\(\int (a+b \log (c (d+e x)^n))^4 \, dx\) [17]

Optimal result

Integrand size = 16, antiderivative size = 131 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=-24 a b^3 n^3 x+24 b^4 n^4 x-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2333, 2332} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=-24 a b^3 n^3 x+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \]

[In]

Int[(a + b*Log[c*(d + e*x)^n])^4,x]

[Out]

-24*a*b^3*n^3*x + 24*b^4*n^4*x - (24*b^4*n^3*(d + e*x)*Log[c*(d + e*x)^n])/e + (12*b^2*n^2*(d + e*x)*(a + b*Lo
g[c*(d + e*x)^n])^2)/e - (4*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^
n])^4)/e

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^4 \, dx,x,d+e x\right )}{e} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac {(4 b n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e} \\ & = -\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}+\frac {\left (12 b^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e} \\ & = \frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac {\left (24 b^3 n^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e} \\ & = -24 a b^3 n^3 x+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac {\left (24 b^4 n^3\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e} \\ & = -24 a b^3 n^3 x+24 b^4 n^4 x-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4-4 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )}{e} \]

[In]

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

[Out]

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^4 - 4*b*n*((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3 - 3*b*n*((d + e*x)*(a
+ b*Log[c*(d + e*x)^n])^2 - 2*b*n*(e*(a - b*n)*x + b*(d + e*x)*Log[c*(d + e*x)^n]))))/e

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(524\) vs. \(2(131)=262\).

Time = 1.36 (sec) , antiderivative size = 525, normalized size of antiderivative = 4.01

method	result	size
parallelrisch	\(\frac {-12 x \ln \left (c \left (e x +d \right )^{n}\right ) a^{2} b^{2} e \,n^{2}+4 x \ln \left (c \left (e x +d \right )^{n}\right ) a^{3} b e n +4 x \ln \left (c \left (e x +d \right )^{n}\right )^{3} a \,b^{3} e n -12 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} a \,b^{3} e \,n^{2}+24 x \ln \left (c \left (e x +d \right )^{n}\right ) a \,b^{3} e \,n^{3}+6 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} a^{2} b^{2} e n +24 a \,b^{3} d \,n^{4}-12 a^{2} b^{2} d \,n^{3}+4 a^{3} b d \,n^{2}+\ln \left (c \left (e x +d \right )^{n}\right )^{4} b^{4} d n -4 \ln \left (c \left (e x +d \right )^{n}\right )^{3} b^{4} d \,n^{2}+12 \ln \left (c \left (e x +d \right )^{n}\right )^{2} b^{4} d \,n^{3}-24 \ln \left (c \left (e x +d \right )^{n}\right ) b^{4} d \,n^{4}+x \,a^{4} e n +24 x \,b^{4} e \,n^{5}+12 x \,a^{2} b^{2} e \,n^{3}+4 \ln \left (c \left (e x +d \right )^{n}\right )^{3} a \,b^{3} d n -12 \ln \left (c \left (e x +d \right )^{n}\right )^{2} a \,b^{3} d \,n^{2}+24 \ln \left (c \left (e x +d \right )^{n}\right ) a \,b^{3} d \,n^{3}-4 x \,a^{3} b e \,n^{2}+6 \ln \left (c \left (e x +d \right )^{n}\right )^{2} a^{2} b^{2} d n -12 \ln \left (c \left (e x +d \right )^{n}\right ) a^{2} b^{2} d \,n^{2}+4 \ln \left (c \left (e x +d \right )^{n}\right ) a^{3} b d n +x \ln \left (c \left (e x +d \right )^{n}\right )^{4} b^{4} e n -4 x \ln \left (c \left (e x +d \right )^{n}\right )^{3} b^{4} e \,n^{2}+12 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} b^{4} e \,n^{3}-24 x \ln \left (c \left (e x +d \right )^{n}\right ) b^{4} e \,n^{4}-24 x a \,b^{3} e \,n^{4}-24 b^{4} d \,n^{5}-a^{4} d n}{e n}\)	\(525\)
risch	\(\text {Expression too large to display}\)	\(15871\)

