\(\int (a+b \log (c (d (e+f x)^m)^n))^4 \, dx\) [404]

Optimal result

Integrand size = 20, antiderivative size = 160 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=-24 a b^3 m^3 n^3 x+24 b^4 m^4 n^4 x-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f} \]

[Out]

-24*a*b^3*m^3*n^3*x+24*b^4*m^4*n^4*x-24*b^4*m^3*n^3*(f*x+e)*ln(c*(d*(f*x+e)^m)^n)/f+12*b^2*m^2*n^2*(f*x+e)*(a+
b*ln(c*(d*(f*x+e)^m)^n))^2/f-4*b*m*n*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n))^3/f+(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n
))^4/f

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2333, 2332, 2495} \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=-24 a b^3 m^3 n^3 x+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+24 b^4 m^4 n^4 x \]

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^4,x]

[Out]

-24*a*b^3*m^3*n^3*x + 24*b^4*m^4*n^4*x - (24*b^4*m^3*n^3*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f + (12*b^2*m^2*n
^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)/f - (4*b*m*n*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^3)/f
+ ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^4)/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 )^4 \, 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 )^4 \, 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 )^4}{f}-\text {Subst}\left (\frac {(4 b m n) \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 {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}+\text {Subst}\left (\frac {\left (12 b^2 m^2 n^2\right ) \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 {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}-\text {Subst}\left (\frac {\left (24 b^3 m^3 n^3\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 ) \\ & = -24 a b^3 m^3 n^3 x+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}-\text {Subst}\left (\frac {\left (24 b^4 m^3 n^3\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 ) \\ & = -24 a b^3 m^3 n^3 x+24 b^4 m^4 n^4 x-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4-4 b m n \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3-3 b m n \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2-2 b m n \left (f (a-b m n) x+b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )\right )}{f} \]

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^4,x]

[Out]

((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^4 - 4*b*m*n*((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. \(641\) vs. \(2(160)=320\).

Time = 5.58 (sec) , antiderivative size = 642, normalized size of antiderivative = 4.01

method	result	size
parallelrisch	\(\frac {-12 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a \,b^{3} e f m n -12 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a^{2} b^{2} e f m n +24 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a \,b^{3} e f \,m^{2} n^{2}+4 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} a \,b^{3} e^{2}+6 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a^{2} b^{2} e^{2}+x \,a^{4} e f -24 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) b^{4} e f \,m^{3} n^{3}+12 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} b^{4} e f \,m^{2} n^{2}+12 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} b^{4} e^{2} m^{2} n^{2}-4 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} b^{4} e^{2} m n -24 \ln \left (f x +e \right ) b^{4} e^{2} m^{4} n^{4}+x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{4} b^{4} e f +24 x \,b^{4} e f \,m^{4} n^{4}-12 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a \,b^{3} e^{2} m n +24 \ln \left (f x +e \right ) a \,b^{3} e^{2} m^{3} n^{3}-12 \ln \left (f x +e \right ) a^{2} b^{2} e^{2} m^{2} n^{2}-24 x a \,b^{3} e f \,m^{3} n^{3}-4 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} b^{4} e f m n +12 x \,a^{2} b^{2} e f \,m^{2} n^{2}-4 x \,a^{3} b e f m n +24 a \,b^{3} e^{2} m^{3} n^{3}-12 a^{2} b^{2} e^{2} m^{2} n^{2}+4 a^{3} b \,e^{2} m n +4 \ln \left (f x +e \right ) a^{3} b \,e^{2} m n +4 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a^{3} b e f +4 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} a \,b^{3} e f +6 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a^{2} b^{2} e f -a^{4} e^{2}-24 b^{4} e^{2} m^{4} n^{4}+{\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{4} b^{4} e^{2}}{e f}\)	\(642\)

[In]

int((a+b*ln(c*(d*(f*x+e)^m)^n))^4,x,method=_RETURNVERBOSE)

[Out]

