\(\int x (d+e x^r)^2 (a+b \log (c x^n)) \, dx\) [381]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b d^2 n x^2-\frac {b e^2 n x^{2 (1+r)}}{4 (1+r)^2}-\frac {2 b d e n x^{2+r}}{(2+r)^2}+\frac {1}{2} \left (d^2 x^2+\frac {e^2 x^{2 (1+r)}}{1+r}+\frac {4 d e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 102, 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} \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{2} \left (d^2 x^2+\frac {4 d e x^{r+2}}{r+2}+\frac {e^2 x^{2 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d^2 n x^2-\frac {2 b d e n x^{r+2}}{(r+2)^2}-\frac {b e^2 n x^{2 (r+1)}}{4 (r+1)^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.14 \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} x^2 \left (b n \left (-d^2-\frac {8 d e x^r}{(2+r)^2}-\frac {e^2 x^{2 r}}{(1+r)^2}\right )+2 a \left (d^2+\frac {4 d e x^r}{2+r}+\frac {e^2 x^{2 r}}{1+r}\right )+2 b \left (d^2+\frac {4 d e x^r}{2+r}+\frac {e^2 x^{2 r}}{1+r}\right ) \log \left (c x^n\right )\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(576\) vs. \(2(96)=192\).

Time = 1.79 (sec) , antiderivative size = 577, normalized size of antiderivative = 5.66

method	result	size
parallelrisch	\(-\frac {6 x^{2} b \,d^{2} n \,r^{3}+13 x^{2} b \,d^{2} n \,r^{2}+12 x^{2} b \,d^{2} n r -16 x^{2} x^{r} a d e -2 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-12 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}-26 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-24 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{2} r -2 x^{2} x^{2 r} a \,e^{2} r^{3}-10 x^{2} x^{2 r} a \,e^{2} r^{2}-16 x^{2} x^{2 r} a \,e^{2} r +4 x^{2} x^{2 r} b \,e^{2} n -8 x^{2} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2}-8 x^{2} b \ln \left (c \,x^{n}\right ) d^{2}-8 a \,d^{2} x^{2}-32 x^{2} x^{r} a d e \,r^{2}-40 x^{2} x^{r} a d e r +8 x^{2} x^{r} b d e n -8 x^{2} x^{r} a d e \,r^{3}+x^{2} b \,d^{2} n \,r^{4}+8 x^{2} x^{r} b d e n \,r^{2}+16 x^{2} x^{r} b d e n r -8 x^{2} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}-32 x^{2} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-40 x^{2} x^{r} \ln \left (c \,x^{n}\right ) b d e r -16 x^{2} x^{r} \ln \left (c \,x^{n}\right ) b d e -8 x^{2} x^{2 r} a \,e^{2}+x^{2} x^{2 r} b \,e^{2} n \,r^{2}+4 x^{2} x^{2 r} b \,e^{2} n r -2 x^{2} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}-10 x^{2} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-16 x^{2} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r -2 x^{2} a \,d^{2} r^{4}-12 x^{2} a \,d^{2} r^{3}-26 x^{2} a \,d^{2} r^{2}-24 x^{2} a \,d^{2} r +4 b \,d^{2} n \,x^{2}}{4 \left (1+r \right )^{2} \left (2+r \right )^{2}}\)	\(577\)
risch	\(\text {Expression too large to display}\)	\(1922\)

[In]

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

[Out]