[In]

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

[Out]

(-12*x*ln(c*(e*x+d)^n)*a^2*b^2*e*n^2+4*x*ln(c*(e*x+d)^n)*a^3*b*e*n+4*x*ln(c*(e*x+d)^n)^3*a*b^3*e*n-12*x*ln(c*(
e*x+d)^n)^2*a*b^3*e*n^2+24*x*ln(c*(e*x+d)^n)*a*b^3*e*n^3+6*x*ln(c*(e*x+d)^n)^2*a^2*b^2*e*n+24*a*b^3*d*n^4-12*a
^2*b^2*d*n^3+4*a^3*b*d*n^2+ln(c*(e*x+d)^n)^4*b^4*d*n-4*ln(c*(e*x+d)^n)^3*b^4*d*n^2+12*ln(c*(e*x+d)^n)^2*b^4*d*
n^3-24*ln(c*(e*x+d)^n)*b^4*d*n^4+x*a^4*e*n+24*x*b^4*e*n^5+12*x*a^2*b^2*e*n^3+4*ln(c*(e*x+d)^n)^3*a*b^3*d*n-12*
ln(c*(e*x+d)^n)^2*a*b^3*d*n^2+24*ln(c*(e*x+d)^n)*a*b^3*d*n^3-4*x*a^3*b*e*n^2+6*ln(c*(e*x+d)^n)^2*a^2*b^2*d*n-1
2*ln(c*(e*x+d)^n)*a^2*b^2*d*n^2+4*ln(c*(e*x+d)^n)*a^3*b*d*n+x*ln(c*(e*x+d)^n)^4*b^4*e*n-4*x*ln(c*(e*x+d)^n)^3*
b^4*e*n^2+12*x*ln(c*(e*x+d)^n)^2*b^4*e*n^3-24*x*ln(c*(e*x+d)^n)*b^4*e*n^4-24*x*a*b^3*e*n^4-24*b^4*d*n^5-a^4*d*
n)/e/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (131) = 262\).

Time = 0.30 (sec) , antiderivative size = 614, normalized size of antiderivative = 4.69 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\frac {b^{4} e x \log \left (c\right )^{4} + {\left (b^{4} e n^{4} x + b^{4} d n^{4}\right )} \log \left (e x + d\right )^{4} - 4 \, {\left (b^{4} e n - a b^{3} e\right )} x \log \left (c\right )^{3} - 4 \, {\left (b^{4} d n^{4} - a b^{3} d n^{3} + {\left (b^{4} e n^{4} - a b^{3} e n^{3}\right )} x - {\left (b^{4} e n^{3} x + b^{4} d n^{3}\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{3} + 6 \, {\left (2 \, b^{4} e n^{2} - 2 \, a b^{3} e n + a^{2} b^{2} e\right )} x \log \left (c\right )^{2} + 6 \, {\left (2 \, b^{4} d n^{4} - 2 \, a b^{3} d n^{3} + a^{2} b^{2} d n^{2} + {\left (b^{4} e n^{2} x + b^{4} d n^{2}\right )} \log \left (c\right )^{2} + {\left (2 \, b^{4} e n^{4} - 2 \, a b^{3} e n^{3} + a^{2} b^{2} e n^{2}\right )} x - 2 \, {\left (b^{4} d n^{3} - a b^{3} d n^{2} + {\left (b^{4} e n^{3} - a b^{3} e n^{2}\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{2} - 4 \, {\left (6 \, b^{4} e n^{3} - 6 \, a b^{3} e n^{2} + 3 \, a^{2} b^{2} e n - a^{3} b e\right )} x \log \left (c\right ) + {\left (24 \, b^{4} e n^{4} - 24 \, a b^{3} e n^{3} + 12 \, a^{2} b^{2} e n^{2} - 4 \, a^{3} b e n + a^{4} e\right )} x - 4 \, {\left (6 \, b^{4} d n^{4} - 6 \, a b^{3} d n^{3} + 3 \, a^{2} b^{2} d n^{2} - a^{3} b d n - {\left (b^{4} e n x + b^{4} d n\right )} \log \left (c\right )^{3} + 3 \, {\left (b^{4} d n^{2} - a b^{3} d n + {\left (b^{4} e n^{2} - a b^{3} e n\right )} x\right )} \log \left (c\right )^{2} + {\left (6 \, b^{4} e n^{4} - 6 \, a b^{3} e n^{3} + 3 \, a^{2} b^{2} e n^{2} - a^{3} b e n\right )} x - 3 \, {\left (2 \, b^{4} d n^{3} - 2 \, a b^{3} d n^{2} + a^{2} b^{2} d n + {\left (2 \, b^{4} e n^{3} - 2 \, a b^{3} e n^{2} + a^{2} b^{2} e n\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (126) = 252\).

