3.4.100 \(\int (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [400]

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

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {250, 2350} \begin {gather*} d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+\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}-b d^3 n x-\frac {3 b d^2 e n x^{r+1}}{(r+1)^2}-\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} \end {gather*}

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*(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)

Rule 250

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 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, 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 = 0.14, size = 149, normalized size = 0.88 \begin {gather*} x \left (b d^3 n \log (x)+d^3 \left (a-b n-b n \log (x)+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^r \left (a-b n+a r+b (1+r) \log \left (c x^n\right )\right )}{(1+r)^2}+\frac {3 d e^2 x^{2 r} \left (a-b n+2 a r+(b+2 b r) \log \left (c x^n\right )\right )}{(1+2 r)^2}+\frac {e^3 x^{3 r} \left (a-b n+3 a r+(b+3 b r) \log \left (c x^n\right )\right )}{(1+3 r)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 4023, normalized size = 23.80

method	result	size
risch	\(\text {Expression too large to display}\)	\(4023\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 220, normalized size = 1.30 \begin {gather*} -b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x + \frac {b e^{3} x^{3 \, r + 1} \log \left (c x^{n}\right )}{3 \, r + 1} + \frac {3 \, b d e^{2} x^{2 \, r + 1} \log \left (c x^{n}\right )}{2 \, r + 1} + \frac {3 \, b d^{2} e x^{r + 1} \log \left (c x^{n}\right )}{r + 1} - \frac {b e^{3} n x^{3 \, r + 1}}{{\left (3 \, r + 1\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 1}}{3 \, r + 1} - \frac {3 \, b d e^{2} n x^{2 \, r + 1}}{{\left (2 \, r + 1\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 1}}{2 \, r + 1} - \frac {3 \, b d^{2} e n x^{r + 1}}{{\left (r + 1\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 1}}{r + 1} \end {gather*}

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]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (167) = 334\).
time = 0.35, size = 838, normalized size = 4.96 \begin {gather*} \frac {{\left (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}\right )} x \log \left (c\right ) + {\left (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\right )} x \log \left (x\right ) - {\left (36 \, {\left (b d^{3} n - a d^{3}\right )} r^{6} + 132 \, {\left (b d^{3} n - a d^{3}\right )} r^{5} + b d^{3} n + 193 \, {\left (b d^{3} n - a d^{3}\right )} r^{4} - a d^{3} + 144 \, {\left (b d^{3} n - a d^{3}\right )} r^{3} + 58 \, {\left (b d^{3} n - a d^{3}\right )} r^{2} + 12 \, {\left (b d^{3} n - a d^{3}\right )} r\right )} x + {\left ({\left (12 \, b r^{5} + 40 \, b r^{4} + 51 \, b r^{3} + 31 \, b r^{2} + 9 \, b r + b\right )} x e^{3} \log \left (c\right ) + {\left (12 \, b n r^{5} + 40 \, b n r^{4} + 51 \, b n r^{3} + 31 \, b n r^{2} + 9 \, b n r + b n\right )} x e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n - 10 \, a\right )} r^{4} - 3 \, {\left (4 \, b n - 17 \, a\right )} r^{3} - {\left (13 \, b n - 31 \, a\right )} r^{2} - b n - 3 \, {\left (2 \, b n - 3 \, a\right )} r + a\right )} x e^{3}\right )} x^{3 \, r} + 3 \, {\left ({\left (18 \, b d r^{5} + 57 \, b d r^{4} + 68 \, b d r^{3} + 38 \, b d r^{2} + 10 \, b d r + b d\right )} x e^{2} \log \left (c\right ) + {\left (18 \, b d n r^{5} + 57 \, b d n r^{4} + 68 \, b d n r^{3} + 38 \, b d n r^{2} + 10 \, b d n r + b d n\right )} x e^{2} \log \left (x\right ) + {\left (18 \, a d r^{5} - 3 \, {\left (3 \, b d n - 19 \, a d\right )} r^{4} - 4 \, {\left (6 \, b d n - 17 \, a d\right )} r^{3} - b d n - 2 \, {\left (11 \, b d n - 19 \, a d\right )} r^{2} + a d - 2 \, {\left (4 \, b d n - 5 \, a d\right )} r\right )} x e^{2}\right )} x^{2 \, r} + 3 \, {\left ({\left (36 \, b d^{2} r^{5} + 96 \, b d^{2} r^{4} + 97 \, b d^{2} r^{3} + 47 \, b d^{2} r^{2} + 11 \, b d^{2} r + b d^{2}\right )} x e \log \left (c\right ) + {\left (36 \, b d^{2} n r^{5} + 96 \, b d^{2} n r^{4} + 97 \, b d^{2} n r^{3} + 47 \, b d^{2} n r^{2} + 11 \, b d^{2} n r + b d^{2} n\right )} x e \log \left (x\right ) + {\left (36 \, a d^{2} r^{5} - 12 \, {\left (3 \, b d^{2} n - 8 \, a d^{2}\right )} r^{4} - b d^{2} n - {\left (60 \, b d^{2} n - 97 \, a d^{2}\right )} r^{3} + a d^{2} - {\left (37 \, b d^{2} n - 47 \, a d^{2}\right )} r^{2} - {\left (10 \, b d^{2} n - 11 \, a d^{2}\right )} r\right )} x e\right )} x^{r}}{36 \, r^{6} + 132 \, r^{5} + 193 \, r^{4} + 144 \, r^{3} + 58 \, r^{2} + 12 \, r + 1} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 9.52, size = 325, normalized size = 1.92 \begin {gather*} 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}\: r \neq - \frac {1}{2} \\\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}\: r \neq - \frac {1}{3} \\\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}\: r \neq - \frac {1}{2} \\\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}\: r \neq - \frac {1}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \end {gather*}

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(r, -1/2)), (log(x), True)) + a*e**3*Piecewise((x**(3*r + 1)/(3*r + 1), Ne(r, -1/3)), (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(r, -1/2)), (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(r, -1/3)), (log(x), True))*log(c*x**n)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (167) = 334\).
time = 3.31, size = 374, normalized size = 2.21 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________