-1/4*(6*x^2*b*d^2*n*r^3+13*x^2*b*d^2*n*r^2+12*x^2*b*d^2*n*r-16*x^2*x^r*a*d*e-2*x^2*(x^r)^2*a*e^2*r^3-10*x^2*(x
^r)^2*a*e^2*r^2-16*x^2*(x^r)^2*a*e^2*r+4*x^2*(x^r)^2*b*e^2*n-2*x^2*ln(c*x^n)*b*d^2*r^4-12*x^2*ln(c*x^n)*b*d^2*
r^3-26*x^2*ln(c*x^n)*b*d^2*r^2-24*x^2*ln(c*x^n)*b*d^2*r-8*x^2*(x^r)^2*ln(c*x^n)*b*e^2-8*x^2*b*ln(c*x^n)*d^2-8*
a*d^2*x^2-32*x^2*x^r*a*d*e*r^2-40*x^2*x^r*a*d*e*r+8*x^2*x^r*b*d*e*n+x^2*(x^r)^2*b*e^2*n*r^2+4*x^2*(x^r)^2*b*e^
2*n*r-2*x^2*(x^r)^2*ln(c*x^n)*b*e^2*r^3-10*x^2*(x^r)^2*ln(c*x^n)*b*e^2*r^2-16*x^2*(x^r)^2*ln(c*x^n)*b*e^2*r-8*
x^2*x^r*a*d*e*r^3+x^2*b*d^2*n*r^4+8*x^2*x^r*b*d*e*n*r^2+16*x^2*x^r*b*d*e*n*r-8*x^2*x^r*ln(c*x^n)*b*d*e*r^3-32*
x^2*x^r*ln(c*x^n)*b*d*e*r^2-40*x^2*x^r*ln(c*x^n)*b*d*e*r-16*x^2*x^r*ln(c*x^n)*b*d*e-2*x^2*a*d^2*r^4-12*x^2*a*d
^2*r^3-26*x^2*a*d^2*r^2-24*x^2*a*d^2*r-8*x^2*(x^r)^2*a*e^2+4*b*d^2*n*x^2)/(1+r)^2/(2+r)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (96) = 192\).

Time = 0.30 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.78 \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \, {\left (b d^{2} r^{4} + 6 \, b d^{2} r^{3} + 13 \, b d^{2} r^{2} + 12 \, b d^{2} r + 4 \, b d^{2}\right )} x^{2} \log \left (c\right ) + 2 \, {\left (b d^{2} n r^{4} + 6 \, b d^{2} n r^{3} + 13 \, b d^{2} n r^{2} + 12 \, b d^{2} n r + 4 \, b d^{2} n\right )} x^{2} \log \left (x\right ) - {\left ({\left (b d^{2} n - 2 \, a d^{2}\right )} r^{4} + 4 \, b d^{2} n + 6 \, {\left (b d^{2} n - 2 \, a d^{2}\right )} r^{3} - 8 \, a d^{2} + 13 \, {\left (b d^{2} n - 2 \, a d^{2}\right )} r^{2} + 12 \, {\left (b d^{2} n - 2 \, a d^{2}\right )} r\right )} x^{2} + {\left (2 \, {\left (b e^{2} r^{3} + 5 \, b e^{2} r^{2} + 8 \, b e^{2} r + 4 \, b e^{2}\right )} x^{2} \log \left (c\right ) + 2 \, {\left (b e^{2} n r^{3} + 5 \, b e^{2} n r^{2} + 8 \, b e^{2} n r + 4 \, b e^{2} n\right )} x^{2} \log \left (x\right ) + {\left (2 \, a e^{2} r^{3} - 4 \, b e^{2} n + 8 \, a e^{2} - {\left (b e^{2} n - 10 \, a e^{2}\right )} r^{2} - 4 \, {\left (b e^{2} n - 4 \, a e^{2}\right )} r\right )} x^{2}\right )} x^{2 \, r} + 8 \, {\left ({\left (b d e r^{3} + 4 \, b d e r^{2} + 5 \, b d e r + 2 \, b d e\right )} x^{2} \log \left (c\right ) + {\left (b d e n r^{3} + 4 \, b d e n r^{2} + 5 \, b d e n r + 2 \, b d e n\right )} x^{2} \log \left (x\right ) + {\left (a d e r^{3} - b d e n + 2 \, a d e - {\left (b d e n - 4 \, a d e\right )} r^{2} - {\left (2 \, b d e n - 5 \, a d e\right )} r\right )} x^{2}\right )} x^{r}}{4 \, {\left (r^{4} + 6 \, r^{3} + 13 \, r^{2} + 12 \, r + 4\right )}} \]