Time = 0.90 (sec) , antiderivative size = 495, normalized size of antiderivative = 3.78 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 4 a^{3} b n x + 4 a^{3} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {12 a^{2} b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {6 a^{2} b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 12 a^{2} b^{2} n^{2} x - 12 a^{2} b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 6 a^{2} b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {24 a b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {12 a b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {4 a b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 24 a b^{3} n^{3} x + 24 a b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 12 a b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + 4 a b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {24 b^{4} d n^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {12 b^{4} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {4 b^{4} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {b^{4} d \log {\left (c \left (d + e x\right )^{n} \right )}^{4}}{e} + 24 b^{4} n^{4} x - 24 b^{4} n^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} + 12 b^{4} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} - 4 b^{4} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} + b^{4} x \log {\left (c \left (d + e x\right )^{n} \right )}^{4} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (131) = 262\).

Time = 0.21 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.82 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=b^{4} x \log \left ({\left (e x + d\right )}^{n} c\right )^{4} + 4 \, a b^{3} x \log \left ({\left (e x + d\right )}^{n} c\right )^{3} - 4 \, a^{3} b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + 6 \, a^{2} b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 4 \, a^{3} b x \log \left ({\left (e x + d\right )}^{n} c\right ) - 6 \, {\left (2 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} a^{2} b^{2} - 4 \, {\left (3 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} - e n {\left (\frac {{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{2}} - \frac {3 \, {\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{2}}\right )}\right )} a b^{3} - {\left (4 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + {\left (e n {\left (\frac {{\left (d \log \left (e x + d\right )^{4} + 4 \, d \log \left (e x + d\right )^{3} + 12 \, d \log \left (e x + d\right )^{2} - 24 \, e x + 24 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{3}} - \frac {4 \, {\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{3}}\right )} + \frac {6 \, {\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )^{2}}{e^{2}}\right )} e n\right )} b^{4} + a^{4} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 758 vs. \(2 (131) = 262\).

