3.392 \(\int x^5 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=147 \[ \frac{1}{6} \left (\frac{18 d^2 e x^{r+6}}{r+6}+d^3 x^6+\frac{9 d e^2 x^{2 (r+3)}}{r+3}+\frac{2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+6}}{(r+6)^2}-\frac{1}{36} b d^3 n x^6-\frac{3 b d e^2 n x^{2 (r+3)}}{4 (r+3)^2}-\frac{b e^3 n x^{3 (r+2)}}{9 (r+2)^2} \]

[Out]

-(b*d^3*n*x^6)/36 - (b*e^3*n*x^(3*(2 + r)))/(9*(2 + r)^2) - (3*b*d*e^2*n*x^(2*(3 + r)))/(4*(3 + r)^2) - (3*b*d
^2*e*n*x^(6 + r))/(6 + r)^2 + ((d^3*x^6 + (2*e^3*x^(3*(2 + r)))/(2 + r) + (9*d*e^2*x^(2*(3 + r)))/(3 + r) + (1
8*d^2*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6

________________________________________________________________________________________

Rubi [A]  time = 0.38107, antiderivative size = 147, 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} \[ \frac{1}{6} \left (\frac{18 d^2 e x^{r+6}}{r+6}+d^3 x^6+\frac{9 d e^2 x^{2 (r+3)}}{r+3}+\frac{2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r+6}}{(r+6)^2}-\frac{1}{36} b d^3 n x^6-\frac{3 b d e^2 n x^{2 (r+3)}}{4 (r+3)^2}-\frac{b e^3 n x^{3 (r+2)}}{9 (r+2)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^6)/36 - (b*e^3*n*x^(3*(2 + r)))/(9*(2 + r)^2) - (3*b*d*e^2*n*x^(2*(3 + r)))/(4*(3 + r)^2) - (3*b*d
^2*e*n*x^(6 + r))/(6 + r)^2 + ((d^3*x^6 + (2*e^3*x^(3*(2 + r)))/(2 + r) + (9*d*e^2*x^(2*(3 + r)))/(3 + r) + (1
8*d^2*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6

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^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{6} \left (d^3 x^6+\frac{2 e^3 x^{3 (2+r)}}{2+r}+\frac{9 d e^2 x^{2 (3+r)}}{3+r}+\frac{18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{6} x^5 \left (d^3+\frac{18 d^2 e x^r}{6+r}+\frac{9 d e^2 x^{2 r}}{3+r}+\frac{2 e^3 x^{3 r}}{2+r}\right ) \, dx\\ &=\frac{1}{6} \left (d^3 x^6+\frac{2 e^3 x^{3 (2+r)}}{2+r}+\frac{9 d e^2 x^{2 (3+r)}}{3+r}+\frac{18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int x^5 \left (d^3+\frac{18 d^2 e x^r}{6+r}+\frac{9 d e^2 x^{2 r}}{3+r}+\frac{2 e^3 x^{3 r}}{2+r}\right ) \, dx\\ &=\frac{1}{6} \left (d^3 x^6+\frac{2 e^3 x^{3 (2+r)}}{2+r}+\frac{9 d e^2 x^{2 (3+r)}}{3+r}+\frac{18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \left (d^3 x^5+\frac{18 d^2 e x^{5+r}}{6+r}+\frac{9 d e^2 x^{5+2 r}}{3+r}+\frac{2 e^3 x^{5+3 r}}{2+r}\right ) \, dx\\ &=-\frac{1}{36} b d^3 n x^6-\frac{b e^3 n x^{3 (2+r)}}{9 (2+r)^2}-\frac{3 b d e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac{3 b d^2 e n x^{6+r}}{(6+r)^2}+\frac{1}{6} \left (d^3 x^6+\frac{2 e^3 x^{3 (2+r)}}{2+r}+\frac{9 d e^2 x^{2 (3+r)}}{3+r}+\frac{18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.377303, size = 172, normalized size = 1.17 \[ \frac{1}{36} x^6 \left (6 a \left (\frac{18 d^2 e x^r}{r+6}+d^3+\frac{9 d e^2 x^{2 r}}{r+3}+\frac{2 e^3 x^{3 r}}{r+2}\right )+6 b \log \left (c x^n\right ) \left (\frac{18 d^2 e x^r}{r+6}+d^3+\frac{9 d e^2 x^{2 r}}{r+3}+\frac{2 e^3 x^{3 r}}{r+2}\right )+b n \left (-\frac{108 d^2 e x^r}{(r+6)^2}-d^3-\frac{27 d e^2 x^{2 r}}{(r+3)^2}-\frac{4 e^3 x^{3 r}}{(r+2)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