(-12*x*ln(c*(d*(f*x+e)^m)^n)^2*a*b^3*e*f*m*n-12*x*ln(c*(d*(f*x+e)^m)^n)*a^2*b^2*e*f*m*n+24*x*ln(c*(d*(f*x+e)^m
)^n)*a*b^3*e*f*m^2*n^2+4*ln(c*(d*(f*x+e)^m)^n)^3*a*b^3*e^2+6*ln(c*(d*(f*x+e)^m)^n)^2*a^2*b^2*e^2+x*a^4*e*f-24*
x*ln(c*(d*(f*x+e)^m)^n)*b^4*e*f*m^3*n^3+12*x*ln(c*(d*(f*x+e)^m)^n)^2*b^4*e*f*m^2*n^2+12*ln(c*(d*(f*x+e)^m)^n)^
2*b^4*e^2*m^2*n^2-4*ln(c*(d*(f*x+e)^m)^n)^3*b^4*e^2*m*n-24*ln(f*x+e)*b^4*e^2*m^4*n^4+x*ln(c*(d*(f*x+e)^m)^n)^4
*b^4*e*f+24*x*b^4*e*f*m^4*n^4-12*ln(c*(d*(f*x+e)^m)^n)^2*a*b^3*e^2*m*n+24*ln(f*x+e)*a*b^3*e^2*m^3*n^3-12*ln(f*
x+e)*a^2*b^2*e^2*m^2*n^2-24*x*a*b^3*e*f*m^3*n^3-4*x*ln(c*(d*(f*x+e)^m)^n)^3*b^4*e*f*m*n+12*x*a^2*b^2*e*f*m^2*n
^2-4*x*a^3*b*e*f*m*n+24*a*b^3*e^2*m^3*n^3-12*a^2*b^2*e^2*m^2*n^2+4*a^3*b*e^2*m*n+4*ln(f*x+e)*a^3*b*e^2*m*n+4*x
*ln(c*(d*(f*x+e)^m)^n)*a^3*b*e*f+4*x*ln(c*(d*(f*x+e)^m)^n)^3*a*b^3*e*f+6*x*ln(c*(d*(f*x+e)^m)^n)^2*a^2*b^2*e*f
-a^4*e^2-24*b^4*e^2*m^4*n^4+ln(c*(d*(f*x+e)^m)^n)^4*b^4*e^2)/e/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (160) = 320\).

Time = 0.32 (sec) , antiderivative size = 1409, normalized size of antiderivative = 8.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\text {Too large to display} \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^4,x, algorithm="fricas")

[Out]

