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

Optimal. Leaf size=149 \[ \frac{1}{4} \left (\frac{12 d^2 e x^{r+4}}{r+4}+d^3 x^4+\frac{6 d e^2 x^{2 (r+2)}}{r+2}+\frac{4 e^3 x^{3 r+4}}{3 r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+4}}{(r+4)^2}-\frac{1}{16} b d^3 n x^4-\frac{3 b d e^2 n x^{2 (r+2)}}{4 (r+2)^2}-\frac{b e^3 n x^{3 r+4}}{(3 r+4)^2} \]

[Out]

-(b*d^3*n*x^4)/16 - (3*b*d*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (3*b*d^2*e*n*x^(4 + r))/(4 + r)^2 - (b*e^3*n*x
^(4 + 3*r))/(4 + 3*r)^2 + ((d^3*x^4 + (6*d*e^2*x^(2*(2 + r)))/(2 + r) + (12*d^2*e*x^(4 + r))/(4 + r) + (4*e^3*
x^(4 + 3*r))/(4 + 3*r))*(a + b*Log[c*x^n]))/4

________________________________________________________________________________________

Rubi [A]  time = 0.385235, antiderivative size = 149, 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} \[ \frac{1}{4} \left (\frac{12 d^2 e x^{r+4}}{r+4}+d^3 x^4+\frac{6 d e^2 x^{2 (r+2)}}{r+2}+\frac{4 e^3 x^{3 r+4}}{3 r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+4}}{(r+4)^2}-\frac{1}{16} b d^3 n x^4-\frac{3 b d e^2 n x^{2 (r+2)}}{4 (r+2)^2}-\frac{b e^3 n x^{3 r+4}}{(3 r+4)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^3*n*x^4)/16 - (3*b*d*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (3*b*d^2*e*n*x^(4 + r))/(4 + r)^2 - (b*e^3*n*x
^(4 + 3*r))/(4 + 3*r)^2 + ((d^3*x^4 + (6*d*e^2*x^(2*(2 + r)))/(2 + r) + (12*d^2*e*x^(4 + r))/(4 + r) + (4*e^3*
x^(4 + 3*r))/(4 + 3*r))*(a + b*Log[c*x^n]))/4

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^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{4} \left (d^3 x^4+\frac{6 d e^2 x^{2 (2+r)}}{2+r}+\frac{12 d^2 e x^{4+r}}{4+r}+\frac{4 e^3 x^{4+3 r}}{4+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{4} x^3 \left (d^3+\frac{12 d^2 e x^r}{4+r}+\frac{6 d e^2 x^{2 r}}{2+r}+\frac{4 e^3 x^{3 r}}{4+3 r}\right ) \, dx\\ &=\frac{1}{4} \left (d^3 x^4+\frac{6 d e^2 x^{2 (2+r)}}{2+r}+\frac{12 d^2 e x^{4+r}}{4+r}+\frac{4 e^3 x^{4+3 r}}{4+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int x^3 \left (d^3+\frac{12 d^2 e x^r}{4+r}+\frac{6 d e^2 x^{2 r}}{2+r}+\frac{4 e^3 x^{3 r}}{4+3 r}\right ) \, dx\\ &=\frac{1}{4} \left (d^3 x^4+\frac{6 d e^2 x^{2 (2+r)}}{2+r}+\frac{12 d^2 e x^{4+r}}{4+r}+\frac{4 e^3 x^{4+3 r}}{4+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \left (d^3 x^3+\frac{4 e^3 x^{3 (1+r)}}{4+3 r}+\frac{12 d^2 e x^{3+r}}{4+r}+\frac{6 d e^2 x^{3+2 r}}{2+r}\right ) \, dx\\ &=-\frac{1}{16} b d^3 n x^4-\frac{3 b d e^2 n x^{2 (2+r)}}{4 (2+r)^2}-\frac{3 b d^2 e n x^{4+r}}{(4+r)^2}-\frac{b e^3 n x^{4+3 r}}{(4+3 r)^2}+\frac{1}{4} \left (d^3 x^4+\frac{6 d e^2 x^{2 (2+r)}}{2+r}+\frac{12 d^2 e x^{4+r}}{4+r}+\frac{4 e^3 x^{4+3 r}}{4+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.34265, size = 178, normalized size = 1.19 \[ \frac{1}{16} x^4 \left (4 a \left (\frac{12 d^2 e x^r}{r+4}+d^3+\frac{6 d e^2 x^{2 r}}{r+2}+\frac{4 e^3 x^{3 r}}{3 r+4}\right )+4 b \log \left (c x^n\right ) \left (\frac{12 d^2 e x^r}{r+4}+d^3+\frac{6 d e^2 x^{2 r}}{r+2}+\frac{4 e^3 x^{3 r}}{3 r+4}\right )+b n \left (-\frac{48 d^2 e x^r}{(r+4)^2}-d^3-\frac{12 d e^2 x^{2 r}}{(r+2)^2}-\frac{16 e^3 x^{3 r}}{(3 r+4)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(b*n*(-d^3 - (48*d^2*e*x^r)/(4 + r)^2 - (12*d*e^2*x^(2*r))/(2 + r)^2 - (16*e^3*x^(3*r))/(4 + 3*r)^2) + 4*
a*(d^3 + (12*d^2*e*x^r)/(4 + r) + (6*d*e^2*x^(2*r))/(2 + r) + (4*e^3*x^(3*r))/(4 + 3*r)) + 4*b*(d^3 + (12*d^2*
e*x^r)/(4 + r) + (6*d*e^2*x^(2*r))/(2 + r) + (4*e^3*x^(3*r))/(4 + 3*r))*Log[c*x^n]))/16

