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

Optimal. Leaf size=149 \[ \frac{1}{2} \left (\frac{6 d^2 e x^{r+2}}{r+2}+d^3 x^2+\frac{3 d e^2 x^{2 (r+1)}}{r+1}+\frac{2 e^3 x^{3 r+2}}{3 r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+2}}{(r+2)^2}-\frac{1}{4} b d^3 n x^2-\frac{3 b d e^2 n x^{2 (r+1)}}{4 (r+1)^2}-\frac{b e^3 n x^{3 r+2}}{(3 r+2)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.348306, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {270, 2334, 12, 14} \[ \frac{1}{2} \left (\frac{6 d^2 e x^{r+2}}{r+2}+d^3 x^2+\frac{3 d e^2 x^{2 (r+1)}}{r+1}+\frac{2 e^3 x^{3 r+2}}{3 r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+2}}{(r+2)^2}-\frac{1}{4} b d^3 n x^2-\frac{3 b d e^2 n x^{2 (r+1)}}{4 (r+1)^2}-\frac{b e^3 n x^{3 r+2}}{(3 r+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{2} \left (d^3 x^2+\frac{3 d e^2 x^{2 (1+r)}}{1+r}+\frac{6 d^2 e x^{2+r}}{2+r}+\frac{2 e^3 x^{2+3 r}}{2+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{2} x \left (d^3+\frac{6 d^2 e x^r}{2+r}+\frac{3 d e^2 x^{2 r}}{1+r}+\frac{2 e^3 x^{3 r}}{2+3 r}\right ) \, dx\\ &=\frac{1}{2} \left (d^3 x^2+\frac{3 d e^2 x^{2 (1+r)}}{1+r}+\frac{6 d^2 e x^{2+r}}{2+r}+\frac{2 e^3 x^{2+3 r}}{2+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int x \left (d^3+\frac{6 d^2 e x^r}{2+r}+\frac{3 d e^2 x^{2 r}}{1+r}+\frac{2 e^3 x^{3 r}}{2+3 r}\right ) \, dx\\ &=\frac{1}{2} \left (d^3 x^2+\frac{3 d e^2 x^{2 (1+r)}}{1+r}+\frac{6 d^2 e x^{2+r}}{2+r}+\frac{2 e^3 x^{2+3 r}}{2+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \left (d^3 x+\frac{6 d^2 e x^{1+r}}{2+r}+\frac{3 d e^2 x^{1+2 r}}{1+r}+\frac{2 e^3 x^{1+3 r}}{2+3 r}\right ) \, dx\\ &=-\frac{1}{4} b d^3 n x^2-\frac{3 b d e^2 n x^{2 (1+r)}}{4 (1+r)^2}-\frac{3 b d^2 e n x^{2+r}}{(2+r)^2}-\frac{b e^3 n x^{2+3 r}}{(2+3 r)^2}+\frac{1}{2} \left (d^3 x^2+\frac{3 d e^2 x^{2 (1+r)}}{1+r}+\frac{6 d^2 e x^{2+r}}{2+r}+\frac{2 e^3 x^{2+3 r}}{2+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.354285, size = 178, normalized size = 1.19 \[ \frac{1}{4} x^2 \left (2 a \left (\frac{6 d^2 e x^r}{r+2}+d^3+\frac{3 d e^2 x^{2 r}}{r+1}+\frac{2 e^3 x^{3 r}}{3 r+2}\right )+2 b \log \left (c x^n\right ) \left (\frac{6 d^2 e x^r}{r+2}+d^3+\frac{3 d e^2 x^{2 r}}{r+1}+\frac{2 e^3 x^{3 r}}{3 r+2}\right )+b n \left (-\frac{12 d^2 e x^r}{(r+2)^2}-d^3-\frac{3 d e^2 x^{2 r}}{(r+1)^2}-\frac{4 e^3 x^{3 r}}{(3 r+2)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(b*n*(-d^3 - (12*d^2*e*x^r)/(2 + r)^2 - (3*d*e^2*x^(2*r))/(1 + r)^2 - (4*e^3*x^(3*r))/(2 + 3*r)^2) + 2*a*
(d^3 + (6*d^2*e*x^r)/(2 + r) + (3*d*e^2*x^(2*r))/(1 + r) + (2*e^3*x^(3*r))/(2 + 3*r)) + 2*b*(d^3 + (6*d^2*e*x^
r)/(2 + r) + (3*d*e^2*x^(2*r))/(1 + r) + (2*e^3*x^(3*r))/(2 + 3*r))*Log[c*x^n]))/4

________________________________________________________________________________________

Maple [C]  time = 0.381, size = 4027, normalized size = 27. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2*(2*e^3*r^2*(x^r)^3+9*d*e^2*r^2*(x^r)^2+6*e^3*r*(x^r)^3+3*d^3*r^3+18*d^2*e*r^2*x^r+24*d*e^2*r*(x^r)^2
