b^3*f*m^3*n^3 - a*b^2*f*m^2*n^2)*x - (b^3*f*m^2*n^2
*x + b^3*e*m^2*n^2)*log(c) - (b^3*f*m^2*n^3*x + b^3*e*m^2*n^3)*log(d))*log(f*x + e)^2 + 3*(2*b^3*f*m^2*n^2 - 2
*a*b^2*f*m*n + a^2*b*f)*x*log(c) + 3*(b^3*f*n^2*x*log(c) - (b^3*f*m*n^3 - a*b^2*f*n^2)*x)*log(d)^2 - (6*b^3*f*
m^3*n^3 - 6*a*b^2*f*m^2*n^2 + 3*a^2*b*f*m*n - a^3*f)*x + 3*(2*b^3*e*m^3*n^3 - 2*a*b^2*e*m^2*n^2 + a^2*b*e*m*n
+ (b^3*f*m*n*x + b^3*e*m*n)*log(c)^2 + (b^3*f*m*n^3*x + b^3*e*m*n^3)*log(d)^2 + (2*b^3*f*m^3*n^3 - 2*a*b^2*f*m
^2*n^2 + a^2*b*f*m*n)*x - 2*(b^3*e*m^2*n^2 - a*b^2*e*m*n + (b^3*f*m^2*n^2 - a*b^2*f*m*n)*x)*log(c) - 2*(b^3*e*
m^2*n^3 - a*b^2*e*m*n^2 + (b^3*f*m^2*n^3 - a*b^2*f*m*n^2)*x - (b^3*f*m*n^2*x + b^3*e*m*n^2)*log(c))*log(d))*lo
g(f*x + e) + 3*(b^3*f*n*x*log(c)^2 - 2*(b^3*f*m*n^2 - a*b^2*f*n)*x*log(c) + (2*b^3*f*m^2*n^3 - 2*a*b^2*f*m*n^2
+ a^2*b*f*n)*x)*log(d))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (117) = 234\).

Time = 1.14 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.98 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - 3 a^{2} b m n x + 3 a^{2} b x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} - \frac {6 a b^{2} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} + \frac {3 a b^{2} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} + 6 a b^{2} m^{2} n^{2} x - 6 a b^{2} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} + 3 a b^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} + \frac {6 b^{3} e m^{2} n^{2} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - \frac {3 b^{3} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} + \frac {b^{3} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3}}{f} - 6 b^{3} m^{3} n^{3} x + 6 b^{3} m^{2} n^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} - 3 b^{3} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} + b^{3} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3} & \text {for}\: f \neq 0 \\x \left (a + b \log {\left (c \left (d e^{m}\right )^{n} \right )}\right )^{3} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*ln(c*(d*(f*x+e)**m)**n))**3,x)

[Out]

Piecewise((a**3*x + 3*a**2*b*e*log(c*(d*(e + f*x)**m)**n)/f - 3*a**2*b*m*n*x + 3*a**2*b*x*log(c*(d*(e + f*x)**
m)**n) - 6*a*b**2*e*m*n*log(c*(d*(e + f*x)**m)**n)/f + 3*a*b**2*e*log(c*(d*(e + f*x)**m)**n)**2/f + 6*a*b**2*m
**2*n**2*x - 6*a*b**2*m*n*x*log(c*(d*(e + f*x)**m)**n) + 3*a*b**2*x*log(c*(d*(e + f*x)**m)**n)**2 + 6*b**3*e*m
**2*n**2*log(c*(d*(e + f*x)**m)**n)/f - 3*b**3*e*m*n*log(c*(d*(e + f*x)**m)**n)**2/f + b**3*e*log(c*(d*(e + f*
x)**m)**n)**3/f - 6*b**3*m**3*n**3*x + 6*b**3*m**2*n**2*x*log(c*(d*(e + f*x)**m)**n) - 3*b**3*m*n*x*log(c*(d*(
e + f*x)**m)**n)**2 + b**3*x*log(c*(d*(e + f*x)**m)**n)**3, Ne(f, 0)), (x*(a + b*log(c*(d*e**m)**n))**3, True)
)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (121) = 242\).

