3.55 \(\int (a+b \log (c (d+e x)^n))^3 \, dx\)
Optimal. Leaf size=99 \[ 6 a b^2 n^2 x-\frac{3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \]
[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
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Rubi [A] time = 0.0524308, antiderivative size = 99, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used =
{2389, 2296, 2295} \[ 6 a b^2 n^2 x-\frac{3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \]
Antiderivative was successfully verified.
[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 2389
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 2296
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 2295
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=\frac{\operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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] time = 0.0215956, size = 85, normalized size = 0.86 \[ \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )}{e} \]
Antiderivative was successfully verified.
[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
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Maple [C] time = 0.185, size = 4872, normalized size = 49.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*(e*x+d)^n))^3,x)
[Out]
6*ln(c)*b^3*n^2*x+3*ln(c)*a^2*b*x+3*ln(c)^2*a*b^2*x-3*ln(c)^2*b^3*n*x+ln(c)^3*b^3*x+3/4*b*(4*a^2*e*x-4*b^2*d*n
^2*ln(e*x+d)^2+8*ln(e*x+d)*a*b*d*n-8*ln(e*x+d)*b^2*d*n^2+8*b^2*e*n^2*x-Pi^2*b^2*e*x*csgn(I*c*(e*x+d)^n)^6-Pi^2
*b^2*e*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+2*Pi^2*b^2*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5+4*ln(c)^2*b^2*e*x+4*
I*ln(c)*Pi*b^2*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+4*I*ln(c)*Pi*b^2*e*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^
2+8*ln(c)*a*b*e*x-8*ln(c)*b^2*e*n*x-Pi^2*b^2*e*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+2*Pi^2*b^2*e*x*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5+8*ln(e*x+d)*ln(c)*b^2*d*n+4*I*Pi*a*b*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+4
*I*Pi*a*b*e*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+4*I*ln(e*x+d)*Pi*b^2*d*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2
+4*I*ln(e*x+d)*Pi*b^2*d*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+2*Pi^2*b^2*e*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)^3+2*Pi^2*b^2*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3+4*I*Pi*b^2*e*n*x*csg
n(I*c*(e*x+d)^n)^3-4*I*ln(e*x+d)*Pi*b^2*d*n*csgn(I*c*(e*x+d)^n)^3-4*I*Pi*b^2*e*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^
n)^2-4*I*Pi*b^2*e*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-8*a*b*e*n*x-Pi^2*b^2*e*x*csgn(I*c)^2*csgn(I*(e*x
+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+4*I*Pi*b^2*e*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-4*I*Pi*a*b*e*x
*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-4*I*ln(e*x+d)*Pi*b^2*d*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c
*(e*x+d)^n)-4*I*ln(c)*Pi*b^2*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-4*I*Pi*a*b*e*x*csgn(I*c*(e*x+
d)^n)^3-4*I*ln(c)*Pi*b^2*e*x*csgn(I*c*(e*x+d)^n)^3-4*Pi^2*b^2*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)^4)/e*ln((e*x+d)^n)+3/2*b^2*(-I*Pi*b*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*Pi*b*e*x*csgn(I*
