3.400 \(\int (d+e x^r)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=169 \[ \frac{3 d^2 e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac{3 d e^2 x^{2 r+1} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+\frac{e^3 x^{3 r+1} \left (a+b \log \left (c x^n\right )\right )}{3 r+1}-\frac{3 b d^2 e n x^{r+1}}{(r+1)^2}-b d^3 n x-\frac{3 b d e^2 n x^{2 r+1}}{(2 r+1)^2}-\frac{b e^3 n x^{3 r+1}}{(3 r+1)^2} \]

[Out]

-(b*d^3*n*x) - (3*b*d^2*e*n*x^(1 + r))/(1 + r)^2 - (3*b*d*e^2*n*x^(1 + 2*r))/(1 + 2*r)^2 - (b*e^3*n*x^(1 + 3*r
))/(1 + 3*r)^2 + d^3*x*(a + b*Log[c*x^n]) + (3*d^2*e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r) + (3*d*e^2*x^(1 + 2
*r)*(a + b*Log[c*x^n]))/(1 + 2*r) + (e^3*x^(1 + 3*r)*(a + b*Log[c*x^n]))/(1 + 3*r)

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Rubi [A]  time = 0.10286, antiderivative size = 141, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {244, 2313} \[ \left (\frac{3 d^2 e x^{r+1}}{r+1}+d^3 x+\frac{3 d e^2 x^{2 r+1}}{2 r+1}+\frac{e^3 x^{3 r+1}}{3 r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+1}}{(r+1)^2}-b d^3 n x-\frac{3 b d e^2 n x^{2 r+1}}{(2 r+1)^2}-\frac{b e^3 n x^{3 r+1}}{(3 r+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^3*n*x) - (3*b*d^2*e*n*x^(1 + r))/(1 + r)^2 - (3*b*d*e^2*n*x^(1 + 2*r))/(1 + 2*r)^2 - (b*e^3*n*x^(1 + 3*r
))/(1 + 3*r)^2 + (d^3*x + (3*d^2*e*x^(1 + r))/(1 + r) + (3*d*e^2*x^(1 + 2*r))/(1 + 2*r) + (e^3*x^(1 + 3*r))/(1
+ 3*r))*(a + b*Log[c*x^n])

Rule 244

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n},
x] && IGtQ[p, 0]

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\left (d^3 x+\frac{3 d^2 e x^{1+r}}{1+r}+\frac{3 d e^2 x^{1+2 r}}{1+2 r}+\frac{e^3 x^{1+3 r}}{1+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d^3+\frac{3 d^2 e x^r}{1+r}+\frac{3 d e^2 x^{2 r}}{1+2 r}+\frac{e^3 x^{3 r}}{1+3 r}\right ) \, dx\\ &=-b d^3 n x-\frac{3 b d^2 e n x^{1+r}}{(1+r)^2}-\frac{3 b d e^2 n x^{1+2 r}}{(1+2 r)^2}-\frac{b e^3 n x^{1+3 r}}{(1+3 r)^2}+\left (d^3 x+\frac{3 d^2 e x^{1+r}}{1+r}+\frac{3 d e^2 x^{1+2 r}}{1+2 r}+\frac{e^3 x^{1+3 r}}{1+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.221951, size = 159, normalized size = 0.94 \[ x \left (\frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r+1}+a d^3+b d^3 \log \left (c x^n\right )-\frac{3 b d^2 e n x^r}{(r+1)^2}-b d^3 n-\frac{3 b d e^2 n x^{2 r}}{(2 r+1)^2}-\frac{b e^3 n x^{3 r}}{(3 r+1)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(a*d^3 - b*d^3*n - (3*b*d^2*e*n*x^r)/(1 + r)^2 - (3*b*d*e^2*n*x^(2*r))/(1 + 2*r)^2 - (b*e^3*n*x^(3*r))/(1 +
3*r)^2 + b*d^3*Log[c*x^n] + (3*d^2*e*x^r*(a + b*Log[c*x^n]))/(1 + r) + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(1
+ 2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(1 + 3*r))