(b^4*f*n^4*x*log(d)^4 + b^4*f*x*log(c)^4 + (b^4*f*m^4*n^4*x + b^4*e*m^4*n^4)*log(f*x + e)^4 - 4*(b^4*f*m*n - a
*b^3*f)*x*log(c)^3 - 4*(b^4*e*m^4*n^4 - a*b^3*e*m^3*n^3 + (b^4*f*m^4*n^4 - a*b^3*f*m^3*n^3)*x - (b^4*f*m^3*n^3
*x + b^4*e*m^3*n^3)*log(c) - (b^4*f*m^3*n^4*x + b^4*e*m^3*n^4)*log(d))*log(f*x + e)^3 + 6*(2*b^4*f*m^2*n^2 - 2
*a*b^3*f*m*n + a^2*b^2*f)*x*log(c)^2 + 4*(b^4*f*n^3*x*log(c) - (b^4*f*m*n^4 - a*b^3*f*n^3)*x)*log(d)^3 + 6*(2*
b^4*e*m^4*n^4 - 2*a*b^3*e*m^3*n^3 + a^2*b^2*e*m^2*n^2 + (b^4*f*m^2*n^2*x + b^4*e*m^2*n^2)*log(c)^2 + (b^4*f*m^
2*n^4*x + b^4*e*m^2*n^4)*log(d)^2 + (2*b^4*f*m^4*n^4 - 2*a*b^3*f*m^3*n^3 + a^2*b^2*f*m^2*n^2)*x - 2*(b^4*e*m^3
*n^3 - a*b^3*e*m^2*n^2 + (b^4*f*m^3*n^3 - a*b^3*f*m^2*n^2)*x)*log(c) - 2*(b^4*e*m^3*n^4 - a*b^3*e*m^2*n^3 + (b
^4*f*m^3*n^4 - a*b^3*f*m^2*n^3)*x - (b^4*f*m^2*n^3*x + b^4*e*m^2*n^3)*log(c))*log(d))*log(f*x + e)^2 - 4*(6*b^
4*f*m^3*n^3 - 6*a*b^3*f*m^2*n^2 + 3*a^2*b^2*f*m*n - a^3*b*f)*x*log(c) + 6*(b^4*f*n^2*x*log(c)^2 - 2*(b^4*f*m*n
^3 - a*b^3*f*n^2)*x*log(c) + (2*b^4*f*m^2*n^4 - 2*a*b^3*f*m*n^3 + a^2*b^2*f*n^2)*x)*log(d)^2 + (24*b^4*f*m^4*n
^4 - 24*a*b^3*f*m^3*n^3 + 12*a^2*b^2*f*m^2*n^2 - 4*a^3*b*f*m*n + a^4*f)*x - 4*(6*b^4*e*m^4*n^4 - 6*a*b^3*e*m^3
*n^3 + 3*a^2*b^2*e*m^2*n^2 - a^3*b*e*m*n - (b^4*f*m*n*x + b^4*e*m*n)*log(c)^3 - (b^4*f*m*n^4*x + b^4*e*m*n^4)*
log(d)^3 + 3*(b^4*e*m^2*n^2 - a*b^3*e*m*n + (b^4*f*m^2*n^2 - a*b^3*f*m*n)*x)*log(c)^2 + 3*(b^4*e*m^2*n^4 - a*b
^3*e*m*n^3 + (b^4*f*m^2*n^4 - a*b^3*f*m*n^3)*x - (b^4*f*m*n^3*x + b^4*e*m*n^3)*log(c))*log(d)^2 + (6*b^4*f*m^4
*n^4 - 6*a*b^3*f*m^3*n^3 + 3*a^2*b^2*f*m^2*n^2 - a^3*b*f*m*n)*x - 3*(2*b^4*e*m^3*n^3 - 2*a*b^3*e*m^2*n^2 + a^2
*b^2*e*m*n + (2*b^4*f*m^3*n^3 - 2*a*b^3*f*m^2*n^2 + a^2*b^2*f*m*n)*x)*log(c) - 3*(2*b^4*e*m^3*n^4 - 2*a*b^3*e*
m^2*n^3 + a^2*b^2*e*m*n^2 + (b^4*f*m*n^2*x + b^4*e*m*n^2)*log(c)^2 + (2*b^4*f*m^3*n^4 - 2*a*b^3*f*m^2*n^3 + a^
2*b^2*f*m*n^2)*x - 2*(b^4*e*m^2*n^3 - a*b^3*e*m*n^2 + (b^4*f*m^2*n^3 - a*b^3*f*m*n^2)*x)*log(c))*log(d))*log(f
*x + e) + 4*(b^4*f*n*x*log(c)^3 - 3*(b^4*f*m*n^2 - a*b^3*f*n)*x*log(c)^2 + 3*(2*b^4*f*m^2*n^3 - 2*a*b^3*f*m*n^
2 + a^2*b^2*f*n)*x*log(c) - (6*b^4*f*m^3*n^4 - 6*a*b^3*f*m^2*n^3 + 3*a^2*b^2*f*m*n^2 - a^3*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. 609 vs. \(2 (155) = 310\).