c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*e*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*e*x*csgn(I*c*(e*x+d)^n)^3+2
*ln(c)*b*e*x+2*ln(e*x+d)*b*d*n-2*b*e*n*x+2*a*e*x)/e*ln((e*x+d)^n)^2-3*a^2*b*n*x+6*b^3*d*n^3/e*ln(e*x+d)+a^3*x-
3/4/e*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*c*(e*x+d)^n)^6+6/e*ln(c)*ln(e*x+d)*a*b^2*d*n+3/2*Pi^2*a*b^2*x*csgn(I*c)^2*
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+b^3*x*ln((e*x+d)^n)^3-3/4/e*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*c)^2*csgn(I*
(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+3/2/e*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)^3+3/2/e*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-3/e*ln(e*x+d)*Pi^2*b^3*
d*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4-3*I/e*ln(e*x+d)*ln(c)*Pi*b^3*d*n*csgn(I*c)*csgn(I*(e*x+d
)^n)*csgn(I*c*(e*x+d)^n)-3*I/e*ln(e*x+d)*Pi*a*b^2*d*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+6*a*b^2*
n^2*x+1/e*b^3*d*n^3*ln(e*x+d)^3+3/e*b^3*d*n^3*ln(e*x+d)^2-3/4*ln(c)*Pi^2*b^3*x*csgn(I*c*(e*x+d)^n)^6+3/4*Pi^2*
b^3*n*x*csgn(I*c*(e*x+d)^n)^6-3/4*Pi^2*a*b^2*x*csgn(I*c*(e*x+d)^n)^6-6*ln(c)*a*b^2*n*x+1/8*I*Pi^3*b^3*x*csgn(I
*c*(e*x+d)^n)^9-3/4*ln(c)*Pi^2*b^3*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+3/2*ln(c)*Pi^2*b^3*x*csgn(I*c)*csgn(I*c
*(e*x+d)^n)^5-3/4*ln(c)*Pi^2*b^3*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+3/2*ln(c)*Pi^2*b^3*x*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)^5+3/4*Pi^2*b^3*n*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4-3/2*Pi^2*b^3*n*x*csgn(I*c)*csgn
(I*c*(e*x+d)^n)^5+3/4*Pi^2*b^3*n*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4-3*I/e*ln(e*x+d)*Pi*a*b^2*d*n*csgn
(I*c*(e*x+d)^n)^3-3/2*I/e*Pi*b^3*d*n^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*ln(e*x+d)^2-3/2*I/e*Pi*b^3*d*n^2*csgn(I
*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*ln(e*x+d)^2-3*I/e*ln(e*x+d)*ln(c)*Pi*b^3*d*n*csgn(I*c*(e*x+d)^n)^3-3*I/e*Pi*
b^3*d*n^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*ln(e*x+d)-3*I/e*Pi*b^3*d*n^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2
*ln(e*x+d)+3*I*ln(c)*Pi*b^3*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-3*I*ln(c)*Pi*a*b^2*x*csgn(I*c)
*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+3*I*Pi*a*b^2*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+3*I/e*
Pi*b^3*d*n^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*ln(e*x+d)+3*I/e*ln(e*x+d)*Pi*a*b^2*d*n*csgn(I*c)*
csgn(I*c*(e*x+d)^n)^2+3*I/e*ln(e*x+d)*Pi*a*b^2*d*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-3/4/e*ln(e*x+d)*Pi^
2*b^3*d*n*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+3/2/e*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5-3/4/e
*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+3/2/e*ln(e*x+d)*Pi^2*b^3*d*n*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^5+3/2*I/e*Pi*b^3*d*n^2*csgn(I*c*(e*x+d)^n)^3*ln(e*x+d)^2+3*I/e*Pi*b^3*d*n^2*csgn(I*c*(e
*x+d)^n)^3*ln(e*x+d)-3*I*Pi*a*b^2*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-3*I*Pi*a*b^2*n*x*csgn(I*(e*x+d)^n)*csgn(
