\(\int (a+b \log (c (d+e x)^n))^3 \, dx\) [18]

Optimal result

Integrand size = 16, antiderivative size = 99 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2333, 2332} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \]

[In]

Int[(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )}{e} \]

[In]

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

[Out]

((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. \(321\) vs. \(2(99)=198\).

Time = 0.62 (sec) , antiderivative size = 322, normalized size of antiderivative = 3.25

method	result	size
parallelrisch	\(\frac {x \ln \left (c \left (e x +d \right )^{n}\right )^{3} b^{3} e n -3 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} b^{3} e \,n^{2}+6 x \ln \left (c \left (e x +d \right )^{n}\right ) b^{3} e \,n^{3}-6 x \,b^{3} e \,n^{4}+3 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} a \,b^{2} e n -6 x \ln \left (c \left (e x +d \right )^{n}\right ) a \,b^{2} e \,n^{2}+6 x a \,b^{2} e \,n^{3}+\ln \left (c \left (e x +d \right )^{n}\right )^{3} b^{3} d n -3 \ln \left (c \left (e x +d \right )^{n}\right )^{2} b^{3} d \,n^{2}+6 \ln \left (c \left (e x +d \right )^{n}\right ) b^{3} d \,n^{3}+6 b^{3} d \,n^{4}+3 x \ln \left (c \left (e x +d \right )^{n}\right ) a^{2} b e n -3 x \,a^{2} b e \,n^{2}+3 \ln \left (c \left (e x +d \right )^{n}\right )^{2} a \,b^{2} d n -6 \ln \left (c \left (e x +d \right )^{n}\right ) a \,b^{2} d \,n^{2}-6 a \,b^{2} d \,n^{3}+x \,a^{3} e n +3 \ln \left (c \left (e x +d \right )^{n}\right ) a^{2} b d n +3 a^{2} b d \,n^{2}-a^{3} d n}{e n}\)	\(322\)
risch	\(\text {Expression too large to display}\)	\(4872\)

[In]

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

[Out]

(x*ln(c*(e*x+d)^n)^3*b^3*e*n-3*x*ln(c*(e*x+d)^n)^2*b^3*e*n^2+6*x*ln(c*(e*x+d)^n)*b^3*e*n^3-6*x*b^3*e*n^4+3*x*l
n(c*(e*x+d)^n)^2*a*b^2*e*n-6*x*ln(c*(e*x+d)^n)*a*b^2*e*n^2+6*x*a*b^2*e*n^3+ln(c*(e*x+d)^n)^3*b^3*d*n-3*ln(c*(e
*x+d)^n)^2*b^3*d*n^2+6*ln(c*(e*x+d)^n)*b^3*d*n^3+6*b^3*d*n^4+3*x*ln(c*(e*x+d)^n)*a^2*b*e*n-3*x*a^2*b*e*n^2+3*l
n(c*(e*x+d)^n)^2*a*b^2*d*n-6*ln(c*(e*x+d)^n)*a*b^2*d*n^2-6*a*b^2*d*n^3+x*a^3*e*n+3*ln(c*(e*x+d)^n)*a^2*b*d*n+3
*a^2*b*d*n^2-a^3*d*n)/e/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (99) = 198\).

Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.27 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {b^{3} e x \log \left (c\right )^{3} + {\left (b^{3} e n^{3} x + b^{3} d n^{3}\right )} \log \left (e x + d\right )^{3} - 3 \, {\left (b^{3} e n - a b^{2} e\right )} x \log \left (c\right )^{2} - 3 \, {\left (b^{3} d n^{3} - a b^{2} d n^{2} + {\left (b^{3} e n^{3} - a b^{2} e n^{2}\right )} x - {\left (b^{3} e n^{2} x + b^{3} d n^{2}\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{2} + 3 \, {\left (2 \, b^{3} e n^{2} - 2 \, a b^{2} e n + a^{2} b e\right )} x \log \left (c\right ) - {\left (6 \, b^{3} e n^{3} - 6 \, a b^{2} e n^{2} + 3 \, a^{2} b e n - a^{3} e\right )} x + 3 \, {\left (2 \, b^{3} d n^{3} - 2 \, a b^{2} d n^{2} + a^{2} b d n + {\left (b^{3} e n x + b^{3} d n\right )} \log \left (c\right )^{2} + {\left (2 \, b^{3} e n^{3} - 2 \, a b^{2} e n^{2} + a^{2} b e n\right )} x - 2 \, {\left (b^{3} d n^{2} - a b^{2} d n + {\left (b^{3} e n^{2} - a b^{2} e n\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \]

[In]

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

[Out]