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Maple [C]  time = 0.386, size = 4023, normalized size = 23.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*(2*e^3*r^2*(x^r)^3+9*d*e^2*r^2*(x^r)^2+3*e^3*r*(x^r)^3+6*d^3*r^3+18*d^2*e*r^2*x^r+12*d*e^2*r*(x^r)^2+e^3*(
x^r)^3+11*d^3*r^2+15*d^2*e*r*x^r+3*d*e^2*(x^r)^2+6*d^3*r+3*d^2*e*x^r+d^3)/(1+3*r)/(1+2*r)/(1+r)*ln(x^n)-1/2*x*
(-2*a*d^3+291*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+3*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-
2*a*e^3*(x^r)^3+I*Pi*b*d^3*csgn(I*c*x^n)^3-2*ln(c)*b*d^3-141*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-30
*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-30*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+72*b*d
^3*n*r^6+264*b*d^3*n*r^5+386*b*d^3*n*r^4-72*a*d^3*r^6-264*a*d^3*r^5-386*a*d^3*r^4+9*I*Pi*b*e^3*r*csgn(I*c*x^n)
^3*(x^r)^3-36*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)+3*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+51*I*Pi*b*e^3*r^3*cs
gn(I*c*x^n)^3*(x^r)^3+114*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-24*a*e^3*r^5*(x^r)^3-80
*a*e^3*r^4*(x^r)^3-6*a*d*e^2*(x^r)^2-6*a*d^2*e*x^r+2*b*e^3*n*(x^r)^3-102*a*e^3*r^3*(x^r)^3-62*a*e^3*r^2*(x^r)^
3-18*a*e^3*r*(x^r)^3-2*ln(c)*b*e^3*(x^r)^3-12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+108*I*Pi*b*d^
2*e*r^5*csgn(I*c*x^n)^3*x^r+12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3+288*b*d^3*n*r^3+116*b*d^3*n*r^2+24*b*d^3
*n*r-72*ln(c)*b*d^3*r^6-264*ln(c)*b*d^3*r^5-386*ln(c)*b*d^3*r^4-288*ln(c)*b*d^3*r^3-116*ln(c)*b*d^3*r^2-24*ln(
c)*b*d^3*r-288*a*d^3*r^3-116*a*d^3*r^2-24*a*d^3*r-60*a*d*e^2*r*(x^r)^2-582*a*d^2*e*r^3*x^r-282*a*d^2*e*r^2*x^r
-66*a*d^2*e*r*x^r+51*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+2*b*d^3*n-288*I*Pi*b*d^2*e*r^4
*csgn(I*c*x^n)^2*csgn(I*c)*x^r+40*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-171*I*Pi*b*d*e^2*
r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-171*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-80*ln(c)*b*e^3*
r^4*(x^r)^3-102*ln(c)*b*e^3*r^3*(x^r)^3-62*ln(c)*b*e^3*r^2*(x^r)^3-18*ln(c)*b*e^3*r*(x^r)^3-6*ln(c)*b*d^2*e*x^
r-6*ln(c)*b*d*e^2*(x^r)^2+26*b*e^3*n*r^2*(x^r)^3+12*b*e^3*n*r*(x^r)^3+6*b*d*e^2*n*(x^r)^2+6*b*d^2*e*n*x^r-408*
a*d*e^2*r^3*(x^r)^2-228*a*d*e^2*r^2*(x^r)^2+8*b*e^3*n*r^4*(x^r)^3+24*b*e^3*n*r^3*(x^r)^3-108*a*d*e^2*r^5*(x^r)
^2-342*a*d*e^2*r^4*(x^r)^2-216*a*d^2*e*r^5*x^r-576*a*d^2*e*r^4*x^r-24*ln(c)*b*e^3*r^5*(x^r)^3+132*b*d*e^2*n*r^
2*(x^r)^2+222*b*d^2*e*n*r^2*x^r-58*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)-40*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*
csgn(I*c)*(x^r)^3+171*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2+288*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r-40*I*P
i*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+141*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-9*I*Pi*b*e^3*r*csgn(I
*c*x^n)^2*csgn(I*c)*(x^r)^3+291*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r+114*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r
)^2-9*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-288*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+14
4*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3+58*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+48*b*d*e^2*n*r*(x^r)^2+60*b*d^2*e*n*r*x^r+5
4*b*d*e^2*n*r^4*(x^r)^2+144*b*d*e^2*n*r^3*(x^r)^2+216*b*d^2*e*n*r^4*x^r+360*b*d^2*e*n*r^3*x^r-108*ln(c)*b*d*e^
2*r^5*(x^r)^2-342*ln(c)*b*d*e^2*r^4*(x^r)^2-216*ln(c)*b*d^2*e*r^5*x^r-576*ln(c)*b*d^2*e*r^4*x^r-33*I*Pi*b*d^2*
e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-582*ln(c)*b*d^2*e*r^3*x^r-2
82*ln(c)*b*d^2*e*r^2*x^r-66*ln(c)*b*d^2*e*r*x^r-408*ln(c)*b*d*e^2*r^3*(x^r)^2-228*ln(c)*b*d*e^2*r^2*(x^r)^2-60
*ln(c)*b*d*e^2*r*(x^r)^2+33*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+204*I*Pi*b*d*e^2*r^3*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-58*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-204*I*Pi*b*d*e^2*r^3*csgn(
I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-141*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-108*I*Pi*b*d^2*e*r^5*csgn(
I*c*x^n)^2*csgn(I*c)*x^r-108*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-291*I*Pi*b*d^2*e*r^3*csgn(I*x^n)
*csgn(I*c*x^n)^2*x^r-291*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-114*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(
I*c*x^n)^2*(x^r)^2-132*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-132*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)
-12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+58*I*Pi*b*d^3
*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-204*I*Pi*b*d*e
