\(\int x (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [394]

Optimal result

Integrand size = 21, antiderivative size = 149 \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b d^3 n x^2-\frac {3 b d e^2 n x^{2 (1+r)}}{4 (1+r)^2}-\frac {3 b d^2 e n x^{2+r}}{(2+r)^2}-\frac {b e^3 n x^{2+3 r}}{(2+3 r)^2}+\frac {1}{2} \left (d^3 x^2+\frac {3 d e^2 x^{2 (1+r)}}{1+r}+\frac {6 d^2 e x^{2+r}}{2+r}+\frac {2 e^3 x^{2+3 r}}{2+3 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {276, 2371, 12, 14} \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{2} \left (d^3 x^2+\frac {6 d^2 e x^{r+2}}{r+2}+\frac {3 d e^2 x^{2 (r+1)}}{r+1}+\frac {2 e^3 x^{3 r+2}}{3 r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d^3 n x^2-\frac {3 b d^2 e n x^{r+2}}{(r+2)^2}-\frac {3 b d e^2 n x^{2 (r+1)}}{4 (r+1)^2}-\frac {b e^3 n x^{3 r+2}}{(3 r+2)^2} \]

[In]

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

[Out]

-1/4*(b*d^3*n*x^2) - (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 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]]

Rule 276

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 2371

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] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.19 \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} x^2 \left (b n \left (-d^3-\frac {12 d^2 e x^r}{(2+r)^2}-\frac {3 d e^2 x^{2 r}}{(1+r)^2}-\frac {4 e^3 x^{3 r}}{(2+3 r)^2}\right )+2 a \left (d^3+\frac {6 d^2 e x^r}{2+r}+\frac {3 d e^2 x^{2 r}}{1+r}+\frac {2 e^3 x^{3 r}}{2+3 r}\right )+2 b \left (d^3+\frac {6 d^2 e x^r}{2+r}+\frac {3 d e^2 x^{2 r}}{1+r}+\frac {2 e^3 x^{3 r}}{2+3 r}\right ) \log \left (c x^n\right )\right ) \]

[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1266\) vs. \(2(143)=286\).

Time = 5.65 (sec) , antiderivative size = 1267, normalized size of antiderivative = 8.50

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1267\)
risch	\(\text {Expression too large to display}\)	\(4027\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1024 vs. \(2 (143) = 286\).