Time = 0.24 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.62 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=-3 \, a^{2} b f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b^{3} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{3} + 3 \, a b^{2} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} + 3 \, a^{2} b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) - 3 \, {\left (2 \, f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f}\right )} a b^{2} - {\left (3 \, f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} - {\left (\frac {{\left (e \log \left (f x + e\right )^{3} + 3 \, e \log \left (f x + e\right )^{2} - 6 \, f x + 6 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f^{2}} - \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m n \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )}{f^{2}}\right )} f m n\right )} b^{3} + a^{3} x \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="maxima")

[Out]

-3*a^2*b*f*m*n*(x/f - e*log(f*x + e)/f^2) + b^3*x*log(((f*x + e)^m*d)^n*c)^3 + 3*a*b^2*x*log(((f*x + e)^m*d)^n
*c)^2 + 3*a^2*b*x*log(((f*x + e)^m*d)^n*c) - 3*(2*f*m*n*(x/f - e*log(f*x + e)/f^2)*log(((f*x + e)^m*d)^n*c) +
(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*m^2*n^2/f)*a*b^2 - (3*f*m*n*(x/f - e*log(f*x + e)/f^2)*log(((f*x
+ e)^m*d)^n*c)^2 - ((e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e))*m^2*n^2/f^2 - 3*(e*log
(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*m*n*log(((f*x + e)^m*d)^n*c)/f^2)*f*m*n)*b^3 + a^3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (121) = 242\).

Time = 0.32 (sec) , antiderivative size = 772, normalized size of antiderivative = 6.38 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=\frac {{\left (f x + e\right )} b^{3} m^{3} n^{3} \log \left (f x + e\right )^{3}}{f} - \frac {3 \, {\left (f x + e\right )} b^{3} m^{3} n^{3} \log \left (f x + e\right )^{2}}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m^{2} n^{3} \log \left (f x + e\right )^{2} \log \left (d\right )}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m^{3} n^{3} \log \left (f x + e\right )}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m^{2} n^{2} \log \left (f x + e\right )^{2} \log \left (c\right )}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{3} \log \left (f x + e\right ) \log \left (d\right )}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m n^{3} \log \left (f x + e\right ) \log \left (d\right )^{2}}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m^{3} n^{3}}{f} + \frac {3 \, {\left (f x + e\right )} a b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2}}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{2} \log \left (f x + e\right ) \log \left (c\right )}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{3} \log \left (d\right )}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m n^{2} \log \left (f x + e\right ) \log \left (c\right ) \log \left (d\right )}{f} - \frac {3 \, {\left (f x + e\right )} b^{3} m n^{3} \log \left (d\right )^{2}}{f} + \frac {{\left (f x + e\right )} b^{3} n^{3} \log \left (d\right )^{3}}{f} - \frac {6 \, {\left (f x + e\right )} a b^{2} m^{2} n^{2} \log \left (f x + e\right )}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{2} \log \left (c\right )}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m n \log \left (f x + e\right ) \log \left (c\right )^{2}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} m n^{2} \log \left (f x + e\right ) \log \left (d\right )}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m n^{2} \log \left (c\right ) \log \left (d\right )}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} n^{2} \log \left (c\right ) \log \left (d\right )^{2}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} m^{2} n^{2}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} m n \log \left (f x + e\right ) \log \left (c\right )}{f} - \frac {3 \, {\left (f x + e\right )} b^{3} m n \log \left (c\right )^{2}}{f} - \frac {6 \, {\left (f x + e\right )} a b^{2} m n^{2} \log \left (d\right )}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} n \log \left (c\right )^{2} \log \left (d\right )}{f} + \frac {3 \, {\left (f x + e\right )} a b^{2} n^{2} \log \left (d\right )^{2}}{f} + \frac {3 \, {\left (f x + e\right )} a^{2} b m n \log \left (f x + e\right )}{f} - \frac {6 \, {\left (f x + e\right )} a b^{2} m n \log \left (c\right )}{f} + \frac {{\left (f x + e\right )} b^{3} \log \left (c\right )^{3}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} n \log \left (c\right ) \log \left (d\right )}{f} - \frac {3 \, {\left (f x + e\right )} a^{2} b m n}{f} + \frac {3 \, {\left (f x + e\right )} a b^{2} \log \left (c\right )^{2}}{f} + \frac {3 \, {\left (f x + e\right )} a^{2} b n \log \left (d\right )}{f} + \frac {3 \, {\left (f x + e\right )} a^{2} b \log \left (c\right )}{f} + \frac {{\left (f x + e\right )} a^{3}}{f} \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="giac")

[Out]