Time = 0.34 (sec) , antiderivative size = 758, normalized size of antiderivative = 5.79 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx=\frac {{\left (e x + d\right )} b^{4} n^{4} \log \left (e x + d\right )^{4}}{e} - \frac {4 \, {\left (e x + d\right )} b^{4} n^{4} \log \left (e x + d\right )^{3}}{e} + \frac {4 \, {\left (e x + d\right )} b^{4} n^{3} \log \left (e x + d\right )^{3} \log \left (c\right )}{e} + \frac {12 \, {\left (e x + d\right )} b^{4} n^{4} \log \left (e x + d\right )^{2}}{e} + \frac {4 \, {\left (e x + d\right )} a b^{3} n^{3} \log \left (e x + d\right )^{3}}{e} - \frac {12 \, {\left (e x + d\right )} b^{4} n^{3} \log \left (e x + d\right )^{2} \log \left (c\right )}{e} + \frac {6 \, {\left (e x + d\right )} b^{4} n^{2} \log \left (e x + d\right )^{2} \log \left (c\right )^{2}}{e} - \frac {24 \, {\left (e x + d\right )} b^{4} n^{4} \log \left (e x + d\right )}{e} - \frac {12 \, {\left (e x + d\right )} a b^{3} n^{3} \log \left (e x + d\right )^{2}}{e} + \frac {24 \, {\left (e x + d\right )} b^{4} n^{3} \log \left (e x + d\right ) \log \left (c\right )}{e} + \frac {12 \, {\left (e x + d\right )} a b^{3} n^{2} \log \left (e x + d\right )^{2} \log \left (c\right )}{e} - \frac {12 \, {\left (e x + d\right )} b^{4} n^{2} \log \left (e x + d\right ) \log \left (c\right )^{2}}{e} + \frac {4 \, {\left (e x + d\right )} b^{4} n \log \left (e x + d\right ) \log \left (c\right )^{3}}{e} + \frac {24 \, {\left (e x + d\right )} b^{4} n^{4}}{e} + \frac {24 \, {\left (e x + d\right )} a b^{3} n^{3} \log \left (e x + d\right )}{e} + \frac {6 \, {\left (e x + d\right )} a^{2} b^{2} n^{2} \log \left (e x + d\right )^{2}}{e} - \frac {24 \, {\left (e x + d\right )} b^{4} n^{3} \log \left (c\right )}{e} - \frac {24 \, {\left (e x + d\right )} a b^{3} n^{2} \log \left (e x + d\right ) \log \left (c\right )}{e} + \frac {12 \, {\left (e x + d\right )} b^{4} n^{2} \log \left (c\right )^{2}}{e} + \frac {12 \, {\left (e x + d\right )} a b^{3} n \log \left (e x + d\right ) \log \left (c\right )^{2}}{e} - \frac {4 \, {\left (e x + d\right )} b^{4} n \log \left (c\right )^{3}}{e} + \frac {{\left (e x + d\right )} b^{4} \log \left (c\right )^{4}}{e} - \frac {24 \, {\left (e x + d\right )} a b^{3} n^{3}}{e} - \frac {12 \, {\left (e x + d\right )} a^{2} b^{2} n^{2} \log \left (e x + d\right )}{e} + \frac {24 \, {\left (e x + d\right )} a b^{3} n^{2} \log \left (c\right )}{e} + \frac {12 \, {\left (e x + d\right )} a^{2} b^{2} n \log \left (e x + d\right ) \log \left (c\right )}{e} - \frac {12 \, {\left (e x + d\right )} a b^{3} n \log \left (c\right )^{2}}{e} + \frac {4 \, {\left (e x + d\right )} a b^{3} \log \left (c\right )^{3}}{e} + \frac {12 \, {\left (e x + d\right )} a^{2} b^{2} n^{2}}{e} + \frac {4 \, {\left (e x + d\right )} a^{3} b n \log \left (e x + d\right )}{e} - \frac {12 \, {\left (e x + d\right )} a^{2} b^{2} n \log \left (c\right )}{e} + \frac {6 \, {\left (e x + d\right )} a^{2} b^{2} \log \left (c\right )^{2}}{e} - \frac {4 \, {\left (e x + d\right )} a^{3} b n}{e} + \frac {4 \, {\left (e x + d\right )} a^{3} b \log \left (c\right )}{e} + \frac {{\left (e x + d\right )} a^{4}}{e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.10 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^4 \, dx={\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {6\,\left (d\,a^2\,b^2-2\,d\,a\,b^3\,n+2\,d\,b^4\,n^2\right )}{e}+6\,b^2\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )\right )+x\,\left (a^4-4\,a^3\,b\,n+12\,a^2\,b^2\,n^2-24\,a\,b^3\,n^3+24\,b^4\,n^4\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^4\,\left (b^4\,x+\frac {b^4\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (\frac {4\,\left (a\,b^3\,d-b^4\,d\,n\right )}{e}+4\,b^3\,x\,\left (a-b\,n\right )\right )-\frac {\ln \left (d+e\,x\right )\,\left (-4\,d\,a^3\,b\,n+12\,d\,a^2\,b^2\,n^2-24\,d\,a\,b^3\,n^3+24\,d\,b^4\,n^4\right )}{e}+4\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right ) \]

[In]

int((a + b*log(c*(d + e*x)^n))^4,x)

[Out]

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