________________________________________________________________________________________

Maple [C]  time = 0.389, 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^3*(d+e*x^r)^3*(a+b*ln(c*x^n)),x)

[Out]

1/4*b*x^4*(4*e^3*r^2*(x^r)^3+18*d*e^2*r^2*(x^r)^2+24*e^3*r*(x^r)^3+3*d^3*r^3+36*d^2*e*r^2*x^r+96*d*e^2*r*(x^r)
^2+32*e^3*(x^r)^3+22*d^3*r^2+120*d^2*e*r*x^r+96*d*e^2*(x^r)^2+48*d^3*r+96*d^2*e*x^r+32*d^3)/(4+3*r)/(2+r)/(4+r
)*ln(x^n)-1/16*x^4*(-4096*a*d^3+24*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-108*I*Pi*b*d*e^2
*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3968*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+6144*
I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+18*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+264*I*P
i*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4096*a*e^3*(x^r)^3+2048*I*Pi*b*d^3*csgn(I*c*x^n)^3-4096*ln(c)*
b*d^3+9*b*d^3*n*r^6+132*b*d^3*n*r^5+772*b*d^3*n*r^4+6528*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-36*a*d^3*r^6
-528*a*d^3*r^5-3088*a*d^3*r^4-48*a*e^3*r^5*(x^r)^3-640*a*e^3*r^4*(x^r)^3-12288*a*d*e^2*(x^r)^2-12288*a*d^2*e*x
^r+1024*b*e^3*n*(x^r)^3-3264*a*e^3*r^3*(x^r)^3-7936*a*e^3*r^2*(x^r)^3-9216*a*e^3*r*(x^r)^3-4096*ln(c)*b*e^3*(x
^r)^3+2304*b*d^3*n*r^3+3712*b*d^3*n*r^2+3072*b*d^3*n*r-36*ln(c)*b*d^3*r^6-528*ln(c)*b*d^3*r^5-3088*ln(c)*b*d^3
*r^4-9216*ln(c)*b*d^3*r^3-14848*ln(c)*b*d^3*r^2-12288*ln(c)*b*d^3*r-9216*a*d^3*r^3-14848*a*d^3*r^2-12288*a*d^3
*r-30720*a*d*e^2*r*(x^r)^2-18624*a*d^2*e*r^3*x^r-36096*a*d^2*e*r^2*x^r-33792*a*d^2*e*r*x^r+1024*b*d^3*n-15360*
I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-16896*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+6144*I*P
i*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-2304*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-640*ln
(c)*b*e^3*r^4*(x^r)^3-3264*ln(c)*b*e^3*r^3*(x^r)^3-7936*ln(c)*b*e^3*r^2*(x^r)^3-9216*ln(c)*b*e^3*r*(x^r)^3-122
88*ln(c)*b*d^2*e*x^r-12288*ln(c)*b*d*e^2*(x^r)^2+832*b*e^3*n*r^2*(x^r)^3+1536*b*e^3*n*r*(x^r)^3+3072*b*d*e^2*n
*(x^r)^2+3072*b*d^2*e*n*x^r-13056*a*d*e^2*r^3*(x^r)^2-29184*a*d*e^2*r^2*(x^r)^2+16*b*e^3*n*r^4*(x^r)^3+192*b*e
^3*n*r^3*(x^r)^3-216*a*d*e^2*r^5*(x^r)^2-2736*a*d*e^2*r^4*(x^r)^2-432*a*d^2*e*r^5*x^r-4608*a*d^2*e*r^4*x^r-48*
ln(c)*b*e^3*r^5*(x^r)^3+4224*b*d*e^2*n*r^2*(x^r)^2+7104*b*d^2*e*n*r^2*x^r+4608*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)*(x^r)^3-216*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r-1368*I*Pi*b*d*e^2*r^4*csgn(I*x^n)
*csgn(I*c*x^n)^2*(x^r)^2+1632*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+6144*I*Pi*b*d*e^2*csg
n(I*c*x^n)^3*(x^r)^2-2048*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+24*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^
r)^3+320*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-6528*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-1804
8*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-18048*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-1632*I
*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-24*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-24*I*Pi*
b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-6144*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6144*I*Pi*b*d^2*
e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-1632*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-15360*I*Pi*b*d*e^2*r*c
sgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+108*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2-320*I*Pi*b*e^3*r^4*csgn(I*x^n)
*csgn(I*c*x^n)^2*(x^r)^3-320*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+18048*I*Pi*b*d^2*e*r^2*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)*x^r+15360*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+16896*I*Pi*b*d
^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+6144*b*d*e^2*n*r*(x^r)^2+7680*b*d^2*e*n*r*x^r+108*b*d*e^2*n*r^4
*(x^r)^2+1152*b*d*e^2*n*r^3*(x^r)^2+432*b*d^2*e*n*r^4*x^r+2880*b*d^2*e*n*r^3*x^r-216*ln(c)*b*d*e^2*r^5*(x^r)^2
-2736*ln(c)*b*d*e^2*r^4*(x^r)^2-432*ln(c)*b*d^2*e*r^5*x^r-4608*ln(c)*b*d^2*e*r^4*x^r+6144*I*Pi*b*d^2*e*csgn(I*
c*x^n)^3*x^r-2304*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-108*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c
)*(x^r)^2+320*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+3968*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(
x^r)^3+4608*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-2048*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-16896*I*Pi*
b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+6144*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+1632*I*P
i*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+2304*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+2048*I*Pi*b*e^3*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)*(x^r)^3+16896*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-18624*ln(c)*b*d^2*e*r^3*x^r-36096*ln(c)*b
*d^2*e*r^2*x^r-33792*ln(c)*b*d^2*e*r*x^r-13056*ln(c)*b*d*e^2*r^3*(x^r)^2-29184*ln(c)*b*d*e^2*r^2*(x^r)^2-30720
*ln(c)*b*d*e^2*r*(x^r)^2+2048*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-7424*I*Pi*b*d^3*r^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2-4608*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+18048*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+
15360*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-18*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-264*I*Pi*b*d^3*r^5*cs
gn(I*x^n)*csgn(I*c*x^n)^2-7424*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)-6144*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c
*x^n)^2-216*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6528*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(
x^r)^2-1368*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+6528*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)*(x^r)^2+9312*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+14592*I*Pi*b*d*e^2*r^2*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-6144*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)-264*I*Pi*b*d^3*r^5*csgn(I*c*
x^n)^2*csgn(I*c)-1544*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-1544*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)
+2048*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3-9312*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+4608*I*Pi*b*d^3
*r^3*csgn(I*c*x^n)^3+7424*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+6144*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+2304*I*Pi*b*d^2*e*r
^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+108*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+21
6*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-6144*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)