Time = 2.56 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - 4 a^{3} b m n x + 4 a^{3} b x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} - \frac {12 a^{2} b^{2} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} + \frac {6 a^{2} b^{2} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} + 12 a^{2} b^{2} m^{2} n^{2} x - 12 a^{2} b^{2} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} + 6 a^{2} b^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} + \frac {24 a b^{3} e m^{2} n^{2} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - \frac {12 a b^{3} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} + \frac {4 a b^{3} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3}}{f} - 24 a b^{3} m^{3} n^{3} x + 24 a b^{3} m^{2} n^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} - 12 a b^{3} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} + 4 a b^{3} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3} - \frac {24 b^{4} e m^{3} n^{3} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} + \frac {12 b^{4} e m^{2} n^{2} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} - \frac {4 b^{4} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3}}{f} + \frac {b^{4} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{4}}{f} + 24 b^{4} m^{4} n^{4} x - 24 b^{4} m^{3} n^{3} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} + 12 b^{4} m^{2} n^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} - 4 b^{4} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3} + b^{4} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{4} & \text {for}\: f \neq 0 \\x \left (a + b \log {\left (c \left (d e^{m}\right )^{n} \right )}\right )^{4} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*ln(c*(d*(f*x+e)**m)**n))**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*e*log(c*(d*(e + f*x)**m)**n)/f - 4*a**3*b*m*n*x + 4*a**3*b*x*log(c*(d*(e + f*x)**
m)**n) - 12*a**2*b**2*e*m*n*log(c*(d*(e + f*x)**m)**n)/f + 6*a**2*b**2*e*log(c*(d*(e + f*x)**m)**n)**2/f + 12*
a**2*b**2*m**2*n**2*x - 12*a**2*b**2*m*n*x*log(c*(d*(e + f*x)**m)**n) + 6*a**2*b**2*x*log(c*(d*(e + f*x)**m)**
n)**2 + 24*a*b**3*e*m**2*n**2*log(c*(d*(e + f*x)**m)**n)/f - 12*a*b**3*e*m*n*log(c*(d*(e + f*x)**m)**n)**2/f +
4*a*b**3*e*log(c*(d*(e + f*x)**m)**n)**3/f - 24*a*b**3*m**3*n**3*x + 24*a*b**3*m**2*n**2*x*log(c*(d*(e + f*x)
**m)**n) - 12*a*b**3*m*n*x*log(c*(d*(e + f*x)**m)**n)**2 + 4*a*b**3*x*log(c*(d*(e + f*x)**m)**n)**3 - 24*b**4*
e*m**3*n**3*log(c*(d*(e + f*x)**m)**n)/f + 12*b**4*e*m**2*n**2*log(c*(d*(e + f*x)**m)**n)**2/f - 4*b**4*e*m*n*
log(c*(d*(e + f*x)**m)**n)**3/f + b**4*e*log(c*(d*(e + f*x)**m)**n)**4/f + 24*b**4*m**4*n**4*x - 24*b**4*m**3*
n**3*x*log(c*(d*(e + f*x)**m)**n) + 12*b**4*m**2*n**2*x*log(c*(d*(e + f*x)**m)**n)**2 - 4*b**4*m*n*x*log(c*(d*
(e + f*x)**m)**n)**3 + b**4*x*log(c*(d*(e + f*x)**m)**n)**4, Ne(f, 0)), (x*(a + b*log(c*(d*e**m)**n))**4, True
))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (160) = 320\).

Time = 0.23 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.49 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=b^{4} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{4} - 4 \, a^{3} b f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + 4 \, a b^{3} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{3} + 6 \, a^{2} b^{2} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} + 4 \, a^{3} b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) - 6 \, {\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^{2} b^{2} - 4 \, {\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 )} a b^{3} - {\left (4 \, 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 )^{3} + {\left ({\left (\frac {{\left (e \log \left (f x + e\right )^{4} + 4 \, e \log \left (f x + e\right )^{3} + 12 \, e \log \left (f x + e\right )^{2} - 24 \, f x + 24 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f^{3}} - \frac {4 \, {\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 n \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )}{f^{3}}\right )} f m n + \frac {6 \, {\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 )^{2}}{f^{2}}\right )} f m n\right )} b^{4} + a^{4} x \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^4,x, algorithm="maxima")

[Out]

b^4*x*log(((f*x + e)^m*d)^n*c)^4 - 4*a^3*b*f*m*n*(x/f - e*log(f*x + e)/f^2) + 4*a*b^3*x*log(((f*x + e)^m*d)^n*
c)^3 + 6*a^2*b^2*x*log(((f*x + e)^m*d)^n*c)^2 + 4*a^3*b*x*log(((f*x + e)^m*d)^n*c) - 6*(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^2*b^2 - 4*(
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)*a*b^3 - (4*f*m*n*(x/f - e*log(f*x + e)/f^2)*log(((f*x + e)^m*d)^n*c)^3 + (((e*log(f*x + e)^4 +
4*e*log(f*x + e)^3 + 12*e*log(f*x + e)^2 - 24*f*x + 24*e*log(f*x + e))*m^2*n^2/f^3 - 4*(e*log(f*x + e)^3 + 3*e
*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e))*m*n*log(((f*x + e)^m*d)^n*c)/f^3)*f*m*n + 6*(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)^2/f^2)*f*m*n)*b^4 + a^4*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (160) = 320\).

