3.399 \(\int x^2 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=148 \[ \frac{1}{3} \left (\frac{9 d^2 e x^{r+3}}{r+3}+d^3 x^3+\frac{9 d e^2 x^{2 r+3}}{2 r+3}+\frac{e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+3}}{(r+3)^2}-\frac{1}{9} b d^3 n x^3-\frac{3 b d e^2 n x^{2 r+3}}{(2 r+3)^2}-\frac{b e^3 n x^{3 (r+1)}}{9 (r+1)^2} \]

[Out]

-(b*d^3*n*x^3)/9 - (b*e^3*n*x^(3*(1 + r)))/(9*(1 + r)^2) - (3*b*d^2*e*n*x^(3 + r))/(3 + r)^2 - (3*b*d*e^2*n*x^
(3 + 2*r))/(3 + 2*r)^2 + ((d^3*x^3 + (e^3*x^(3*(1 + r)))/(1 + r) + (9*d^2*e*x^(3 + r))/(3 + r) + (9*d*e^2*x^(3
+ 2*r))/(3 + 2*r))*(a + b*Log[c*x^n]))/3

________________________________________________________________________________________

Rubi [A]  time = 0.380695, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ \frac{1}{3} \left (\frac{9 d^2 e x^{r+3}}{r+3}+d^3 x^3+\frac{9 d e^2 x^{2 r+3}}{2 r+3}+\frac{e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+3}}{(r+3)^2}-\frac{1}{9} b d^3 n x^3-\frac{3 b d e^2 n x^{2 r+3}}{(2 r+3)^2}-\frac{b e^3 n x^{3 (r+1)}}{9 (r+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^3)/9 - (b*e^3*n*x^(3*(1 + r)))/(9*(1 + r)^2) - (3*b*d^2*e*n*x^(3 + r))/(3 + r)^2 - (3*b*d*e^2*n*x^
(3 + 2*r))/(3 + 2*r)^2 + ((d^3*x^3 + (e^3*x^(3*(1 + r)))/(1 + r) + (9*d^2*e*x^(3 + r))/(3 + r) + (9*d*e^2*x^(3
+ 2*r))/(3 + 2*r))*(a + b*Log[c*x^n]))/3

Rule 270

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

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(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] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{3} \left (d^3 x^3+\frac{e^3 x^{3 (1+r)}}{1+r}+\frac{9 d^2 e x^{3+r}}{3+r}+\frac{9 d e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{3} x^2 \left (d^3+\frac{9 d^2 e x^r}{3+r}+\frac{9 d e^2 x^{2 r}}{3+2 r}+\frac{e^3 x^{3 r}}{1+r}\right ) \, dx\\ &=\frac{1}{3} \left (d^3 x^3+\frac{e^3 x^{3 (1+r)}}{1+r}+\frac{9 d^2 e x^{3+r}}{3+r}+\frac{9 d e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int x^2 \left (d^3+\frac{9 d^2 e x^r}{3+r}+\frac{9 d e^2 x^{2 r}}{3+2 r}+\frac{e^3 x^{3 r}}{1+r}\right ) \, dx\\ &=\frac{1}{3} \left (d^3 x^3+\frac{e^3 x^{3 (1+r)}}{1+r}+\frac{9 d^2 e x^{3+r}}{3+r}+\frac{9 d e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int \left (d^3 x^2+\frac{9 d e^2 x^{2 (1+r)}}{3+2 r}+\frac{9 d^2 e x^{2+r}}{3+r}+\frac{e^3 x^{2+3 r}}{1+r}\right ) \, dx\\ &=-\frac{1}{9} b d^3 n x^3-\frac{b e^3 n x^{3 (1+r)}}{9 (1+r)^2}-\frac{3 b d^2 e n x^{3+r}}{(3+r)^2}-\frac{3 b d e^2 n x^{3+2 r}}{(3+2 r)^2}+\frac{1}{3} \left (d^3 x^3+\frac{e^3 x^{3 (1+r)}}{1+r}+\frac{9 d^2 e x^{3+r}}{3+r}+\frac{9 d e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.343478, size = 176, normalized size = 1.19 \[ \frac{1}{9} x^3 \left (3 a \left (\frac{9 d^2 e x^r}{r+3}+d^3+\frac{9 d e^2 x^{2 r}}{2 r+3}+\frac{e^3 x^{3 r}}{r+1}\right )+3 b \log \left (c x^n\right ) \left (\frac{9 d^2 e x^r}{r+3}+d^3+\frac{9 d e^2 x^{2 r}}{2 r+3}+\frac{e^3 x^{3 r}}{r+1}\right )+b n \left (-\frac{27 d^2 e x^r}{(r+3)^2}-d^3-\frac{27 d e^2 x^{2 r}}{(2 r+3)^2}-\frac{e^3 x^{3 r}}{(r+1)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