^2-6144*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+216*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+1368*I*Pi*b*d*
e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-3968*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-3968*I*Pi*b*e^3*r^2*cs
gn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+4608*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+7424*I*Pi*b*d^3*r^2*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-9312*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-14592*I*Pi*b*d*e^2*r^2*c
sgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-14592*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+1544*I*Pi*b*d^3*r^
4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+9312*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r+14592*I*Pi*b*d*e^2*r^2*csgn(I*
c*x^n)^3*(x^r)^2-4608*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-2048*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^
n)^2-2048*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+18*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+264*I*Pi*b*d^3*r^5*csgn(I*c*x
^n)^3+1544*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3-4608*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-4608*I*Pi*b*d^3*r^3*
csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+1368*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)*(x^r)^2)/(4+3*r)^2/(2+r)^2/(4+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^3*(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 = 1.43212, size = 2461, normalized size = 16.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*(9*b*d^3*r^6 + 132*b*d^3*r^5 + 772*b*d^3*r^4 + 2304*b*d^3*r^3 + 3712*b*d^3*r^2 + 3072*b*d^3*r + 1024*b
*d^3)*x^4*log(c) + 4*(9*b*d^3*n*r^6 + 132*b*d^3*n*r^5 + 772*b*d^3*n*r^4 + 2304*b*d^3*n*r^3 + 3712*b*d^3*n*r^2
+ 3072*b*d^3*n*r + 1024*b*d^3*n)*x^4*log(x) - (9*(b*d^3*n - 4*a*d^3)*r^6 + 132*(b*d^3*n - 4*a*d^3)*r^5 + 1024*
b*d^3*n + 772*(b*d^3*n - 4*a*d^3)*r^4 - 4096*a*d^3 + 2304*(b*d^3*n - 4*a*d^3)*r^3 + 3712*(b*d^3*n - 4*a*d^3)*r
^2 + 3072*(b*d^3*n - 4*a*d^3)*r)*x^4 + 16*((3*b*e^3*r^5 + 40*b*e^3*r^4 + 204*b*e^3*r^3 + 496*b*e^3*r^2 + 576*b
*e^3*r + 256*b*e^3)*x^4*log(c) + (3*b*e^3*n*r^5 + 40*b*e^3*n*r^4 + 204*b*e^3*n*r^3 + 496*b*e^3*n*r^2 + 576*b*e
^3*n*r + 256*b*e^3*n)*x^4*log(x) + (3*a*e^3*r^5 - 64*b*e^3*n - (b*e^3*n - 40*a*e^3)*r^4 + 256*a*e^3 - 12*(b*e^
3*n - 17*a*e^3)*r^3 - 4*(13*b*e^3*n - 124*a*e^3)*r^2 - 96*(b*e^3*n - 6*a*e^3)*r)*x^4)*x^(3*r) + 12*(2*(9*b*d*e
^2*r^5 + 114*b*d*e^2*r^4 + 544*b*d*e^2*r^3 + 1216*b*d*e^2*r^2 + 1280*b*d*e^2*r + 512*b*d*e^2)*x^4*log(c) + 2*(
9*b*d*e^2*n*r^5 + 114*b*d*e^2*n*r^4 + 544*b*d*e^2*n*r^3 + 1216*b*d*e^2*n*r^2 + 1280*b*d*e^2*n*r + 512*b*d*e^2*
n)*x^4*log(x) + (18*a*d*e^2*r^5 - 256*b*d*e^2*n - 3*(3*b*d*e^2*n - 76*a*d*e^2)*r^4 + 1024*a*d*e^2 - 32*(3*b*d*
e^2*n - 34*a*d*e^2)*r^3 - 32*(11*b*d*e^2*n - 76*a*d*e^2)*r^2 - 512*(b*d*e^2*n - 5*a*d*e^2)*r)*x^4)*x^(2*r) + 4
8*((9*b*d^2*e*r^5 + 96*b*d^2*e*r^4 + 388*b*d^2*e*r^3 + 752*b*d^2*e*r^2 + 704*b*d^2*e*r + 256*b*d^2*e)*x^4*log(
c) + (9*b*d^2*e*n*r^5 + 96*b*d^2*e*n*r^4 + 388*b*d^2*e*n*r^3 + 752*b*d^2*e*n*r^2 + 704*b*d^2*e*n*r + 256*b*d^2
*e*n)*x^4*log(x) + (9*a*d^2*e*r^5 - 64*b*d^2*e*n - 3*(3*b*d^2*e*n - 32*a*d^2*e)*r^4 + 256*a*d^2*e - 4*(15*b*d^
2*e*n - 97*a*d^2*e)*r^3 - 4*(37*b*d^2*e*n - 188*a*d^2*e)*r^2 - 32*(5*b*d^2*e*n - 22*a*d^2*e)*r)*x^4)*x^r)/(9*r
^6 + 132*r^5 + 772*r^4 + 2304*r^3 + 3712*r^2 + 3072*r + 1024)