Time = 0.34 (sec) , antiderivative size = 1024, normalized size of antiderivative = 6.87 \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \, {\left (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}\right )} x^{2} \log \left (c\right ) + 2 \, {\left (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\right )} x^{2} \log \left (x\right ) - {\left (9 \, {\left (b d^{3} n - 2 \, a d^{3}\right )} r^{6} + 66 \, {\left (b d^{3} n - 2 \, a d^{3}\right )} r^{5} + 16 \, b d^{3} n + 193 \, {\left (b d^{3} n - 2 \, a d^{3}\right )} r^{4} - 32 \, a d^{3} + 288 \, {\left (b d^{3} n - 2 \, a d^{3}\right )} r^{3} + 232 \, {\left (b d^{3} n - 2 \, a d^{3}\right )} r^{2} + 96 \, {\left (b d^{3} n - 2 \, a d^{3}\right )} r\right )} x^{2} + 4 \, {\left ({\left (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}\right )} x^{2} \log \left (c\right ) + {\left (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\right )} x^{2} \log \left (x\right ) + {\left (3 \, a e^{3} r^{5} - 4 \, b e^{3} n - {\left (b e^{3} n - 20 \, a e^{3}\right )} r^{4} + 8 \, a e^{3} - 3 \, {\left (2 \, b e^{3} n - 17 \, a e^{3}\right )} r^{3} - {\left (13 \, b e^{3} n - 62 \, a e^{3}\right )} r^{2} - 12 \, {\left (b e^{3} n - 3 \, a e^{3}\right )} r\right )} x^{2}\right )} x^{3 \, r} + 3 \, {\left (2 \, {\left (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}\right )} x^{2} \log \left (c\right ) + 2 \, {\left (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\right )} x^{2} \log \left (x\right ) + {\left (18 \, a d e^{2} r^{5} - 16 \, b d e^{2} n - 3 \, {\left (3 \, b d e^{2} n - 38 \, a d e^{2}\right )} r^{4} + 32 \, a d e^{2} - 16 \, {\left (3 \, b d e^{2} n - 17 \, a d e^{2}\right )} r^{3} - 8 \, {\left (11 \, b d e^{2} n - 38 \, a d e^{2}\right )} r^{2} - 32 \, {\left (2 \, b d e^{2} n - 5 \, a d e^{2}\right )} r\right )} x^{2}\right )} x^{2 \, r} + 12 \, {\left ({\left (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\right )} x^{2} \log \left (c\right ) + {\left (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\right )} x^{2} \log \left (x\right ) + {\left (9 \, a d^{2} e r^{5} - 4 \, b d^{2} e n - 3 \, {\left (3 \, b d^{2} e n - 16 \, a d^{2} e\right )} r^{4} + 8 \, a d^{2} e - {\left (30 \, b d^{2} e n - 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n - 94 \, a d^{2} e\right )} r^{2} - 4 \, {\left (5 \, b d^{2} e n - 11 \, a d^{2} e\right )} r\right )} x^{2}\right )} x^{r}}{4 \, {\left (9 \, r^{6} + 66 \, r^{5} + 193 \, r^{4} + 288 \, r^{3} + 232 \, r^{2} + 96 \, r + 16\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 86.83 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.40 \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{3} x^{2}}{2} + 3 a d^{2} e \left (\begin {cases} \frac {x^{2} x^{r}}{r + 2} & \text {for}\: r \neq -2 \\x^{2} x^{r} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2} x^{2 r}}{2 r + 2} & \text {for}\: r \neq -1 \\x^{2} x^{2 r} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{2} x^{3 r}}{3 r + 2} & \text {for}\: r \neq - \frac {2}{3} \\x^{2} x^{3 r} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n x^{2}}{4} + \frac {b d^{3} x^{2} \log {\left (c x^{n} \right )}}{2} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r + 2}}{r + 2} & \text {for}\: r \neq -2 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r + 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq -2 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r + 2}}{r + 2} & \text {for}\: r \neq -2 \\\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^{2 r + 2}}{2 r + 2} & \text {for}\: r \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r + 2} & \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 e^{2} \left (\begin {cases} \frac {x^{2 r + 2}}{2 r + 2} & \text {for}\: 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^{3 r + 2}}{3 r + 2} & \text {for}\: r \neq - \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 r + 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq - \frac {2}{3} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r + 2}}{3 r + 2} & \text {for}\: r \neq - \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]

[In]

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

[Out]