+4*e^3*(x^r)^3+11*d^3*r^2+30*d^2*e*r*x^r+12*d*e^2*(x^r)^2+12*d^3*r+12*d^2*e*x^r+4*d^3)/(2+3*r)/(1+r)/(2+r)*ln(
x^n)-1/4*x^2*(582*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+456*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*(x^r)^2-32*a*d^3+408*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-288*I*
Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-288*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)-9*I*Pi*b*d^3*r^6*csgn(I*
x^n)*csgn(I*c*x^n)^2-32*a*e^3*(x^r)^3+16*I*Pi*b*d^3*csgn(I*c*x^n)^3-32*ln(c)*b*d^3+456*I*Pi*b*d*e^2*r^2*csgn(I
*c*x^n)^3*(x^r)^2-72*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-72*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c
)*(x^r)^3+564*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+9*b*d^3*n*r^6+66*b*d^3*n*r^5+193*b*d^3*n*r^4-18*a*d^3*r^6-1
32*a*d^3*r^5-386*a*d^3*r^4-232*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+6*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^
r)^3-232*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)-96*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-96*I*Pi*b*d^3*r*
csgn(I*c*x^n)^2*csgn(I*c)+16*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12*a*e^3*r^5*(x^r)^3-80*a*e^3*r^4*
(x^r)^3-96*a*d*e^2*(x^r)^2-96*a*d^2*e*x^r+16*b*e^3*n*(x^r)^3-204*a*e^3*r^3*(x^r)^3-248*a*e^3*r^2*(x^r)^3-144*a
*e^3*r*(x^r)^3-32*ln(c)*b*e^3*(x^r)^3+288*b*d^3*n*r^3+232*b*d^3*n*r^2+96*b*d^3*n*r-18*ln(c)*b*d^3*r^6-132*ln(c
)*b*d^3*r^5-386*ln(c)*b*d^3*r^4-576*ln(c)*b*d^3*r^3-464*ln(c)*b*d^3*r^2-192*ln(c)*b*d^3*r-576*a*d^3*r^3-464*a*
d^3*r^2-192*a*d^3*r-480*a*d*e^2*r*(x^r)^2-1164*a*d^2*e*r^3*x^r-1128*a*d^2*e*r^2*x^r-528*a*d^2*e*r*x^r+6*I*Pi*b
*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+16*b*d^3*n-54*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^
2*x^r-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)^2-408*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*
c)*(x^r)^2-171*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+124*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)*(x^r)^3+264*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-6*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3
-48*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-48*I*P
i*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+27*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2-582*I*Pi*b*d^2*e*r^3*c
sgn(I*c*x^n)^2*csgn(I*c)*x^r-456*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-80*ln(c)*b*e^3*r^4*(x^r)
^3-204*ln(c)*b*e^3*r^3*(x^r)^3-248*ln(c)*b*e^3*r^2*(x^r)^3-144*ln(c)*b*e^3*r*(x^r)^3-96*ln(c)*b*d^2*e*x^r-96*l
n(c)*b*d*e^2*(x^r)^2+52*b*e^3*n*r^2*(x^r)^3+48*b*e^3*n*r*(x^r)^3+48*b*d*e^2*n*(x^r)^2+48*b*d^2*e*n*x^r-816*a*d
*e^2*r^3*(x^r)^2-912*a*d*e^2*r^2*(x^r)^2+4*b*e^3*n*r^4*(x^r)^3+24*b*e^3*n*r^3*(x^r)^3-54*a*d*e^2*r^5*(x^r)^2-3
42*a*d*e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r-576*a*d^2*e*r^4*x^r-12*ln(c)*b*e^3*r^5*(x^r)^3+264*b*d*e^2*n*r^2*(x
^r)^2+444*b*d^2*e*n*r^2*x^r-564*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-564*I*Pi*b*d^2*e*r^2*csgn(I*c