(x^6*(b*n*(-d^3 - (108*d^2*e*x^r)/(6 + r)^2 - (27*d*e^2*x^(2*r))/(3 + r)^2 - (4*e^3*x^(3*r))/(2 + r)^2) + 6*a*
(d^3 + (18*d^2*e*x^r)/(6 + r) + (9*d*e^2*x^(2*r))/(3 + r) + (2*e^3*x^(3*r))/(2 + r)) + 6*b*(d^3 + (18*d^2*e*x^
r)/(6 + r) + (9*d*e^2*x^(2*r))/(3 + r) + (2*e^3*x^(3*r))/(2 + r))*Log[c*x^n]))/36

________________________________________________________________________________________

Maple [C]  time = 0.398, size = 4021, normalized size = 27.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d+e*x^r)^3*(a+b*ln(c*x^n)),x)

[Out]

1/6*x^6*b*(2*e^3*r^2*(x^r)^3+9*d*e^2*r^2*(x^r)^2+18*e^3*r*(x^r)^3+d^3*r^3+18*d^2*e*r^2*x^r+72*d*e^2*r*(x^r)^2+
36*e^3*(x^r)^3+11*d^3*r^2+90*d^2*e*r*x^r+108*d*e^2*(x^r)^2+36*d^3*r+108*d^2*e*x^r+36*d^3)/(2+r)/(3+r)/(6+r)*ln
(x^n)-1/36*x^6*(-7776*a*d^3+3672*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-15228*I*Pi*b*d^2
*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+3888*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3-7776*a*e^3*(x^r)^3+2592*I*Pi*b*d^
3*r^3*csgn(I*c*x^n)^3+6264*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+7776*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-3888*I*Pi*b*d^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2-7776*ln(c)*b*d^3-5238*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-12312*I*Pi*b*d*
e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+21384*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-11664*I*Pi*b*d*e^2*csgn(I
*x^n)*csgn(I*c*x^n)^2*(x^r)^2-11664*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+b*d^3*n*r^6+22*b*d^3*n*r^5+
193*b*d^3*n*r^4-6*a*d^3*r^6-132*a*d^3*r^5-1158*a*d^3*r^4-3888*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+6
*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3+120*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+5832*I*Pi*b*e^3*r*csgn(I*c*
x^n)^3*(x^r)^3-3888*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+11664*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+21
384*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+513*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*(x^r)^2-12*a*e^3*r^5*(x^r)^3-240*a*e^3*r^4*(x^r)^3-23328*a*d*e^2*(x^r)^2-23328*a*d^2*e*x^r+1296*b*e^3*n*
(x^r)^3-1836*a*e^3*r^3*(x^r)^3-6696*a*e^3*r^2*(x^r)^3-11664*a*e^3*r*(x^r)^3-7776*ln(c)*b*e^3*(x^r)^3+864*b*d^3
*n*r^3+2088*b*d^3*n*r^2+2592*b*d^3*n*r-6*ln(c)*b*d^3*r^6-132*ln(c)*b*d^3*r^5-1158*ln(c)*b*d^3*r^4-5184*ln(c)*b
*d^3*r^3-12528*ln(c)*b*d^3*r^2-15552*ln(c)*b*d^3*r-5184*a*d^3*r^3-12528*a*d^3*r^2-15552*a*d^3*r-38880*a*d*e^2*
r*(x^r)^2-10476*a*d^2*e*r^3*x^r-30456*a*d^2*e*r^2*x^r-42768*a*d^2*e*r*x^r-3672*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^
2*csgn(I*c)*(x^r)^2-513*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+6*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)*(x^r)^3+1296*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-3672*I*Pi*b*d*e^2*r
^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-918*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-6*I*Pi*b*e^3*r^5*c
sgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-6*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+27*I*Pi*b*d*e^2*r^5*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)^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)^3-1836*ln(c)*b*e^3*r^3*(x^r)^3-6696*ln(c)*b*e^3*r^2*(x^r)^3-11664*