a*d**3*x**2/2 + 3*a*d**2*e*Piecewise((x**2*x**r/(r + 2), Ne(r, -2)), (x**2*x**r*log(x), True)) + 3*a*d*e**2*Pi
ecewise((x**2*x**(2*r)/(2*r + 2), Ne(r, -1)), (x**2*x**(2*r)*log(x), True)) + a*e**3*Piecewise((x**2*x**(3*r)/
(3*r + 2), Ne(r, -2/3)), (x**2*x**(3*r)*log(x), True)) - b*d**3*n*x**2/4 + b*d**3*x**2*log(c*x**n)/2 - 3*b*d**
2*e*n*Piecewise((Piecewise((x**(r + 2)/(r + 2), Ne(r, -2)), (log(x), True))/(r + 2), (r > -oo) & (r < oo) & Ne
(r, -2)), (log(x)**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r + 2)/(r + 2), Ne(r, -2)), (log(x), True))*log(c*x
**n) - 3*b*d*e**2*n*Piecewise((Piecewise((x**(2*r + 2)/(2*r + 2), Ne(r, -1)), (log(x), True))/(2*r + 2), (r >
-oo) & (r < oo) & Ne(r, -1)), (log(x)**2/2, True)) + 3*b*d*e**2*Piecewise((x**(2*r + 2)/(2*r + 2), Ne(r, -1)),
(log(x), True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r + 2)/(3*r + 2), Ne(r, -2/3)), (log(x), T
rue))/(3*r + 2), (r > -oo) & (r < oo) & Ne(r, -2/3)), (log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r + 2)/(3
*r + 2), Ne(r, -2/3)), (log(x), True))*log(c*x**n)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.49 \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} \, b d^{3} n x^{2} + \frac {1}{2} \, b d^{3} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d^{3} x^{2} + \frac {b e^{3} x^{3 \, r + 2} \log \left (c x^{n}\right )}{3 \, r + 2} + \frac {3 \, b d e^{2} x^{2 \, r + 2} \log \left (c x^{n}\right )}{2 \, {\left (r + 1\right )}} + \frac {3 \, b d^{2} e x^{r + 2} \log \left (c x^{n}\right )}{r + 2} - \frac {b e^{3} n x^{3 \, r + 2}}{{\left (3 \, r + 2\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 2}}{3 \, r + 2} - \frac {3 \, b d e^{2} n x^{2 \, r + 2}}{4 \, {\left (r + 1\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 2}}{2 \, {\left (r + 1\right )}} - \frac {3 \, b d^{2} e n x^{r + 2}}{{\left (r + 2\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 2}}{r + 2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (143) = 286\).