(b^3*e*x*log(c)^3 + (b^3*e*n^3*x + b^3*d*n^3)*log(e*x + d)^3 - 3*(b^3*e*n - a*b^2*e)*x*log(c)^2 - 3*(b^3*d*n^3
- a*b^2*d*n^2 + (b^3*e*n^3 - a*b^2*e*n^2)*x - (b^3*e*n^2*x + b^3*d*n^2)*log(c))*log(e*x + d)^2 + 3*(2*b^3*e*n
^2 - 2*a*b^2*e*n + a^2*b*e)*x*log(c) - (6*b^3*e*n^3 - 6*a*b^2*e*n^2 + 3*a^2*b*e*n - a^3*e)*x + 3*(2*b^3*d*n^3
- 2*a*b^2*d*n^2 + a^2*b*d*n + (b^3*e*n*x + b^3*d*n)*log(c)^2 + (2*b^3*e*n^3 - 2*a*b^2*e*n^2 + a^2*b*e*n)*x - 2
*(b^3*d*n^2 - a*b^2*d*n + (b^3*e*n^2 - a*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. 294 vs. \(2 (95) = 190\).

Time = 0.50 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.97 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 3 a^{2} b n x + 3 a^{2} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {6 a b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 6 a b^{2} n^{2} x - 6 a b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {6 b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {3 b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 6 b^{3} n^{3} x + 6 b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 3 b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (99) = 198\).

Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.85 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=b^{3} x \log \left ({\left (e x + d\right )}^{n} c\right )^{3} - 3 \, a^{2} b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + 3 \, a b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b x \log \left ({\left (e x + d\right )}^{n} c\right ) - 3 \, {\left (2 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} a b^{2} - {\left (3 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} - e n {\left (\frac {{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{2}} - \frac {3 \, {\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{2}}\right )}\right )} b^{3} + a^{3} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (99) = 198\).

Time = 0.31 (sec) , antiderivative size = 399, normalized size of antiderivative = 4.03 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {{\left (e x + d\right )} b^{3} n^{3} \log \left (e x + d\right )^{3}}{e} - \frac {3 \, {\left (e x + d\right )} b^{3} n^{3} \log \left (e x + d\right )^{2}}{e} + \frac {3 \, {\left (e x + d\right )} b^{3} n^{2} \log \left (e x + d\right )^{2} \log \left (c\right )}{e} + \frac {6 \, {\left (e x + d\right )} b^{3} n^{3} \log \left (e x + d\right )}{e} + \frac {3 \, {\left (e x + d\right )} a b^{2} n^{2} \log \left (e x + d\right )^{2}}{e} - \frac {6 \, {\left (e x + d\right )} b^{3} n^{2} \log \left (e x + d\right ) \log \left (c\right )}{e} + \frac {3 \, {\left (e x + d\right )} b^{3} n \log \left (e x + d\right ) \log \left (c\right )^{2}}{e} - \frac {6 \, {\left (e x + d\right )} b^{3} n^{3}}{e} - \frac {6 \, {\left (e x + d\right )} a b^{2} n^{2} \log \left (e x + d\right )}{e} + \frac {6 \, {\left (e x + d\right )} b^{3} n^{2} \log \left (c\right )}{e} + \frac {6 \, {\left (e x + d\right )} a b^{2} n \log \left (e x + d\right ) \log \left (c\right )}{e} - \frac {3 \, {\left (e x + d\right )} b^{3} n \log \left (c\right )^{2}}{e} + \frac {{\left (e x + d\right )} b^{3} \log \left (c\right )^{3}}{e} + \frac {6 \, {\left (e x + d\right )} a b^{2} n^{2}}{e} + \frac {3 \, {\left (e x + d\right )} a^{2} b n \log \left (e x + d\right )}{e} - \frac {6 \, {\left (e x + d\right )} a b^{2} n \log \left (c\right )}{e} + \frac {3 \, {\left (e x + d\right )} a b^{2} \log \left (c\right )^{2}}{e} - \frac {3 \, {\left (e x + d\right )} a^{2} b n}{e} + \frac {3 \, {\left (e x + d\right )} a^{2} b \log \left (c\right )}{e} + \frac {{\left (e x + d\right )} a^{3}}{e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.38 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=x\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {3\,\left (a\,b^2\,d-b^3\,d\,n\right )}{e}+3\,b^2\,x\,\left (a-b\,n\right )\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,d\,a^2\,b\,n-6\,d\,a\,b^2\,n^2+6\,d\,b^3\,n^3\right )}{e}+3\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right ) \]

[In]

int((a + b*log(c*(d + e*x)^n))^3,x)

[Out]

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