(x^3*(b*n*(-d^3 - (27*d^2*e*x^r)/(3 + r)^2 - (27*d*e^2*x^(2*r))/(3 + 2*r)^2 - (e^3*x^(3*r))/(1 + r)^2) + 3*a*(
d^3 + (9*d^2*e*x^r)/(3 + r) + (9*d*e^2*x^(2*r))/(3 + 2*r) + (e^3*x^(3*r))/(1 + r)) + 3*b*(d^3 + (9*d^2*e*x^r)/
(3 + r) + (9*d*e^2*x^(2*r))/(3 + 2*r) + (e^3*x^(3*r))/(1 + r))*Log[c*x^n]))/9

________________________________________________________________________________________

Maple [C]  time = 0.395, size = 4027, normalized size = 27.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d+e*x^r)^3*(a+b*ln(c*x^n)),x)

[Out]

1/3*b*x^3*(2*e^3*r^2*(x^r)^3+9*d*e^2*r^2*(x^r)^2+9*e^3*r*(x^r)^3+2*d^3*r^3+18*d^2*e*r^2*x^r+36*d*e^2*r*(x^r)^2
+9*e^3*(x^r)^3+11*d^3*r^2+45*d^2*e*r*x^r+27*d*e^2*(x^r)^2+18*d^3*r+27*d^2*e*x^r+9*d^3)/(1+r)/(3+2*r)/(3+r)*ln(
x^n)-1/18*x^3*(-486*a*d^3-486*a*e^3*(x^r)^3-486*ln(c)*b*d^3+1566*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)+972*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1296*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)+8*b*d^3*n*r^6+88*b*d^3*n*r^5+386*b*d^3*n*r^4-24*a*d^3*r^6-264*a*d^3*r^5-1158*a*d^3*r^4+120*I*Pi*b*e^3*r^4
*csgn(I*c*x^n)^3*(x^r)^3+513*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+729*I*Pi*b*d^2*e*csg
n(I*c*x^n)^3*x^r+459*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+729*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-24*a*e^3*
r^5*(x^r)^3-240*a*e^3*r^4*(x^r)^3-1458*a*d*e^2*(x^r)^2-1458*a*d^2*e*x^r+162*b*e^3*n*(x^r)^3-918*a*e^3*r^3*(x^r
)^3-1674*a*e^3*r^2*(x^r)^3-1458*a*e^3*r*(x^r)^3-486*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-837*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^
3-837*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+729*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+12*I*Pi*b*e^3*
r^5*csgn(I*c*x^n)^3*(x^r)^3+864*b*d^3*n*r^3+1044*b*d^3*n*r^2+648*b*d^3*n*r-24*ln(c)*b*d^3*r^6-264*ln(c)*b*d^3*
r^5-1158*ln(c)*b*d^3*r^4-2592*ln(c)*b*d^3*r^3-3132*ln(c)*b*d^3*r^2-1944*ln(c)*b*d^3*r-2592*a*d^3*r^3-3132*a*d^
3*r^2-1944*a*d^3*r-4860*a*d*e^2*r*(x^r)^2-5238*a*d^2*e*r^3*x^r-7614*a*d^2*e*r^2*x^r-5346*a*d^2*e*r*x^r-513*I*P
i*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+162*b*d^3*n+972*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+243*I*Pi*b*d^3*cs
gn(I*c*x^n)^3-120*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-120*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I
*c)*(x^r)^3-240*ln(c)*b*e^3*r^4*(x^r)^3-918*ln(c)*b*e^3*r^3*(x^r)^3-1674*ln(c)*b*e^3*r^2*(x^r)^3-1458*ln(c)*b*
e^3*r*(x^r)^3-1458*ln(c)*b*d^2*e*x^r-1458*ln(c)*b*d*e^2*(x^r)^2+234*b*e^3*n*r^2*(x^r)^3+324*b*e^3*n*r*(x^r)^3+