(f*x + e)*b^3*m^3*n^3*log(f*x + e)^3/f - 3*(f*x + e)*b^3*m^3*n^3*log(f*x + e)^2/f + 3*(f*x + e)*b^3*m^2*n^3*lo
g(f*x + e)^2*log(d)/f + 6*(f*x + e)*b^3*m^3*n^3*log(f*x + e)/f + 3*(f*x + e)*b^3*m^2*n^2*log(f*x + e)^2*log(c)
/f - 6*(f*x + e)*b^3*m^2*n^3*log(f*x + e)*log(d)/f + 3*(f*x + e)*b^3*m*n^3*log(f*x + e)*log(d)^2/f - 6*(f*x +
e)*b^3*m^3*n^3/f + 3*(f*x + e)*a*b^2*m^2*n^2*log(f*x + e)^2/f - 6*(f*x + e)*b^3*m^2*n^2*log(f*x + e)*log(c)/f
+ 6*(f*x + e)*b^3*m^2*n^3*log(d)/f + 6*(f*x + e)*b^3*m*n^2*log(f*x + e)*log(c)*log(d)/f - 3*(f*x + e)*b^3*m*n^
3*log(d)^2/f + (f*x + e)*b^3*n^3*log(d)^3/f - 6*(f*x + e)*a*b^2*m^2*n^2*log(f*x + e)/f + 6*(f*x + e)*b^3*m^2*n
^2*log(c)/f + 3*(f*x + e)*b^3*m*n*log(f*x + e)*log(c)^2/f + 6*(f*x + e)*a*b^2*m*n^2*log(f*x + e)*log(d)/f - 6*
(f*x + e)*b^3*m*n^2*log(c)*log(d)/f + 3*(f*x + e)*b^3*n^2*log(c)*log(d)^2/f + 6*(f*x + e)*a*b^2*m^2*n^2/f + 6*
(f*x + e)*a*b^2*m*n*log(f*x + e)*log(c)/f - 3*(f*x + e)*b^3*m*n*log(c)^2/f - 6*(f*x + e)*a*b^2*m*n^2*log(d)/f
+ 3*(f*x + e)*b^3*n*log(c)^2*log(d)/f + 3*(f*x + e)*a*b^2*n^2*log(d)^2/f + 3*(f*x + e)*a^2*b*m*n*log(f*x + e)/
f - 6*(f*x + e)*a*b^2*m*n*log(c)/f + (f*x + e)*b^3*log(c)^3/f + 6*(f*x + e)*a*b^2*n*log(c)*log(d)/f - 3*(f*x +
e)*a^2*b*m*n/f + 3*(f*x + e)*a*b^2*log(c)^2/f + 3*(f*x + e)*a^2*b*n*log(d)/f + 3*(f*x + e)*a^2*b*log(c)/f + (
f*x + e)*a^3/f

Mupad [B] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.00 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx=x\,\left (a^3-3\,a^2\,b\,m\,n+6\,a\,b^2\,m^2\,n^2-6\,b^3\,m^3\,n^3\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^2\,\left (\frac {3\,\left (a\,b^2\,e-b^3\,e\,m\,n\right )}{f}+3\,b^2\,x\,\left (a-b\,m\,n\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,e}{f}\right )+\frac {\ln \left (e+f\,x\right )\,\left (3\,e\,a^2\,b\,m\,n-6\,e\,a\,b^2\,m^2\,n^2+6\,e\,b^3\,m^3\,n^3\right )}{f}+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\,\left (3\,b\,f\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\,x^2+3\,b\,e\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\,x\right )}{e+f\,x} \]

[In]

int((a + b*log(c*(d*(e + f*x)^m)^n))^3,x)

[Out]

x*(a^3 - 6*b^3*m^3*n^3 + 6*a*b^2*m^2*n^2 - 3*a^2*b*m*n) + log(c*(d*(e + f*x)^m)^n)^2*((3*(a*b^2*e - b^3*e*m*n)
)/f + 3*b^2*x*(a - b*m*n)) + log(c*(d*(e + f*x)^m)^n)^3*(b^3*x + (b^3*e)/f) + (log(e + f*x)*(6*b^3*e*m^3*n^3 -
6*a*b^2*e*m^2*n^2 + 3*a^2*b*e*m*n))/f + (log(c*(d*(e + f*x)^m)^n)*(3*b*e*x*(a^2 + 2*b^2*m^2*n^2 - 2*a*b*m*n)
+ 3*b*f*x^2*(a^2 + 2*b^2*m^2*n^2 - 2*a*b*m*n)))/(e + f*x)