ln(c)*b*e^3*r*(x^r)^3-23328*ln(c)*b*d^2*e*x^r-23328*ln(c)*b*d*e^2*(x^r)^2+468*b*e^3*n*r^2*(x^r)^3+1296*b*e^3*n
*r*(x^r)^3+3888*b*d*e^2*n*(x^r)^2+3888*b*d^2*e*n*x^r-7344*a*d*e^2*r^3*(x^r)^2-24624*a*d*e^2*r^2*(x^r)^2+4*b*e^
3*n*r^4*(x^r)^3+72*b*e^3*n*r^3*(x^r)^3-54*a*d*e^2*r^5*(x^r)^2-1026*a*d*e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r-172
8*a*d^2*e*r^4*x^r-12*ln(c)*b*e^3*r^5*(x^r)^3+3888*I*Pi*b*d^3*csgn(I*c*x^n)^3+2376*b*d*e^2*n*r^2*(x^r)^2+3996*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+3672*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-3348*I
*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-3348*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+513*I*
Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2+12312*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+1522
8*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+19440*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*(x^r)^2+5238*I*Pi*b*d^2*e*r^3*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+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+918*I*Pi*b*e^3*r^3*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*(x^r)^3+11664*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+11664*I*Pi*b*d^2*
e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+2592*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+6264*I*Pi*b*
d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+7776*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+5184*b*d*e^2
*n*r*(x^r)^2+6480*b*d^2*e*n*r*x^r+27*b*d*e^2*n*r^4*(x^r)^2+432*b*d*e^2*n*r^3*(x^r)^2+108*b*d^2*e*n*r^4*x^r+108
0*b*d^2*e*n*r^3*x^r-54*ln(c)*b*d*e^2*r^5*(x^r)^2-1026*ln(c)*b*d*e^2*r^4*(x^r)^2-108*ln(c)*b*d^2*e*r^5*x^r-1728
*ln(c)*b*d^2*e*r^4*x^r+918*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+3348*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3-
12312*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+5832*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
*(x^r)^3+54*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+5238*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r+12312*I*Pi*b*d*e^2*
r^2*csgn(I*c*x^n)^3*(x^r)^2-5832*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-5832*I*Pi*b*e^3*r*csgn(I*c*x
^n)^2*csgn(I*c)*(x^r)^3-10476*ln(c)*b*d^2*e*r^3*x^r-30456*ln(c)*b*d^2*e*r^2*x^r-42768*ln(c)*b*d^2*e*r*x^r-7344
*ln(c)*b*d*e^2*r^3*(x^r)^2-24624*ln(c)*b*d*e^2*r^2*(x^r)^2-38880*ln(c)*b*d*e^2*r*(x^r)^2-3888*I*Pi*b*d^3*csgn(
I*c*x^n)^2*csgn(I*c)+3*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+54*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*
x^r-7776*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)+3888*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-27*I*Pi*b*
d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3348*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-
5238*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-3*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d^
3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-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)-11664*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-11664*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^
r-918*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-6264*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-6264*