486*b*d*e^2*n*(x^r)^2+486*b*d^2*e*n*x^r-3672*a*d*e^2*r^3*(x^r)^2-6156*a*d*e^2*r^2*(x^r)^2+8*b*e^3*n*r^4*(x^r)^
3+72*b*e^3*n*r^3*(x^r)^3-108*a*d*e^2*r^5*(x^r)^2-1026*a*d*e^2*r^4*(x^r)^2-216*a*d^2*e*r^5*x^r-1728*a*d^2*e*r^4
*x^r-24*ln(c)*b*e^3*r^5*(x^r)^3+2673*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-729*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*
x^n)^2*(x^r)^2-729*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-729*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2
*x^r+1188*b*d*e^2*n*r^2*(x^r)^2+1998*b*d^2*e*n*r^2*x^r+864*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+513*I*Pi*b*d*e
^2*r^4*csgn(I*c*x^n)^3*(x^r)^2+120*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-864*I*Pi*b*d^2*e
*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-864*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r+729*I*Pi*b*e^3*r*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+459*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-1836*I*P
i*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2619*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-3078*I*P
i*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+243*
I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+3078*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)*(x^r)^2+3807*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-729*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn
(I*c)*(x^r)^3+3807*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+1296*b*d*e^2*n*r*(x^r)^2+1620*b*d^2*e*n*r*x^r+54*b*d*e
^2*n*r^4*(x^r)^2+432*b*d*e^2*n*r^3*(x^r)^2+216*b*d^2*e*n*r^4*x^r+1080*b*d^2*e*n*r^3*x^r-108*ln(c)*b*d*e^2*r^5*
(x^r)^2-1026*ln(c)*b*d*e^2*r^4*(x^r)^2-216*ln(c)*b*d^2*e*r^5*x^r-1728*ln(c)*b*d^2*e*r^4*x^r-12*I*Pi*b*d^3*r^6*
csgn(I*x^n)*csgn(I*c*x^n)^2+837*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3-243*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n
)^2*(x^r)^3-729*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-459*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r
)^3-459*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+243*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-52
38*ln(c)*b*d^2*e*r^3*x^r-7614*ln(c)*b*d^2*e*r^2*x^r-5346*ln(c)*b*d^2*e*r*x^r-3672*ln(c)*b*d*e^2*r^3*(x^r)^2-61
56*ln(c)*b*d*e^2*r^2*(x^r)^2-4860*ln(c)*b*d*e^2*r*(x^r)^2+3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-729*I*
Pi*b*e^3*r*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)^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-1566*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)-972*I*Pi*b*d^3*r*csgn(I