*x^n)^2*csgn(I*c)*x^r-240*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-240*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^
2*csgn(I*c)*(x^r)^2-40*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-102*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-102*I*P
i*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-40*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-48*I*Pi*b*
d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-48*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+48*I*Pi*b*d^2*e*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*x^r-27*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-288*I*Pi*b*d^2*e*r^4*csgn(
I*x^n)*csgn(I*c*x^n)^2*x^r-264*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-264*I*Pi*b*d^2*e*r*csgn(I*c*x^n)
^2*csgn(I*c)*x^r+48*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+408*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n
)^3*(x^r)^2-124*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+192*b*d*e^2*n*r*(x^r)^2+240*b*d^2*e*n*r*x^r
+27*b*d*e^2*n*r^4*(x^r)^2+144*b*d*e^2*n*r^3*(x^r)^2+108*b*d^2*e*n*r^4*x^r+360*b*d^2*e*n*r^3*x^r-54*ln(c)*b*d*e
^2*r^5*(x^r)^2-342*ln(c)*b*d*e^2*r^4*(x^r)^2-108*ln(c)*b*d^2*e*r^5*x^r-576*ln(c)*b*d^2*e*r^4*x^r-456*I*Pi*b*d*
e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+72*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+102*I*Pi
*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-582*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+48
*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+102*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+124*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^
3*(x^r)^3+72*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3+54*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r-16*I*Pi*b*e^3*csgn(I
*c*x^n)^2*csgn(I*c)*(x^r)^3+48*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-16*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*
(x^r)^3+232*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+96*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)+9*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1164*ln(c)*b*d^2*e*r^3*x^r-1128*ln(c)*b*d^2*e*r^2*x^
r-528*ln(c)*b*d^2*e*r*x^r-816*ln(c)*b*d*e^2*r^3*(x^r)^2-912*ln(c)*b*d*e^2*r^2*(x^r)^2-480*ln(c)*b*d*e^2*r*(x^r
)^2+54*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-27*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^
2*(x^r)^2-66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-66*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)+240*I*Pi*b
*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+16*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-408*I*Pi*b*d*e^2*r^
3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+40*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+27*I*Pi*b*d*e^2*r^5*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+66*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-16*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+9*