^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+31*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-I*Pi*b*
e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-12*I*Pi*b*d^3*r*csgn(I*c*
x^n)^2*csgn(I*c)+40*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-36*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+31*I*
Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+141*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+30*I*Pi*b*d*
e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+30*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+33*I*Pi*b*d^2*e*r*
csgn(I*c*x^n)^3*x^r-3*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-31*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(
I*c)*(x^r)^3+204*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+144*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)+36*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
*(x^r)^3-54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+12*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)+193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+54*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*(x^r)^2+171*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+288*I*Pi*b*d^2*e*r^4*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*x^r+12*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*b*e^3*
csgn(I*c*x^n)^3*(x^r)^3-12*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-144*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^
n)^2-144*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)+36*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3-I*Pi*b*d^3*csgn(I*x^n)*csg
n(I*c*x^n)^2+108*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+132*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-3
*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-3*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-3*I*Pi*b*d^2*e*
csgn(I*c*x^n)^2*csgn(I*c)*x^r+3*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-51*I*Pi*b*e^3*r^3*csgn(I*
x^n)*csgn(I*c*x^n)^2*(x^r)^3-51*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-31*I*Pi*b*e^3*r^2*csgn(I*x^n)
*csgn(I*c*x^n)^2*(x^r)^3-33*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+3*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)*(x^r)^2+132*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+193*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^
3-193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-193*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-54*I*Pi*b*d*e^2*
r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-114*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+9*I*Pi*b*e^3*r*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3)/(1+3*r)^2/(1+2*r)^2/(1+r)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.4291, size = 2237, normalized size = 13.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((36*b*d^3*r^6 + 132*b*d^3*r^5 + 193*b*d^3*r^4 + 144*b*d^3*r^3 + 58*b*d^3*r^2 + 12*b*d^3*r + b*d^3)*x*log(c) +
(36*b*d^3*n*r^6 + 132*b*d^3*n*r^5 + 193*b*d^3*n*r^4 + 144*b*d^3*n*r^3 + 58*b*d^3*n*r^2 + 12*b*d^3*n*r + b*d^3
*n)*x*log(x) - (36*(b*d^3*n - a*d^3)*r^6 + 132*(b*d^3*n - a*d^3)*r^5 + b*d^3*n + 193*(b*d^3*n - a*d^3)*r^4 - a
*d^3 + 144*(b*d^3*n - a*d^3)*r^3 + 58*(b*d^3*n - a*d^3)*r^2 + 12*(b*d^3*n - a*d^3)*r)*x + ((12*b*e^3*r^5 + 40*
b*e^3*r^4 + 51*b*e^3*r^3 + 31*b*e^3*r^2 + 9*b*e^3*r + b*e^3)*x*log(c) + (12*b*e^3*n*r^5 + 40*b*e^3*n*r^4 + 51*
b*e^3*n*r^3 + 31*b*e^3*n*r^2 + 9*b*e^3*n*r + b*e^3*n)*x*log(x) + (12*a*e^3*r^5 - b*e^3*n - 4*(b*e^3*n - 10*a*e
^3)*r^4 + a*e^3 - 3*(4*b*e^3*n - 17*a*e^3)*r^3 - (13*b*e^3*n - 31*a*e^3)*r^2 - 3*(2*b*e^3*n - 3*a*e^3)*r)*x)*x
^(3*r) + 3*((18*b*d*e^2*r^5 + 57*b*d*e^2*r^4 + 68*b*d*e^2*r^3 + 38*b*d*e^2*r^2 + 10*b*d*e^2*r + b*d*e^2)*x*log
(c) + (18*b*d*e^2*n*r^5 + 57*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 + 38*b*d*e^2*n*r^2 + 10*b*d*e^2*n*r + b*d*e^2*n)
*x*log(x) + (18*a*d*e^2*r^5 - b*d*e^2*n - 3*(3*b*d*e^2*n - 19*a*d*e^2)*r^4 + a*d*e^2 - 4*(6*b*d*e^2*n - 17*a*d
*e^2)*r^3 - 2*(11*b*d*e^2*n - 19*a*d*e^2)*r^2 - 2*(4*b*d*e^2*n - 5*a*d*e^2)*r)*x)*x^(2*r) + 3*((36*b*d^2*e*r^5
+ 96*b*d^2*e*r^4 + 97*b*d^2*e*r^3 + 47*b*d^2*e*r^2 + 11*b*d^2*e*r + b*d^2*e)*x*log(c) + (36*b*d^2*e*n*r^5 + 9
6*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 + 47*b*d^2*e*n*r^2 + 11*b*d^2*e*n*r + b*d^2*e*n)*x*log(x) + (36*a*d^2*e*r^5
- b*d^2*e*n - 12*(3*b*d^2*e*n - 8*a*d^2*e)*r^4 + a*d^2*e - (60*b*d^2*e*n - 97*a*d^2*e)*r^3 - (37*b*d^2*e*n -
47*a*d^2*e)*r^2 - (10*b*d^2*e*n - 11*a*d^2*e)*r)*x)*x^r)/(36*r^6 + 132*r^5 + 193*r^4 + 144*r^3 + 58*r^2 + 12*r
+ 1)