*x^n)*csgn(I*c*x^n)^2-1296*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1296*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn
(I*c)-12*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-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)+1836*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-12*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+1296*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3+2619*I*Pi*
b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+1566*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+864*I*Pi*b*d^2*e*r^4*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-2619*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+579*I*Pi*b*d^3*r^
4*csgn(I*c*x^n)^3-579*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-579*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*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+837*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(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^6*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+579*I*Pi*b*d^3*r^4*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)-243*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-972*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I
*c)-1566*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+2430*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+108*I*Pi*b*d^2
*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+1836*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)
^2+243*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3-243*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+12*I*Pi*b*d^3*r^6*csgn(I*c*
x^n)^3+132*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-243*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-2430*I*Pi*b*d*e^2*r*csgn(
I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2430*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-1836*I*Pi*b*d*e^2*r^3*csg
n(I*c*x^n)^2*csgn(I*c)*(x^r)^2-513*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+132*I*Pi*b*d^3*r^5*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2619*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r-3807*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2*x^r-3807*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+729*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)*(x^r)^2+729*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-54*I*Pi*b*d*e^2*r^5*csgn(I*x^n)
*csgn(I*c*x^n)^2*(x^r)^2-2673*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-2673*I*Pi*b*d^2*e*r*csgn(I*c*x^n)