I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+582*I*Pi*b*d^2*e*r^3*csg
n(I*c*x^n)^3*x^r+96*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-16*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-124*I*Pi*b*e^3*r^2*
csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+288*I*Pi*b*d^3*r^3*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+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-54*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r+193*I*Pi*b*
d^3*r^4*csgn(I*c*x^n)^3+16*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+288*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3+232*I*Pi*b*d^
3*r^2*csgn(I*c*x^n)^3-9*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-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)+564*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+24
0*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+264*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*x^r)/(2+3*r)^2/(1+r)^2/(2+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*(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 = 1.45181, size = 2353, normalized size = 15.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(9*b*d^3*r^6 + 66*b*d^3*r^5 + 193*b*d^3*r^4 + 288*b*d^3*r^3 + 232*b*d^3*r^2 + 96*b*d^3*r + 16*b*d^3)*x^
2*log(c) + 2*(9*b*d^3*n*r^6 + 66*b*d^3*n*r^5 + 193*b*d^3*n*r^4 + 288*b*d^3*n*r^3 + 232*b*d^3*n*r^2 + 96*b*d^3*
n*r + 16*b*d^3*n)*x^2*log(x) - (9*(b*d^3*n - 2*a*d^3)*r^6 + 66*(b*d^3*n - 2*a*d^3)*r^5 + 16*b*d^3*n + 193*(b*d
^3*n - 2*a*d^3)*r^4 - 32*a*d^3 + 288*(b*d^3*n - 2*a*d^3)*r^3 + 232*(b*d^3*n - 2*a*d^3)*r^2 + 96*(b*d^3*n - 2*a
*d^3)*r)*x^2 + 4*((3*b*e^3*r^5 + 20*b*e^3*r^4 + 51*b*e^3*r^3 + 62*b*e^3*r^2 + 36*b*e^3*r + 8*b*e^3)*x^2*log(c)
+ (3*b*e^3*n*r^5 + 20*b*e^3*n*r^4 + 51*b*e^3*n*r^3 + 62*b*e^3*n*r^2 + 36*b*e^3*n*r + 8*b*e^3*n)*x^2*log(x) +
(3*a*e^3*r^5 - 4*b*e^3*n - (b*e^3*n - 20*a*e^3)*r^4 + 8*a*e^3 - 3*(2*b*e^3*n - 17*a*e^3)*r^3 - (13*b*e^3*n - 6
2*a*e^3)*r^2 - 12*(b*e^3*n - 3*a*e^3)*r)*x^2)*x^(3*r) + 3*(2*(9*b*d*e^2*r^5 + 57*b*d*e^2*r^4 + 136*b*d*e^2*r^3
+ 152*b*d*e^2*r^2 + 80*b*d*e^2*r + 16*b*d*e^2)*x^2*log(c) + 2*(9*b*d*e^2*n*r^5 + 57*b*d*e^2*n*r^4 + 136*b*d*e
^2*n*r^3 + 152*b*d*e^2*n*r^2 + 80*b*d*e^2*n*r + 16*b*d*e^2*n)*x^2*log(x) + (18*a*d*e^2*r^5 - 16*b*d*e^2*n - 3*
(3*b*d*e^2*n - 38*a*d*e^2)*r^4 + 32*a*d*e^2 - 16*(3*b*d*e^2*n - 17*a*d*e^2)*r^3 - 8*(11*b*d*e^2*n - 38*a*d*e^2
)*r^2 - 32*(2*b*d*e^2*n - 5*a*d*e^2)*r)*x^2)*x^(2*r) + 12*((9*b*d^2*e*r^5 + 48*b*d^2*e*r^4 + 97*b*d^2*e*r^3 +
94*b*d^2*e*r^2 + 44*b*d^2*e*r + 8*b*d^2*e)*x^2*log(c) + (9*b*d^2*e*n*r^5 + 48*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3
+ 94*b*d^2*e*n*r^2 + 44*b*d^2*e*n*r + 8*b*d^2*e*n)*x^2*log(x) + (9*a*d^2*e*r^5 - 4*b*d^2*e*n - 3*(3*b*d^2*e*n
- 16*a*d^2*e)*r^4 + 8*a*d^2*e - (30*b*d^2*e*n - 97*a*d^2*e)*r^3 - (37*b*d^2*e*n - 94*a*d^2*e)*r^2 - 4*(5*b*d^
2*e*n - 11*a*d^2*e)*r)*x^2)*x^r)/(9*r^6 + 66*r^5 + 193*r^4 + 288*r^3 + 232*r^2 + 96*r + 16)

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

[Out]

Timed out