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Sympy [A]  time = 26.8578, size = 325, normalized size = 1.92 \begin{align*} a d^{3} x + 3 a d^{2} e \left (\begin{cases} \frac{x^{r + 1}}{r + 1} & \text{for}\: r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + 3 a d e^{2} \left (\begin{cases} \frac{x^{2 r + 1}}{2 r + 1} & \text{for}\: 2 r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + a e^{3} \left (\begin{cases} \frac{x^{3 r + 1}}{3 r + 1} & \text{for}\: 3 r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) - b d^{3} n x + b d^{3} x \log{\left (c x^{n} \right )} - 3 b d^{2} e n \left (\begin{cases} \frac{\begin{cases} \frac{x x^{r}}{r + 1} & \text{for}\: r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{r + 1} & \text{for}\: r > -\infty \wedge r < \infty \wedge r \neq -1 \\\frac{\log{\left (x \right )}^{2}}{2} & \text{otherwise} \end{cases}\right ) + 3 b d^{2} e \left (\begin{cases} \frac{x^{r + 1}}{r + 1} & \text{for}\: r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} - 3 b d e^{2} n \left (\begin{cases} \frac{\begin{cases} \frac{x x^{2 r}}{2 r + 1} & \text{for}\: r \neq - \frac{1}{2} \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{2 r + 1} & \text{for}\: r > -\infty \wedge r < \infty \wedge r \neq - \frac{1}{2} \\\frac{\log{\left (x \right )}^{2}}{2} & \text{otherwise} \end{cases}\right ) + 3 b d e^{2} \left (\begin{cases} \frac{x^{2 r + 1}}{2 r + 1} & \text{for}\: 2 r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} - b e^{3} n \left (\begin{cases} \frac{\begin{cases} \frac{x x^{3 r}}{3 r + 1} & \text{for}\: r \neq - \frac{1}{3} \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{3 r + 1} & \text{for}\: r > -\infty \wedge r < \infty \wedge r \neq - \frac{1}{3} \\\frac{\log{\left (x \right )}^{2}}{2} & \text{otherwise} \end{cases}\right ) + b e^{3} \left (\begin{cases} \frac{x^{3 r + 1}}{3 r + 1} & \text{for}\: 3 r \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**3*x + 3*a*d**2*e*Piecewise((x**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True)) + 3*a*d*e**2*Piecewise((x**(2
*r + 1)/(2*r + 1), Ne(2*r, -1)), (log(x), True)) + a*e**3*Piecewise((x**(3*r + 1)/(3*r + 1), Ne(3*r, -1)), (lo
g(x), True)) - b*d**3*n*x + b*d**3*x*log(c*x**n) - 3*b*d**2*e*n*Piecewise((Piecewise((x*x**r/(r + 1), Ne(r, -1
)), (log(x), True))/(r + 1), (r > -oo) & (r < oo) & Ne(r, -1)), (log(x)**2/2, True)) + 3*b*d**2*e*Piecewise((x
**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n*Piecewise((Piecewise((x*x**(2*r)/(2*
r + 1), Ne(r, -1/2)), (log(x), True))/(2*r + 1), (r > -oo) & (r < oo) & Ne(r, -1/2)), (log(x)**2/2, True)) + 3
*b*d*e**2*Piecewise((x**(2*r + 1)/(2*r + 1), Ne(2*r, -1)), (log(x), True))*log(c*x**n) - b*e**3*n*Piecewise((P
iecewise((x*x**(3*r)/(3*r + 1), Ne(r, -1/3)), (log(x), True))/(3*r + 1), (r > -oo) & (r < oo) & Ne(r, -1/3)),
(log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r + 1)/(3*r + 1), Ne(3*r, -1)), (log(x), True))*log(c*x**n)