^2*csgn(I*c)*x^r+2430*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+2673*I*Pi*b*d^2*e*r*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)*x^r)/(1+r)^2/(3+2*r)^2/(3+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(x^2*(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 = 1.4805, size = 2376, normalized size = 16.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/9*(3*(4*b*d^3*r^6 + 44*b*d^3*r^5 + 193*b*d^3*r^4 + 432*b*d^3*r^3 + 522*b*d^3*r^2 + 324*b*d^3*r + 81*b*d^3)*x
^3*log(c) + 3*(4*b*d^3*n*r^6 + 44*b*d^3*n*r^5 + 193*b*d^3*n*r^4 + 432*b*d^3*n*r^3 + 522*b*d^3*n*r^2 + 324*b*d^
3*n*r + 81*b*d^3*n)*x^3*log(x) - (4*(b*d^3*n - 3*a*d^3)*r^6 + 44*(b*d^3*n - 3*a*d^3)*r^5 + 81*b*d^3*n + 193*(b
*d^3*n - 3*a*d^3)*r^4 - 243*a*d^3 + 432*(b*d^3*n - 3*a*d^3)*r^3 + 522*(b*d^3*n - 3*a*d^3)*r^2 + 324*(b*d^3*n -
3*a*d^3)*r)*x^3 + (3*(4*b*e^3*r^5 + 40*b*e^3*r^4 + 153*b*e^3*r^3 + 279*b*e^3*r^2 + 243*b*e^3*r + 81*b*e^3)*x^
3*log(c) + 3*(4*b*e^3*n*r^5 + 40*b*e^3*n*r^4 + 153*b*e^3*n*r^3 + 279*b*e^3*n*r^2 + 243*b*e^3*n*r + 81*b*e^3*n)
*x^3*log(x) + (12*a*e^3*r^5 - 81*b*e^3*n - 4*(b*e^3*n - 30*a*e^3)*r^4 + 243*a*e^3 - 9*(4*b*e^3*n - 51*a*e^3)*r
^3 - 9*(13*b*e^3*n - 93*a*e^3)*r^2 - 81*(2*b*e^3*n - 9*a*e^3)*r)*x^3)*x^(3*r) + 27*((2*b*d*e^2*r^5 + 19*b*d*e^
2*r^4 + 68*b*d*e^2*r^3 + 114*b*d*e^2*r^2 + 90*b*d*e^2*r + 27*b*d*e^2)*x^3*log(c) + (2*b*d*e^2*n*r^5 + 19*b*d*e
^2*n*r^4 + 68*b*d*e^2*n*r^3 + 114*b*d*e^2*n*r^2 + 90*b*d*e^2*n*r + 27*b*d*e^2*n)*x^3*log(x) + (2*a*d*e^2*r^5 -
9*b*d*e^2*n - (b*d*e^2*n - 19*a*d*e^2)*r^4 + 27*a*d*e^2 - 4*(2*b*d*e^2*n - 17*a*d*e^2)*r^3 - 2*(11*b*d*e^2*n
- 57*a*d*e^2)*r^2 - 6*(4*b*d*e^2*n - 15*a*d*e^2)*r)*x^3)*x^(2*r) + 27*((4*b*d^2*e*r^5 + 32*b*d^2*e*r^4 + 97*b*
d^2*e*r^3 + 141*b*d^2*e*r^2 + 99*b*d^2*e*r + 27*b*d^2*e)*x^3*log(c) + (4*b*d^2*e*n*r^5 + 32*b*d^2*e*n*r^4 + 97
*b*d^2*e*n*r^3 + 141*b*d^2*e*n*r^2 + 99*b*d^2*e*n*r + 27*b*d^2*e*n)*x^3*log(x) + (4*a*d^2*e*r^5 - 9*b*d^2*e*n
- 4*(b*d^2*e*n - 8*a*d^2*e)*r^4 + 27*a*d^2*e - (20*b*d^2*e*n - 97*a*d^2*e)*r^3 - (37*b*d^2*e*n - 141*a*d^2*e)*
r^2 - 3*(10*b*d^2*e*n - 33*a*d^2*e)*r)*x^3)*x^r)/(4*r^6 + 44*r^5 + 193*r^4 + 432*r^3 + 522*r^2 + 324*r + 81)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d+e*x**r)**3*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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Giac [B]  time = 1.31604, size = 2144, normalized size = 14.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/9*(12*b*d^3*n*r^6*x^3*log(x) + 108*b*d^2*n*r^5*x^3*x^r*e*log(x) - 4*b*d^3*n*r^6*x^3 + 12*b*d^3*r^6*x^3*log(c
) + 108*b*d^2*r^5*x^3*x^r*e*log(c) + 132*b*d^3*n*r^5*x^3*log(x) + 54*b*d*n*r^5*x^3*x^(2*r)*e^2*log(x) + 864*b*
d^2*n*r^4*x^3*x^r*e*log(x) - 44*b*d^3*n*r^5*x^3 + 12*a*d^3*r^6*x^3 - 108*b*d^2*n*r^4*x^3*x^r*e + 108*a*d^2*r^5