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

[Out]

Timed out

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Giac [B]  time = 1.39621, 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^3*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/16*(36*b*d^3*n*r^6*x^4*log(x) + 432*b*d^2*n*r^5*x^4*x^r*e*log(x) - 9*b*d^3*n*r^6*x^4 + 36*b*d^3*r^6*x^4*log(
c) + 432*b*d^2*r^5*x^4*x^r*e*log(c) + 528*b*d^3*n*r^5*x^4*log(x) + 216*b*d*n*r^5*x^4*x^(2*r)*e^2*log(x) + 4608
*b*d^2*n*r^4*x^4*x^r*e*log(x) - 132*b*d^3*n*r^5*x^4 + 36*a*d^3*r^6*x^4 - 432*b*d^2*n*r^4*x^4*x^r*e + 432*a*d^2
*r^5*x^4*x^r*e + 528*b*d^3*r^5*x^4*log(c) + 216*b*d*r^5*x^4*x^(2*r)*e^2*log(c) + 4608*b*d^2*r^4*x^4*x^r*e*log(
c) + 3088*b*d^3*n*r^4*x^4*log(x) + 48*b*n*r^5*x^4*x^(3*r)*e^3*log(x) + 2736*b*d*n*r^4*x^4*x^(2*r)*e^2*log(x) +
18624*b*d^2*n*r^3*x^4*x^r*e*log(x) - 772*b*d^3*n*r^4*x^4 + 528*a*d^3*r^5*x^4 - 108*b*d*n*r^4*x^4*x^(2*r)*e^2
+ 216*a*d*r^5*x^4*x^(2*r)*e^2 - 2880*b*d^2*n*r^3*x^4*x^r*e + 4608*a*d^2*r^4*x^4*x^r*e + 3088*b*d^3*r^4*x^4*log
(c) + 48*b*r^5*x^4*x^(3*r)*e^3*log(c) + 2736*b*d*r^4*x^4*x^(2*r)*e^2*log(c) + 18624*b*d^2*r^3*x^4*x^r*e*log(c)
+ 9216*b*d^3*n*r^3*x^4*log(x) + 640*b*n*r^4*x^4*x^(3*r)*e^3*log(x) + 13056*b*d*n*r^3*x^4*x^(2*r)*e^2*log(x) +
36096*b*d^2*n*r^2*x^4*x^r*e*log(x) - 2304*b*d^3*n*r^3*x^4 + 3088*a*d^3*r^4*x^4 - 16*b*n*r^4*x^4*x^(3*r)*e^3 +
48*a*r^5*x^4*x^(3*r)*e^3 - 1152*b*d*n*r^3*x^4*x^(2*r)*e^2 + 2736*a*d*r^4*x^4*x^(2*r)*e^2 - 7104*b*d^2*n*r^2*x
^4*x^r*e + 18624*a*d^2*r^3*x^4*x^r*e + 9216*b*d^3*r^3*x^4*log(c) + 640*b*r^4*x^4*x^(3*r)*e^3*log(c) + 13056*b*