I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)-7776*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-15228*I*Pi*b*d^2*e*r^2*
csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+864*I*Pi*b*d^2*e*r^4*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)*csgn(I*c)*(x^r)^2+66*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3+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)+11664*I*Pi*b
*d^2*e*csgn(I*c*x^n)^3*x^r+15228*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+19440*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^
r)^2+3888*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+3*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)+66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+579*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)-19440*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-21384*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2
*x^r-21384*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*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-513*
I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-19440*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-
2592*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-2592*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c))/(2+r)^2/(3+r)^2
/(6+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^5*(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 = 1.39827, size = 2402, normalized size = 16.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/36*(6*(b*d^3*r^6 + 22*b*d^3*r^5 + 193*b*d^3*r^4 + 864*b*d^3*r^3 + 2088*b*d^3*r^2 + 2592*b*d^3*r + 1296*b*d^3
)*x^6*log(c) + 6*(b*d^3*n*r^6 + 22*b*d^3*n*r^5 + 193*b*d^3*n*r^4 + 864*b*d^3*n*r^3 + 2088*b*d^3*n*r^2 + 2592*b
*d^3*n*r + 1296*b*d^3*n)*x^6*log(x) - ((b*d^3*n - 6*a*d^3)*r^6 + 22*(b*d^3*n - 6*a*d^3)*r^5 + 1296*b*d^3*n + 1
93*(b*d^3*n - 6*a*d^3)*r^4 - 7776*a*d^3 + 864*(b*d^3*n - 6*a*d^3)*r^3 + 2088*(b*d^3*n - 6*a*d^3)*r^2 + 2592*(b
*d^3*n - 6*a*d^3)*r)*x^6 + 4*(3*(b*e^3*r^5 + 20*b*e^3*r^4 + 153*b*e^3*r^3 + 558*b*e^3*r^2 + 972*b*e^3*r + 648*
b*e^3)*x^6*log(c) + 3*(b*e^3*n*r^5 + 20*b*e^3*n*r^4 + 153*b*e^3*n*r^3 + 558*b*e^3*n*r^2 + 972*b*e^3*n*r + 648*
b*e^3*n)*x^6*log(x) + (3*a*e^3*r^5 - 324*b*e^3*n - (b*e^3*n - 60*a*e^3)*r^4 + 1944*a*e^3 - 9*(2*b*e^3*n - 51*a
*e^3)*r^3 - 9*(13*b*e^3*n - 186*a*e^3)*r^2 - 324*(b*e^3*n - 9*a*e^3)*r)*x^6)*x^(3*r) + 27*(2*(b*d*e^2*r^5 + 19
*b*d*e^2*r^4 + 136*b*d*e^2*r^3 + 456*b*d*e^2*r^2 + 720*b*d*e^2*r + 432*b*d*e^2)*x^6*log(c) + 2*(b*d*e^2*n*r^5
+ 19*b*d*e^2*n*r^4 + 136*b*d*e^2*n*r^3 + 456*b*d*e^2*n*r^2 + 720*b*d*e^2*n*r + 432*b*d*e^2*n)*x^6*log(x) + (2*
a*d*e^2*r^5 - 144*b*d*e^2*n - (b*d*e^2*n - 38*a*d*e^2)*r^4 + 864*a*d*e^2 - 16*(b*d*e^2*n - 17*a*d*e^2)*r^3 - 8
*(11*b*d*e^2*n - 114*a*d*e^2)*r^2 - 96*(2*b*d*e^2*n - 15*a*d*e^2)*r)*x^6)*x^(2*r) + 108*((b*d^2*e*r^5 + 16*b*d
^2*e*r^4 + 97*b*d^2*e*r^3 + 282*b*d^2*e*r^2 + 396*b*d^2*e*r + 216*b*d^2*e)*x^6*log(c) + (b*d^2*e*n*r^5 + 16*b*
d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 + 282*b*d^2*e*n*r^2 + 396*b*d^2*e*n*r + 216*b*d^2*e*n)*x^6*log(x) + (a*d^2*e*r^
5 - 36*b*d^2*e*n - (b*d^2*e*n - 16*a*d^2*e)*r^4 + 216*a*d^2*e - (10*b*d^2*e*n - 97*a*d^2*e)*r^3 - (37*b*d^2*e*
n - 282*a*d^2*e)*r^2 - 12*(5*b*d^2*e*n - 33*a*d^2*e)*r)*x^6)*x^r)/(r^6 + 22*r^5 + 193*r^4 + 864*r^3 + 2088*r^2
+ 2592*r + 1296)