*x^3*x^r*e + 132*b*d^3*r^5*x^3*log(c) + 54*b*d*r^5*x^3*x^(2*r)*e^2*log(c) + 864*b*d^2*r^4*x^3*x^r*e*log(c) + 5
79*b*d^3*n*r^4*x^3*log(x) + 12*b*n*r^5*x^3*x^(3*r)*e^3*log(x) + 513*b*d*n*r^4*x^3*x^(2*r)*e^2*log(x) + 2619*b*
d^2*n*r^3*x^3*x^r*e*log(x) - 193*b*d^3*n*r^4*x^3 + 132*a*d^3*r^5*x^3 - 27*b*d*n*r^4*x^3*x^(2*r)*e^2 + 54*a*d*r
^5*x^3*x^(2*r)*e^2 - 540*b*d^2*n*r^3*x^3*x^r*e + 864*a*d^2*r^4*x^3*x^r*e + 579*b*d^3*r^4*x^3*log(c) + 12*b*r^5
*x^3*x^(3*r)*e^3*log(c) + 513*b*d*r^4*x^3*x^(2*r)*e^2*log(c) + 2619*b*d^2*r^3*x^3*x^r*e*log(c) + 1296*b*d^3*n*
r^3*x^3*log(x) + 120*b*n*r^4*x^3*x^(3*r)*e^3*log(x) + 1836*b*d*n*r^3*x^3*x^(2*r)*e^2*log(x) + 3807*b*d^2*n*r^2
*x^3*x^r*e*log(x) - 432*b*d^3*n*r^3*x^3 + 579*a*d^3*r^4*x^3 - 4*b*n*r^4*x^3*x^(3*r)*e^3 + 12*a*r^5*x^3*x^(3*r)
*e^3 - 216*b*d*n*r^3*x^3*x^(2*r)*e^2 + 513*a*d*r^4*x^3*x^(2*r)*e^2 - 999*b*d^2*n*r^2*x^3*x^r*e + 2619*a*d^2*r^
3*x^3*x^r*e + 1296*b*d^3*r^3*x^3*log(c) + 120*b*r^4*x^3*x^(3*r)*e^3*log(c) + 1836*b*d*r^3*x^3*x^(2*r)*e^2*log(
c) + 3807*b*d^2*r^2*x^3*x^r*e*log(c) + 1566*b*d^3*n*r^2*x^3*log(x) + 459*b*n*r^3*x^3*x^(3*r)*e^3*log(x) + 3078
*b*d*n*r^2*x^3*x^(2*r)*e^2*log(x) + 2673*b*d^2*n*r*x^3*x^r*e*log(x) - 522*b*d^3*n*r^2*x^3 + 1296*a*d^3*r^3*x^3
- 36*b*n*r^3*x^3*x^(3*r)*e^3 + 120*a*r^4*x^3*x^(3*r)*e^3 - 594*b*d*n*r^2*x^3*x^(2*r)*e^2 + 1836*a*d*r^3*x^3*x
^(2*r)*e^2 - 810*b*d^2*n*r*x^3*x^r*e + 3807*a*d^2*r^2*x^3*x^r*e + 1566*b*d^3*r^2*x^3*log(c) + 459*b*r^3*x^3*x^
(3*r)*e^3*log(c) + 3078*b*d*r^2*x^3*x^(2*r)*e^2*log(c) + 2673*b*d^2*r*x^3*x^r*e*log(c) + 972*b*d^3*n*r*x^3*log
(x) + 837*b*n*r^2*x^3*x^(3*r)*e^3*log(x) + 2430*b*d*n*r*x^3*x^(2*r)*e^2*log(x) + 729*b*d^2*n*x^3*x^r*e*log(x)
- 324*b*d^3*n*r*x^3 + 1566*a*d^3*r^2*x^3 - 117*b*n*r^2*x^3*x^(3*r)*e^3 + 459*a*r^3*x^3*x^(3*r)*e^3 - 648*b*d*n
*r*x^3*x^(2*r)*e^2 + 3078*a*d*r^2*x^3*x^(2*r)*e^2 - 243*b*d^2*n*x^3*x^r*e + 2673*a*d^2*r*x^3*x^r*e + 972*b*d^3
*r*x^3*log(c) + 837*b*r^2*x^3*x^(3*r)*e^3*log(c) + 2430*b*d*r*x^3*x^(2*r)*e^2*log(c) + 729*b*d^2*x^3*x^r*e*log
(c) + 243*b*d^3*n*x^3*log(x) + 729*b*n*r*x^3*x^(3*r)*e^3*log(x) + 729*b*d*n*x^3*x^(2*r)*e^2*log(x) - 81*b*d^3*
n*x^3 + 972*a*d^3*r*x^3 - 162*b*n*r*x^3*x^(3*r)*e^3 + 837*a*r^2*x^3*x^(3*r)*e^3 - 243*b*d*n*x^3*x^(2*r)*e^2 +
2430*a*d*r*x^3*x^(2*r)*e^2 + 729*a*d^2*x^3*x^r*e + 243*b*d^3*x^3*log(c) + 729*b*r*x^3*x^(3*r)*e^3*log(c) + 729
*b*d*x^3*x^(2*r)*e^2*log(c) + 243*b*n*x^3*x^(3*r)*e^3*log(x) + 243*a*d^3*x^3 - 81*b*n*x^3*x^(3*r)*e^3 + 729*a*
r*x^3*x^(3*r)*e^3 + 729*a*d*x^3*x^(2*r)*e^2 + 243*b*x^3*x^(3*r)*e^3*log(c) + 243*a*x^3*x^(3*r)*e^3)/(4*r^6 + 4
4*r^5 + 193*r^4 + 432*r^3 + 522*r^2 + 324*r + 81)