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Giac [B]  time = 1.33496, size = 505, normalized size = 2.99 \begin{align*} \frac{3 \, b d^{2} n r x x^{r} e \log \left (x\right )}{r^{2} + 2 \, r + 1} + b d^{3} n x \log \left (x\right ) + \frac{6 \, b d n r x x^{2 \, r} e^{2} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + \frac{3 \, b d^{2} n x x^{r} e \log \left (x\right )}{r^{2} + 2 \, r + 1} - b d^{3} n x - \frac{3 \, b d^{2} n x x^{r} e}{r^{2} + 2 \, r + 1} + b d^{3} x \log \left (c\right ) + \frac{3 \, b d^{2} x x^{r} e \log \left (c\right )}{r + 1} + \frac{3 \, b n r x x^{3 \, r} e^{3} \log \left (x\right )}{9 \, r^{2} + 6 \, r + 1} + \frac{3 \, b d n x x^{2 \, r} e^{2} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + a d^{3} x - \frac{3 \, b d n x x^{2 \, r} e^{2}}{4 \, r^{2} + 4 \, r + 1} + \frac{3 \, a d^{2} x x^{r} e}{r + 1} + \frac{3 \, b d x x^{2 \, r} e^{2} \log \left (c\right )}{2 \, r + 1} + \frac{b n x x^{3 \, r} e^{3} \log \left (x\right )}{9 \, r^{2} + 6 \, r + 1} - \frac{b n x x^{3 \, r} e^{3}}{9 \, r^{2} + 6 \, r + 1} + \frac{3 \, a d x x^{2 \, r} e^{2}}{2 \, r + 1} + \frac{b x x^{3 \, r} e^{3} \log \left (c\right )}{3 \, r + 1} + \frac{a x x^{3 \, r} e^{3}}{3 \, r + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*d^2*n*r*x*x^r*e*log(x)/(r^2 + 2*r + 1) + b*d^3*n*x*log(x) + 6*b*d*n*r*x*x^(2*r)*e^2*log(x)/(4*r^2 + 4*r +
1) + 3*b*d^2*n*x*x^r*e*log(x)/(r^2 + 2*r + 1) - b*d^3*n*x - 3*b*d^2*n*x*x^r*e/(r^2 + 2*r + 1) + b*d^3*x*log(c)
+ 3*b*d^2*x*x^r*e*log(c)/(r + 1) + 3*b*n*r*x*x^(3*r)*e^3*log(x)/(9*r^2 + 6*r + 1) + 3*b*d*n*x*x^(2*r)*e^2*log
(x)/(4*r^2 + 4*r + 1) + a*d^3*x - 3*b*d*n*x*x^(2*r)*e^2/(4*r^2 + 4*r + 1) + 3*a*d^2*x*x^r*e/(r + 1) + 3*b*d*x*
x^(2*r)*e^2*log(c)/(2*r + 1) + b*n*x*x^(3*r)*e^3*log(x)/(9*r^2 + 6*r + 1) - b*n*x*x^(3*r)*e^3/(9*r^2 + 6*r + 1
) + 3*a*d*x*x^(2*r)*e^2/(2*r + 1) + b*x*x^(3*r)*e^3*log(c)/(3*r + 1) + a*x*x^(3*r)*e^3/(3*r + 1)