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

[Out]

Timed out

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Giac [B]  time = 1.62734, size = 2141, normalized size = 14.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/36*(6*b*d^3*n*r^6*x^6*log(x) + 108*b*d^2*n*r^5*x^6*x^r*e*log(x) - b*d^3*n*r^6*x^6 + 6*b*d^3*r^6*x^6*log(c) +
108*b*d^2*r^5*x^6*x^r*e*log(c) + 132*b*d^3*n*r^5*x^6*log(x) + 54*b*d*n*r^5*x^6*x^(2*r)*e^2*log(x) + 1728*b*d^
2*n*r^4*x^6*x^r*e*log(x) - 22*b*d^3*n*r^5*x^6 + 6*a*d^3*r^6*x^6 - 108*b*d^2*n*r^4*x^6*x^r*e + 108*a*d^2*r^5*x^
6*x^r*e + 132*b*d^3*r^5*x^6*log(c) + 54*b*d*r^5*x^6*x^(2*r)*e^2*log(c) + 1728*b*d^2*r^4*x^6*x^r*e*log(c) + 115
8*b*d^3*n*r^4*x^6*log(x) + 12*b*n*r^5*x^6*x^(3*r)*e^3*log(x) + 1026*b*d*n*r^4*x^6*x^(2*r)*e^2*log(x) + 10476*b
*d^2*n*r^3*x^6*x^r*e*log(x) - 193*b*d^3*n*r^4*x^6 + 132*a*d^3*r^5*x^6 - 27*b*d*n*r^4*x^6*x^(2*r)*e^2 + 54*a*d*
r^5*x^6*x^(2*r)*e^2 - 1080*b*d^2*n*r^3*x^6*x^r*e + 1728*a*d^2*r^4*x^6*x^r*e + 1158*b*d^3*r^4*x^6*log(c) + 12*b
*r^5*x^6*x^(3*r)*e^3*log(c) + 1026*b*d*r^4*x^6*x^(2*r)*e^2*log(c) + 10476*b*d^2*r^3*x^6*x^r*e*log(c) + 5184*b*
d^3*n*r^3*x^6*log(x) + 240*b*n*r^4*x^6*x^(3*r)*e^3*log(x) + 7344*b*d*n*r^3*x^6*x^(2*r)*e^2*log(x) + 30456*b*d^
2*n*r^2*x^6*x^r*e*log(x) - 864*b*d^3*n*r^3*x^6 + 1158*a*d^3*r^4*x^6 - 4*b*n*r^4*x^6*x^(3*r)*e^3 + 12*a*r^5*x^6
*x^(3*r)*e^3 - 432*b*d*n*r^3*x^6*x^(2*r)*e^2 + 1026*a*d*r^4*x^6*x^(2*r)*e^2 - 3996*b*d^2*n*r^2*x^6*x^r*e + 104
76*a*d^2*r^3*x^6*x^r*e + 5184*b*d^3*r^3*x^6*log(c) + 240*b*r^4*x^6*x^(3*r)*e^3*log(c) + 7344*b*d*r^3*x^6*x^(2*
r)*e^2*log(c) + 30456*b*d^2*r^2*x^6*x^r*e*log(c) + 12528*b*d^3*n*r^2*x^6*log(x) + 1836*b*n*r^3*x^6*x^(3*r)*e^3
*log(x) + 24624*b*d*n*r^2*x^6*x^(2*r)*e^2*log(x) + 42768*b*d^2*n*r*x^6*x^r*e*log(x) - 2088*b*d^3*n*r^2*x^6 + 5
184*a*d^3*r^3*x^6 - 72*b*n*r^3*x^6*x^(3*r)*e^3 + 240*a*r^4*x^6*x^(3*r)*e^3 - 2376*b*d*n*r^2*x^6*x^(2*r)*e^2 +
7344*a*d*r^3*x^6*x^(2*r)*e^2 - 6480*b*d^2*n*r*x^6*x^r*e + 30456*a*d^2*r^2*x^6*x^r*e + 12528*b*d^3*r^2*x^6*log(
c) + 1836*b*r^3*x^6*x^(3*r)*e^3*log(c) + 24624*b*d*r^2*x^6*x^(2*r)*e^2*log(c) + 42768*b*d^2*r*x^6*x^r*e*log(c)
+ 15552*b*d^3*n*r*x^6*log(x) + 6696*b*n*r^2*x^6*x^(3*r)*e^3*log(x) + 38880*b*d*n*r*x^6*x^(2*r)*e^2*log(x) + 2
3328*b*d^2*n*x^6*x^r*e*log(x) - 2592*b*d^3*n*r*x^6 + 12528*a*d^3*r^2*x^6 - 468*b*n*r^2*x^6*x^(3*r)*e^3 + 1836*
a*r^3*x^6*x^(3*r)*e^3 - 5184*b*d*n*r*x^6*x^(2*r)*e^2 + 24624*a*d*r^2*x^6*x^(2*r)*e^2 - 3888*b*d^2*n*x^6*x^r*e
+ 42768*a*d^2*r*x^6*x^r*e + 15552*b*d^3*r*x^6*log(c) + 6696*b*r^2*x^6*x^(3*r)*e^3*log(c) + 38880*b*d*r*x^6*x^(
2*r)*e^2*log(c) + 23328*b*d^2*x^6*x^r*e*log(c) + 7776*b*d^3*n*x^6*log(x) + 11664*b*n*r*x^6*x^(3*r)*e^3*log(x)
+ 23328*b*d*n*x^6*x^(2*r)*e^2*log(x) - 1296*b*d^3*n*x^6 + 15552*a*d^3*r*x^6 - 1296*b*n*r*x^6*x^(3*r)*e^3 + 669
6*a*r^2*x^6*x^(3*r)*e^3 - 3888*b*d*n*x^6*x^(2*r)*e^2 + 38880*a*d*r*x^6*x^(2*r)*e^2 + 23328*a*d^2*x^6*x^r*e + 7
776*b*d^3*x^6*log(c) + 11664*b*r*x^6*x^(3*r)*e^3*log(c) + 23328*b*d*x^6*x^(2*r)*e^2*log(c) + 7776*b*n*x^6*x^(3
*r)*e^3*log(x) + 7776*a*d^3*x^6 - 1296*b*n*x^6*x^(3*r)*e^3 + 11664*a*r*x^6*x^(3*r)*e^3 + 23328*a*d*x^6*x^(2*r)
*e^2 + 7776*b*x^6*x^(3*r)*e^3*log(c) + 7776*a*x^6*x^(3*r)*e^3)/(r^6 + 22*r^5 + 193*r^4 + 864*r^3 + 2088*r^2 +
2592*r + 1296)