Time = 0.35 (sec) , antiderivative size = 1697, normalized size of antiderivative = 10.61 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\text {Too large to display} \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^4,x, algorithm="giac")

[Out]

(f*x + e)*b^4*m^4*n^4*log(f*x + e)^4/f - 4*(f*x + e)*b^4*m^4*n^4*log(f*x + e)^3/f + 4*(f*x + e)*b^4*m^3*n^4*lo
g(f*x + e)^3*log(d)/f + 12*(f*x + e)*b^4*m^4*n^4*log(f*x + e)^2/f + 4*(f*x + e)*b^4*m^3*n^3*log(f*x + e)^3*log
(c)/f - 12*(f*x + e)*b^4*m^3*n^4*log(f*x + e)^2*log(d)/f + 6*(f*x + e)*b^4*m^2*n^4*log(f*x + e)^2*log(d)^2/f -
24*(f*x + e)*b^4*m^4*n^4*log(f*x + e)/f + 4*(f*x + e)*a*b^3*m^3*n^3*log(f*x + e)^3/f - 12*(f*x + e)*b^4*m^3*n
^3*log(f*x + e)^2*log(c)/f + 24*(f*x + e)*b^4*m^3*n^4*log(f*x + e)*log(d)/f + 12*(f*x + e)*b^4*m^2*n^3*log(f*x
+ e)^2*log(c)*log(d)/f - 12*(f*x + e)*b^4*m^2*n^4*log(f*x + e)*log(d)^2/f + 4*(f*x + e)*b^4*m*n^4*log(f*x + e
)*log(d)^3/f + 24*(f*x + e)*b^4*m^4*n^4/f - 12*(f*x + e)*a*b^3*m^3*n^3*log(f*x + e)^2/f + 24*(f*x + e)*b^4*m^3
*n^3*log(f*x + e)*log(c)/f + 6*(f*x + e)*b^4*m^2*n^2*log(f*x + e)^2*log(c)^2/f - 24*(f*x + e)*b^4*m^3*n^4*log(
d)/f + 12*(f*x + e)*a*b^3*m^2*n^3*log(f*x + e)^2*log(d)/f - 24*(f*x + e)*b^4*m^2*n^3*log(f*x + e)*log(c)*log(d
)/f + 12*(f*x + e)*b^4*m^2*n^4*log(d)^2/f + 12*(f*x + e)*b^4*m*n^3*log(f*x + e)*log(c)*log(d)^2/f - 4*(f*x + e
)*b^4*m*n^4*log(d)^3/f + (f*x + e)*b^4*n^4*log(d)^4/f + 24*(f*x + e)*a*b^3*m^3*n^3*log(f*x + e)/f - 24*(f*x +
e)*b^4*m^3*n^3*log(c)/f + 12*(f*x + e)*a*b^3*m^2*n^2*log(f*x + e)^2*log(c)/f - 12*(f*x + e)*b^4*m^2*n^2*log(f*
x + e)*log(c)^2/f - 24*(f*x + e)*a*b^3*m^2*n^3*log(f*x + e)*log(d)/f + 24*(f*x + e)*b^4*m^2*n^3*log(c)*log(d)/
f + 12*(f*x + e)*b^4*m*n^2*log(f*x + e)*log(c)^2*log(d)/f + 12*(f*x + e)*a*b^3*m*n^3*log(f*x + e)*log(d)^2/f -
12*(f*x + e)*b^4*m*n^3*log(c)*log(d)^2/f + 4*(f*x + e)*b^4*n^3*log(c)*log(d)^3/f - 24*(f*x + e)*a*b^3*m^3*n^3
/f + 6*(f*x + e)*a^2*b^2*m^2*n^2*log(f*x + e)^2/f - 24*(f*x + e)*a*b^3*m^2*n^2*log(f*x + e)*log(c)/f + 12*(f*x
+ e)*b^4*m^2*n^2*log(c)^2/f + 4*(f*x + e)*b^4*m*n*log(f*x + e)*log(c)^3/f + 24*(f*x + e)*a*b^3*m^2*n^3*log(d)
/f + 24*(f*x + e)*a*b^3*m*n^2*log(f*x + e)*log(c)*log(d)/f - 12*(f*x + e)*b^4*m*n^2*log(c)^2*log(d)/f - 12*(f*
x + e)*a*b^3*m*n^3*log(d)^2/f + 6*(f*x + e)*b^4*n^2*log(c)^2*log(d)^2/f + 4*(f*x + e)*a*b^3*n^3*log(d)^3/f - 1
2*(f*x + e)*a^2*b^2*m^2*n^2*log(f*x + e)/f + 24*(f*x + e)*a*b^3*m^2*n^2*log(c)/f + 12*(f*x + e)*a*b^3*m*n*log(
f*x + e)*log(c)^2/f - 4*(f*x + e)*b^4*m*n*log(c)^3/f + 12*(f*x + e)*a^2*b^2*m*n^2*log(f*x + e)*log(d)/f - 24*(
f*x + e)*a*b^3*m*n^2*log(c)*log(d)/f + 4*(f*x + e)*b^4*n*log(c)^3*log(d)/f + 12*(f*x + e)*a*b^3*n^2*log(c)*log
(d)^2/f + 12*(f*x + e)*a^2*b^2*m^2*n^2/f + 12*(f*x + e)*a^2*b^2*m*n*log(f*x + e)*log(c)/f - 12*(f*x + e)*a*b^3
*m*n*log(c)^2/f + (f*x + e)*b^4*log(c)^4/f - 12*(f*x + e)*a^2*b^2*m*n^2*log(d)/f + 12*(f*x + e)*a*b^3*n*log(c)
^2*log(d)/f + 6*(f*x + e)*a^2*b^2*n^2*log(d)^2/f + 4*(f*x + e)*a^3*b*m*n*log(f*x + e)/f - 12*(f*x + e)*a^2*b^2
*m*n*log(c)/f + 4*(f*x + e)*a*b^3*log(c)^3/f + 12*(f*x + e)*a^2*b^2*n*log(c)*log(d)/f - 4*(f*x + e)*a^3*b*m*n/
f + 6*(f*x + e)*a^2*b^2*log(c)^2/f + 4*(f*x + e)*a^3*b*n*log(d)/f + 4*(f*x + e)*a^3*b*log(c)/f + (f*x + e)*a^4
/f