d*r^3*x^4*x^(2*r)*e^2*log(c) + 36096*b*d^2*r^2*x^4*x^r*e*log(c) + 14848*b*d^3*n*r^2*x^4*log(x) + 3264*b*n*r^3*
x^4*x^(3*r)*e^3*log(x) + 29184*b*d*n*r^2*x^4*x^(2*r)*e^2*log(x) + 33792*b*d^2*n*r*x^4*x^r*e*log(x) - 3712*b*d^
3*n*r^2*x^4 + 9216*a*d^3*r^3*x^4 - 192*b*n*r^3*x^4*x^(3*r)*e^3 + 640*a*r^4*x^4*x^(3*r)*e^3 - 4224*b*d*n*r^2*x^
4*x^(2*r)*e^2 + 13056*a*d*r^3*x^4*x^(2*r)*e^2 - 7680*b*d^2*n*r*x^4*x^r*e + 36096*a*d^2*r^2*x^4*x^r*e + 14848*b
*d^3*r^2*x^4*log(c) + 3264*b*r^3*x^4*x^(3*r)*e^3*log(c) + 29184*b*d*r^2*x^4*x^(2*r)*e^2*log(c) + 33792*b*d^2*r
*x^4*x^r*e*log(c) + 12288*b*d^3*n*r*x^4*log(x) + 7936*b*n*r^2*x^4*x^(3*r)*e^3*log(x) + 30720*b*d*n*r*x^4*x^(2*
r)*e^2*log(x) + 12288*b*d^2*n*x^4*x^r*e*log(x) - 3072*b*d^3*n*r*x^4 + 14848*a*d^3*r^2*x^4 - 832*b*n*r^2*x^4*x^
(3*r)*e^3 + 3264*a*r^3*x^4*x^(3*r)*e^3 - 6144*b*d*n*r*x^4*x^(2*r)*e^2 + 29184*a*d*r^2*x^4*x^(2*r)*e^2 - 3072*b
*d^2*n*x^4*x^r*e + 33792*a*d^2*r*x^4*x^r*e + 12288*b*d^3*r*x^4*log(c) + 7936*b*r^2*x^4*x^(3*r)*e^3*log(c) + 30
720*b*d*r*x^4*x^(2*r)*e^2*log(c) + 12288*b*d^2*x^4*x^r*e*log(c) + 4096*b*d^3*n*x^4*log(x) + 9216*b*n*r*x^4*x^(
3*r)*e^3*log(x) + 12288*b*d*n*x^4*x^(2*r)*e^2*log(x) - 1024*b*d^3*n*x^4 + 12288*a*d^3*r*x^4 - 1536*b*n*r*x^4*x
^(3*r)*e^3 + 7936*a*r^2*x^4*x^(3*r)*e^3 - 3072*b*d*n*x^4*x^(2*r)*e^2 + 30720*a*d*r*x^4*x^(2*r)*e^2 + 12288*a*d
^2*x^4*x^r*e + 4096*b*d^3*x^4*log(c) + 9216*b*r*x^4*x^(3*r)*e^3*log(c) + 12288*b*d*x^4*x^(2*r)*e^2*log(c) + 40
96*b*n*x^4*x^(3*r)*e^3*log(x) + 4096*a*d^3*x^4 - 1024*b*n*x^4*x^(3*r)*e^3 + 9216*a*r*x^4*x^(3*r)*e^3 + 12288*a
*d*x^4*x^(2*r)*e^2 + 4096*b*x^4*x^(3*r)*e^3*log(c) + 4096*a*x^4*x^(3*r)*e^3)/(9*r^6 + 132*r^5 + 772*r^4 + 2304
*r^3 + 3712*r^2 + 3072*r + 1024)