I*c*(e*x+d)^n)^2-3/2*I*Pi*a^2*b*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-3/2*I*ln(c)^2*Pi*b^3*x*csgn(
I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-3*I*ln(c)*Pi*b^3*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-3*I*ln(c)*Pi*b
^3*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-3*I*Pi*b^3*n^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)+3*I*ln(c)*Pi*a*b^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*ln(c)*Pi*a*b^2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)^2-1/8*I*Pi^3*b^3*x*csgn(I*(e*x+d)^n)^3*csgn(I*c*(e*x+d)^n)^6+3/8*I*Pi^3*b^3*x*csgn(I*(e*x+d)^n)^2*csgn(I*c
*(e*x+d)^n)^7-3/8*I*Pi^3*b^3*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^8-3/2*I*ln(c)^2*Pi*b^3*x*csgn(I*c*(e*x+d)
^n)^3-3*I*Pi*b^3*n^2*x*csgn(I*c*(e*x+d)^n)^3-3/2*I*Pi*a^2*b*x*csgn(I*c*(e*x+d)^n)^3-3/2*Pi^2*b^3*n*x*csgn(I*(e
*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-3/4*Pi^2*a*b^2*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+3/2*Pi^2*a*b^2*x*csgn(I*c)*c
sgn(I*c*(e*x+d)^n)^5-3/4*Pi^2*a*b^2*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+3/2*Pi^2*a*b^2*x*csgn(I*(e*x+d
)^n)*csgn(I*c*(e*x+d)^n)^5-3/e*ln(c)*b^3*d*n^2*ln(e*x+d)^2+3/e*ln(c)^2*ln(e*x+d)*b^3*d*n-6/e*ln(c)*ln(e*x+d)*b
^3*d*n^2-3/e*a*b^2*d*n^2*ln(e*x+d)^2-6/e*ln(e*x+d)*a*b^2*d*n^2+3/e*ln(e*x+d)*a^2*b*d*n-1/8*I*Pi^3*b^3*x*csgn(I
*c)^3*csgn(I*c*(e*x+d)^n)^6+3/8*I*Pi^3*b^3*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^7-3/8*I*Pi^3*b^3*x*csgn(I*c)*csgn
(I*c*(e*x+d)^n)^8-6*b^3*n^3*x+3*I*ln(c)*Pi*b^3*n*x*csgn(I*c*(e*x+d)^n)^3+3*I*Pi*b^3*n^2*x*csgn(I*c)*csgn(I*c*(
e*x+d)^n)^2+3*I*Pi*b^3*n^2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-3*I*ln(c)*Pi*a*b^2*x*csgn(I*c*(e*x+d)^n)^
3+3*I*Pi*a*b^2*n*x*csgn(I*c*(e*x+d)^n)^3+3/2*I*Pi*a^2*b*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3/2*I*Pi*a^2*b*x*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/8*I*Pi^3*b^3*x*csgn(I*c)^3*csgn(I*(e*x+d)^n)^3*csgn(I*c*(e*x+d)^n)^3-3/
8*I*Pi^3*b^3*x*csgn(I*c)^3*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+3/8*I*Pi^3*b^3*x*csgn(I*c)^3*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)^5-3/8*I*Pi^3*b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^3*csgn(I*c*(e*x+d)^n)^4+9/8*I*Pi^3*
b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^5-9/8*I*Pi^3*b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csg
n(I*c*(e*x+d)^n)^6+3/8*I*Pi^3*b^3*x*csgn(I*c)*csgn(I*(e*x+d)^n)^3*csgn(I*c*(e*x+d)^n)^5-9/8*I*Pi^3*b^3*x*csgn(
I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^6+9/8*I*Pi^3*b^3*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)^7+3/2*Pi^2*a*b^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-3*Pi^2*a*b^2*x*csgn(I*c)*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)^4-3/4*ln(c)*Pi^2*b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+3/2*ln(
c)*Pi^2*b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+3/2*ln(c)*Pi^2*b^3*x*csgn(I*c)*csgn(I*(e*x+d
)^n)^2*csgn(I*c*(e*x+d)^n)^3-3*ln(c)*Pi^2*b^3*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4+3/4*Pi^2*b^3