Time = 0.37 (sec) , antiderivative size = 1611, normalized size of antiderivative = 10.81 \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(12*b*e^3*n*r^5*x^2*x^(3*r)*log(x) + 54*b*d*e^2*n*r^5*x^2*x^(2*r)*log(x) + 108*b*d^2*e*n*r^5*x^2*x^r*log(x
) + 18*b*d^3*n*r^6*x^2*log(x) - 9*b*d^3*n*r^6*x^2 + 12*b*e^3*r^5*x^2*x^(3*r)*log(c) + 54*b*d*e^2*r^5*x^2*x^(2*
r)*log(c) + 108*b*d^2*e*r^5*x^2*x^r*log(c) + 18*b*d^3*r^6*x^2*log(c) + 80*b*e^3*n*r^4*x^2*x^(3*r)*log(x) + 342
*b*d*e^2*n*r^4*x^2*x^(2*r)*log(x) + 576*b*d^2*e*n*r^4*x^2*x^r*log(x) + 132*b*d^3*n*r^5*x^2*log(x) - 4*b*e^3*n*
r^4*x^2*x^(3*r) + 12*a*e^3*r^5*x^2*x^(3*r) - 27*b*d*e^2*n*r^4*x^2*x^(2*r) + 54*a*d*e^2*r^5*x^2*x^(2*r) - 108*b
*d^2*e*n*r^4*x^2*x^r + 108*a*d^2*e*r^5*x^2*x^r - 66*b*d^3*n*r^5*x^2 + 18*a*d^3*r^6*x^2 + 80*b*e^3*r^4*x^2*x^(3
*r)*log(c) + 342*b*d*e^2*r^4*x^2*x^(2*r)*log(c) + 576*b*d^2*e*r^4*x^2*x^r*log(c) + 132*b*d^3*r^5*x^2*log(c) +
204*b*e^3*n*r^3*x^2*x^(3*r)*log(x) + 816*b*d*e^2*n*r^3*x^2*x^(2*r)*log(x) + 1164*b*d^2*e*n*r^3*x^2*x^r*log(x)
+ 386*b*d^3*n*r^4*x^2*log(x) - 24*b*e^3*n*r^3*x^2*x^(3*r) + 80*a*e^3*r^4*x^2*x^(3*r) - 144*b*d*e^2*n*r^3*x^2*x
^(2*r) + 342*a*d*e^2*r^4*x^2*x^(2*r) - 360*b*d^2*e*n*r^3*x^2*x^r + 576*a*d^2*e*r^4*x^2*x^r - 193*b*d^3*n*r^4*x
^2 + 132*a*d^3*r^5*x^2 + 204*b*e^3*r^3*x^2*x^(3*r)*log(c) + 816*b*d*e^2*r^3*x^2*x^(2*r)*log(c) + 1164*b*d^2*e*
r^3*x^2*x^r*log(c) + 386*b*d^3*r^4*x^2*log(c) + 248*b*e^3*n*r^2*x^2*x^(3*r)*log(x) + 912*b*d*e^2*n*r^2*x^2*x^(
2*r)*log(x) + 1128*b*d^2*e*n*r^2*x^2*x^r*log(x) + 576*b*d^3*n*r^3*x^2*log(x) - 52*b*e^3*n*r^2*x^2*x^(3*r) + 20
4*a*e^3*r^3*x^2*x^(3*r) - 264*b*d*e^2*n*r^2*x^2*x^(2*r) + 816*a*d*e^2*r^3*x^2*x^(2*r) - 444*b*d^2*e*n*r^2*x^2*
x^r + 1164*a*d^2*e*r^3*x^2*x^r - 288*b*d^3*n*r^3*x^2 + 386*a*d^3*r^4*x^2 + 248*b*e^3*r^2*x^2*x^(3*r)*log(c) +
912*b*d*e^2*r^2*x^2*x^(2*r)*log(c) + 1128*b*d^2*e*r^2*x^2*x^r*log(c) + 576*b*d^3*r^3*x^2*log(c) + 144*b*e^3*n*
r*x^2*x^(3*r)*log(x) + 480*b*d*e^2*n*r*x^2*x^(2*r)*log(x) + 528*b*d^2*e*n*r*x^2*x^r*log(x) + 464*b*d^3*n*r^2*x
^2*log(x) - 48*b*e^3*n*r*x^2*x^(3*r) + 248*a*e^3*r^2*x^2*x^(3*r) - 192*b*d*e^2*n*r*x^2*x^(2*r) + 912*a*d*e^2*r
^2*x^2*x^(2*r) - 240*b*d^2*e*n*r*x^2*x^r + 1128*a*d^2*e*r^2*x^2*x^r - 232*b*d^3*n*r^2*x^2 + 576*a*d^3*r^3*x^2
+ 144*b*e^3*r*x^2*x^(3*r)*log(c) + 480*b*d*e^2*r*x^2*x^(2*r)*log(c) + 528*b*d^2*e*r*x^2*x^r*log(c) + 464*b*d^3
*r^2*x^2*log(c) + 32*b*e^3*n*x^2*x^(3*r)*log(x) + 96*b*d*e^2*n*x^2*x^(2*r)*log(x) + 96*b*d^2*e*n*x^2*x^r*log(x
) + 192*b*d^3*n*r*x^2*log(x) - 16*b*e^3*n*x^2*x^(3*r) + 144*a*e^3*r*x^2*x^(3*r) - 48*b*d*e^2*n*x^2*x^(2*r) + 4
80*a*d*e^2*r*x^2*x^(2*r) - 48*b*d^2*e*n*x^2*x^r + 528*a*d^2*e*r*x^2*x^r - 96*b*d^3*n*r*x^2 + 464*a*d^3*r^2*x^2
+ 32*b*e^3*x^2*x^(3*r)*log(c) + 96*b*d*e^2*x^2*x^(2*r)*log(c) + 96*b*d^2*e*x^2*x^r*log(c) + 192*b*d^3*r*x^2*l
og(c) + 32*b*d^3*n*x^2*log(x) + 32*a*e^3*x^2*x^(3*r) + 96*a*d*e^2*x^2*x^(2*r) + 96*a*d^2*e*x^2*x^r - 16*b*d^3*
n*x^2 + 192*a*d^3*r*x^2 + 32*b*d^3*x^2*log(c) + 32*a*d^3*x^2)/(9*r^6 + 66*r^5 + 193*r^4 + 288*r^3 + 232*r^2 +
96*r + 16)

Mupad [F(-1)]

Timed out. \[ \int x \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

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

[Out]

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