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Giac [B]  time = 1.36158, size = 2144, normalized size = 14.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(18*b*d^3*n*r^6*x^2*log(x) + 108*b*d^2*n*r^5*x^2*x^r*e*log(x) - 9*b*d^3*n*r^6*x^2 + 18*b*d^3*r^6*x^2*log(c
) + 108*b*d^2*r^5*x^2*x^r*e*log(c) + 132*b*d^3*n*r^5*x^2*log(x) + 54*b*d*n*r^5*x^2*x^(2*r)*e^2*log(x) + 576*b*
d^2*n*r^4*x^2*x^r*e*log(x) - 66*b*d^3*n*r^5*x^2 + 18*a*d^3*r^6*x^2 - 108*b*d^2*n*r^4*x^2*x^r*e + 108*a*d^2*r^5
*x^2*x^r*e + 132*b*d^3*r^5*x^2*log(c) + 54*b*d*r^5*x^2*x^(2*r)*e^2*log(c) + 576*b*d^2*r^4*x^2*x^r*e*log(c) + 3
86*b*d^3*n*r^4*x^2*log(x) + 12*b*n*r^5*x^2*x^(3*r)*e^3*log(x) + 342*b*d*n*r^4*x^2*x^(2*r)*e^2*log(x) + 1164*b*
d^2*n*r^3*x^2*x^r*e*log(x) - 193*b*d^3*n*r^4*x^2 + 132*a*d^3*r^5*x^2 - 27*b*d*n*r^4*x^2*x^(2*r)*e^2 + 54*a*d*r
^5*x^2*x^(2*r)*e^2 - 360*b*d^2*n*r^3*x^2*x^r*e + 576*a*d^2*r^4*x^2*x^r*e + 386*b*d^3*r^4*x^2*log(c) + 12*b*r^5
*x^2*x^(3*r)*e^3*log(c) + 342*b*d*r^4*x^2*x^(2*r)*e^2*log(c) + 1164*b*d^2*r^3*x^2*x^r*e*log(c) + 576*b*d^3*n*r
^3*x^2*log(x) + 80*b*n*r^4*x^2*x^(3*r)*e^3*log(x) + 816*b*d*n*r^3*x^2*x^(2*r)*e^2*log(x) + 1128*b*d^2*n*r^2*x^
2*x^r*e*log(x) - 288*b*d^3*n*r^3*x^2 + 386*a*d^3*r^4*x^2 - 4*b*n*r^4*x^2*x^(3*r)*e^3 + 12*a*r^5*x^2*x^(3*r)*e^
3 - 144*b*d*n*r^3*x^2*x^(2*r)*e^2 + 342*a*d*r^4*x^2*x^(2*r)*e^2 - 444*b*d^2*n*r^2*x^2*x^r*e + 1164*a*d^2*r^3*x
^2*x^r*e + 576*b*d^3*r^3*x^2*log(c) + 80*b*r^4*x^2*x^(3*r)*e^3*log(c) + 816*b*d*r^3*x^2*x^(2*r)*e^2*log(c) + 1
128*b*d^2*r^2*x^2*x^r*e*log(c) + 464*b*d^3*n*r^2*x^2*log(x) + 204*b*n*r^3*x^2*x^(3*r)*e^3*log(x) + 912*b*d*n*r
^2*x^2*x^(2*r)*e^2*log(x) + 528*b*d^2*n*r*x^2*x^r*e*log(x) - 232*b*d^3*n*r^2*x^2 + 576*a*d^3*r^3*x^2 - 24*b*n*
r^3*x^2*x^(3*r)*e^3 + 80*a*r^4*x^2*x^(3*r)*e^3 - 264*b*d*n*r^2*x^2*x^(2*r)*e^2 + 816*a*d*r^3*x^2*x^(2*r)*e^2 -
240*b*d^2*n*r*x^2*x^r*e + 1128*a*d^2*r^2*x^2*x^r*e + 464*b*d^3*r^2*x^2*log(c) + 204*b*r^3*x^2*x^(3*r)*e^3*log
(c) + 912*b*d*r^2*x^2*x^(2*r)*e^2*log(c) + 528*b*d^2*r*x^2*x^r*e*log(c) + 192*b*d^3*n*r*x^2*log(x) + 248*b*n*r
^2*x^2*x^(3*r)*e^3*log(x) + 480*b*d*n*r*x^2*x^(2*r)*e^2*log(x) + 96*b*d^2*n*x^2*x^r*e*log(x) - 96*b*d^3*n*r*x^
2 + 464*a*d^3*r^2*x^2 - 52*b*n*r^2*x^2*x^(3*r)*e^3 + 204*a*r^3*x^2*x^(3*r)*e^3 - 192*b*d*n*r*x^2*x^(2*r)*e^2 +
912*a*d*r^2*x^2*x^(2*r)*e^2 - 48*b*d^2*n*x^2*x^r*e + 528*a*d^2*r*x^2*x^r*e + 192*b*d^3*r*x^2*log(c) + 248*b*r
^2*x^2*x^(3*r)*e^3*log(c) + 480*b*d*r*x^2*x^(2*r)*e^2*log(c) + 96*b*d^2*x^2*x^r*e*log(c) + 32*b*d^3*n*x^2*log(
x) + 144*b*n*r*x^2*x^(3*r)*e^3*log(x) + 96*b*d*n*x^2*x^(2*r)*e^2*log(x) - 16*b*d^3*n*x^2 + 192*a*d^3*r*x^2 - 4
8*b*n*r*x^2*x^(3*r)*e^3 + 248*a*r^2*x^2*x^(3*r)*e^3 - 48*b*d*n*x^2*x^(2*r)*e^2 + 480*a*d*r*x^2*x^(2*r)*e^2 + 9
6*a*d^2*x^2*x^r*e + 32*b*d^3*x^2*log(c) + 144*b*r*x^2*x^(3*r)*e^3*log(c) + 96*b*d*x^2*x^(2*r)*e^2*log(c) + 32*
b*n*x^2*x^(3*r)*e^3*log(x) + 32*a*d^3*x^2 - 16*b*n*x^2*x^(3*r)*e^3 + 144*a*r*x^2*x^(3*r)*e^3 + 96*a*d*x^2*x^(2
*r)*e^2 + 32*b*x^2*x^(3*r)*e^3*log(c) + 32*a*x^2*x^(3*r)*e^3)/(9*r^6 + 66*r^5 + 193*r^4 + 288*r^3 + 232*r^2 +
96*r + 16)