*n*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2-3/2*Pi^2*b^3*n*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^3-3/2*Pi^2*b^3*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3+3*Pi^2*b^3*n*x*csgn(I*c
)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4-3/4*Pi^2*a*b^2*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)
^2+3/2*I*ln(c)^2*Pi*b^3*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3/2*I*ln(c)^2*Pi*b^3*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e
*x+d)^n)^2+3/2*I/e*Pi*b^3*d*n^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*ln(e*x+d)^2+3*I/e*ln(e*x+d)*ln
(c)*Pi*b^3*d*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I/e*ln(e*x+d)*ln(c)*Pi*b^3*d*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*
x+d)^n)^2
________________________________________________________________________________________
Maxima [B] time = 1.29702, size = 381, normalized size = 3.85 \begin{align*} 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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
________________________________________________________________________________________
Fricas [B] time = 2.01939, size = 699, normalized size = 7.06 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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 [A] time = 3.26153, size = 527, normalized size = 5.32 \begin{align*} \begin{cases} a^{3} x + \frac{3 a^{2} b d n \log{\left (d + e x \right )}}{e} + 3 a^{2} b n x \log{\left (d + e x \right )} - 3 a^{2} b n x + 3 a^{2} b x \log{\left (c \right )} + \frac{3 a b^{2} d n^{2} \log{\left (d + e x \right )}^{2}}{e} - \frac{6 a b^{2} d n^{2} \log{\left (d + e x \right )}}{e} + \frac{6 a b^{2} d n \log{\left (c \right )} \log{\left (d + e x \right )}}{e} + 3 a b^{2} n^{2} x \log{\left (d + e x \right )}^{2} - 6 a b^{2} n^{2} x \log{\left (d + e x \right )} + 6 a b^{2} n^{2} x + 6 a b^{2} n x \log{\left (c \right )} \log{\left (d + e x \right )} - 6 a b^{2} n x \log{\left (c \right )} + 3 a b^{2} x \log{\left (c \right )}^{2} + \frac{b^{3} d n^{3} \log{\left (d + e x \right )}^{3}}{e} - \frac{3 b^{3} d n^{3} \log{\left (d + e x \right )}^{2}}{e} + \frac{6 b^{3} d n^{3} \log{\left (d + e x \right )}}{e} + \frac{3 b^{3} d n^{2} \log{\left (c \right )} \log{\left (d + e x \right )}^{2}}{e} - \frac{6 b^{3} d n^{2} \log{\left (c \right )} \log{\left (d + e x \right )}}{e} + \frac{3 b^{3} d n \log{\left (c \right )}^{2} \log{\left (d + e x \right )}}{e} + b^{3} n^{3} x \log{\left (d + e x \right )}^{3} - 3 b^{3} n^{3} x \log{\left (d + e x \right )}^{2} + 6 b^{3} n^{3} x \log{\left (d + e x \right )} - 6 b^{3} n^{3} x + 3 b^{3} n^{2} x \log{\left (c \right )} \log{\left (d + e x \right )}^{2} - 6 b^{3} n^{2} x \log{\left (c \right )} \log{\left (d + e x \right )} + 6 b^{3} n^{2} x \log{\left (c \right )} + 3 b^{3} n x \log{\left (c \right )}^{2} \log{\left (d + e x \right )} - 3 b^{3} n x \log{\left (c \right )}^{2} + b^{3} x \log{\left (c \right )}^{3} & \text{for}\: e \neq 0 \\x \left (a + b \log{\left (c d^{n} \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*ln(c*(e*x+d)**n))**3,x)
[Out]
Piecewise((a**3*x + 3*a**2*b*d*n*log(d + e*x)/e + 3*a**2*b*n*x*log(d + e*x) - 3*a**2*b*n*x + 3*a**2*b*x*log(c)
+ 3*a*b**2*d*n**2*log(d + e*x)**2/e - 6*a*b**2*d*n**2*log(d + e*x)/e + 6*a*b**2*d*n*log(c)*log(d + e*x)/e + 3