Mupad [B] (verification not implemented)

Time = 1.84 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.38 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx={\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^3\,\left (\frac {4\,\left (a\,b^3\,e-b^4\,e\,m\,n\right )}{f}+4\,b^3\,x\,\left (a-b\,m\,n\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^4\,\left (b^4\,x+\frac {b^4\,e}{f}\right )+x\,\left (a^4-4\,a^3\,b\,m\,n+12\,a^2\,b^2\,m^2\,n^2-24\,a\,b^3\,m^3\,n^3+24\,b^4\,m^4\,n^4\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^2\,\left (\frac {6\,\left (e\,a^2\,b^2-2\,e\,a\,b^3\,m\,n+2\,e\,b^4\,m^2\,n^2\right )}{f}+6\,b^2\,x\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\right )-\frac {\ln \left (e+f\,x\right )\,\left (-4\,e\,a^3\,b\,m\,n+12\,e\,a^2\,b^2\,m^2\,n^2-24\,e\,a\,b^3\,m^3\,n^3+24\,e\,b^4\,m^4\,n^4\right )}{f}+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\,\left (4\,b\,f\,\left (a^3-3\,a^2\,b\,m\,n+6\,a\,b^2\,m^2\,n^2-6\,b^3\,m^3\,n^3\right )\,x^2+4\,b\,e\,\left (a^3-3\,a^2\,b\,m\,n+6\,a\,b^2\,m^2\,n^2-6\,b^3\,m^3\,n^3\right )\,x\right )}{e+f\,x} \]

[In]

int((a + b*log(c*(d*(e + f*x)^m)^n))^4,x)

[Out]

log(c*(d*(e + f*x)^m)^n)^3*((4*(a*b^3*e - b^4*e*m*n))/f + 4*b^3*x*(a - b*m*n)) + log(c*(d*(e + f*x)^m)^n)^4*(b
^4*x + (b^4*e)/f) + x*(a^4 + 24*b^4*m^4*n^4 - 24*a*b^3*m^3*n^3 - 4*a^3*b*m*n + 12*a^2*b^2*m^2*n^2) + log(c*(d*
(e + f*x)^m)^n)^2*((6*(a^2*b^2*e + 2*b^4*e*m^2*n^2 - 2*a*b^3*e*m*n))/f + 6*b^2*x*(a^2 + 2*b^2*m^2*n^2 - 2*a*b*
m*n)) - (log(e + f*x)*(24*b^4*e*m^4*n^4 - 24*a*b^3*e*m^3*n^3 - 4*a^3*b*e*m*n + 12*a^2*b^2*e*m^2*n^2))/f + (log
(c*(d*(e + f*x)^m)^n)*(4*b*f*x^2*(a^3 - 6*b^3*m^3*n^3 + 6*a*b^2*m^2*n^2 - 3*a^2*b*m*n) + 4*b*e*x*(a^3 - 6*b^3*
m^3*n^3 + 6*a*b^2*m^2*n^2 - 3*a^2*b*m*n)))/(e + f*x)