*a*b**2*n**2*x*log(d + e*x)**2 - 6*a*b**2*n**2*x*log(d + e*x) + 6*a*b**2*n**2*x + 6*a*b**2*n*x*log(c)*log(d +
e*x) - 6*a*b**2*n*x*log(c) + 3*a*b**2*x*log(c)**2 + b**3*d*n**3*log(d + e*x)**3/e - 3*b**3*d*n**3*log(d + e*x)
**2/e + 6*b**3*d*n**3*log(d + e*x)/e + 3*b**3*d*n**2*log(c)*log(d + e*x)**2/e - 6*b**3*d*n**2*log(c)*log(d + e
*x)/e + 3*b**3*d*n*log(c)**2*log(d + e*x)/e + b**3*n**3*x*log(d + e*x)**3 - 3*b**3*n**3*x*log(d + e*x)**2 + 6*
b**3*n**3*x*log(d + e*x) - 6*b**3*n**3*x + 3*b**3*n**2*x*log(c)*log(d + e*x)**2 - 6*b**3*n**2*x*log(c)*log(d +
e*x) + 6*b**3*n**2*x*log(c) + 3*b**3*n*x*log(c)**2*log(d + e*x) - 3*b**3*n*x*log(c)**2 + b**3*x*log(c)**3, Ne
(e, 0)), (x*(a + b*log(c*d**n))**3, True))
________________________________________________________________________________________
Giac [B] time = 1.36014, size = 552, normalized size = 5.58 \begin{align*}{\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} - 3 \,{\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 3 \,{\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) + 6 \,{\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) + 3 \,{\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 6 \,{\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 3 \,{\left (x e + d\right )} b^{3} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} - 6 \,{\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} - 6 \,{\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 6 \,{\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (c\right ) + 6 \,{\left (x e + d\right )} a b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) - 3 \,{\left (x e + d\right )} b^{3} n e^{\left (-1\right )} \log \left (c\right )^{2} +{\left (x e + d\right )} b^{3} e^{\left (-1\right )} \log \left (c\right )^{3} + 6 \,{\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} + 3 \,{\left (x e + d\right )} a^{2} b n e^{\left (-1\right )} \log \left (x e + d\right ) - 6 \,{\left (x e + d\right )} a b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 3 \,{\left (x e + d\right )} a b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 3 \,{\left (x e + d\right )} a^{2} b n e^{\left (-1\right )} + 3 \,{\left (x e + d\right )} a^{2} b e^{\left (-1\right )} \log \left (c\right ) +{\left (x e + d\right )} a^{3} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")
[Out]
(x*e + d)*b^3*n^3*e^(-1)*log(x*e + d)^3 - 3*(x*e + d)*b^3*n^3*e^(-1)*log(x*e + d)^2 + 3*(x*e + d)*b^3*n^2*e^(-
1)*log(x*e + d)^2*log(c) + 6*(x*e + d)*b^3*n^3*e^(-1)*log(x*e + d) + 3*(x*e + d)*a*b^2*n^2*e^(-1)*log(x*e + d)
^2 - 6*(x*e + d)*b^3*n^2*e^(-1)*log(x*e + d)*log(c) + 3*(x*e + d)*b^3*n*e^(-1)*log(x*e + d)*log(c)^2 - 6*(x*e
+ d)*b^3*n^3*e^(-1) - 6*(x*e + d)*a*b^2*n^2*e^(-1)*log(x*e + d) + 6*(x*e + d)*b^3*n^2*e^(-1)*log(c) + 6*(x*e +
d)*a*b^2*n*e^(-1)*log(x*e + d)*log(c) - 3*(x*e + d)*b^3*n*e^(-1)*log(c)^2 + (x*e + d)*b^3*e^(-1)*log(c)^3 + 6
*(x*e + d)*a*b^2*n^2*e^(-1) + 3*(x*e + d)*a^2*b*n*e^(-1)*log(x*e + d) - 6*(x*e + d)*a*b^2*n*e^(-1)*log(c) + 3*
(x*e + d)*a*b^2*e^(-1)*log(c)^2 - 3*(x*e + d)*a^2*b*n*e^(-1) + 3*(x*e + d)*a^2*b*e^(-1)*log(c) + (x*e + d)*a^3
*e^(-1)