[In]

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

[Out]

1/4*(2*(b*d^2*r^4 + 6*b*d^2*r^3 + 13*b*d^2*r^2 + 12*b*d^2*r + 4*b*d^2)*x^2*log(c) + 2*(b*d^2*n*r^4 + 6*b*d^2*n
*r^3 + 13*b*d^2*n*r^2 + 12*b*d^2*n*r + 4*b*d^2*n)*x^2*log(x) - ((b*d^2*n - 2*a*d^2)*r^4 + 4*b*d^2*n + 6*(b*d^2
*n - 2*a*d^2)*r^3 - 8*a*d^2 + 13*(b*d^2*n - 2*a*d^2)*r^2 + 12*(b*d^2*n - 2*a*d^2)*r)*x^2 + (2*(b*e^2*r^3 + 5*b
*e^2*r^2 + 8*b*e^2*r + 4*b*e^2)*x^2*log(c) + 2*(b*e^2*n*r^3 + 5*b*e^2*n*r^2 + 8*b*e^2*n*r + 4*b*e^2*n)*x^2*log
(x) + (2*a*e^2*r^3 - 4*b*e^2*n + 8*a*e^2 - (b*e^2*n - 10*a*e^2)*r^2 - 4*(b*e^2*n - 4*a*e^2)*r)*x^2)*x^(2*r) +
8*((b*d*e*r^3 + 4*b*d*e*r^2 + 5*b*d*e*r + 2*b*d*e)*x^2*log(c) + (b*d*e*n*r^3 + 4*b*d*e*n*r^2 + 5*b*d*e*n*r + 2
*b*d*e*n)*x^2*log(x) + (a*d*e*r^3 - b*d*e*n + 2*a*d*e - (b*d*e*n - 4*a*d*e)*r^2 - (2*b*d*e*n - 5*a*d*e)*r)*x^2
)*x^r)/(r^4 + 6*r^3 + 13*r^2 + 12*r + 4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1622 vs. \(2 (97) = 194\).

Time = 1.73 (sec) , antiderivative size = 1622, normalized size of antiderivative = 15.90 \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a*d**2*x**2/2 + 2*a*d*e*log(c*x**n)/n - a*e**2/(2*x**2) - b*d**2*n*x**2/4 + b*d**2*x**2*log(c*x**n)
/2 + b*d*e*log(c*x**n)**2/n - b*e**2*n/(4*x**2) - b*e**2*log(c*x**n)/(2*x**2), Eq(r, -2)), (a*d**2*x**2/2 + 2*
a*d*e*x + a*e**2*log(c*x**n)/n - b*d**2*n*x**2/4 + b*d**2*x**2*log(c*x**n)/2 - 2*b*d*e*n*x + 2*b*d*e*x*log(c*x
**n) + b*e**2*log(c*x**n)**2/(2*n), Eq(r, -1)), (2*a*d**2*r**4*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) +
12*a*d**2*r**3*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 26*a*d**2*r**2*x**2/(4*r**4 + 24*r**3 + 52*r**
2 + 48*r + 16) + 24*a*d**2*r*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*a*d**2*x**2/(4*r**4 + 24*r**3 +
52*r**2 + 48*r + 16) + 8*a*d*e*r**3*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 32*a*d*e*r**2*x**2*x
**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 40*a*d*e*r*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) +
16*a*d*e*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 2*a*e**2*r**3*x**2*x**(2*r)/(4*r**4 + 24*r**3 +
52*r**2 + 48*r + 16) + 10*a*e**2*r**2*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 16*a*e**2*r*x*
*2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*a*e**2*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48
*r + 16) - b*d**2*n*r**4*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 6*b*d**2*n*r**3*x**2/(4*r**4 + 24*r**
3 + 52*r**2 + 48*r + 16) - 13*b*d**2*n*r**2*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 12*b*d**2*n*r*x**2
/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 4*b*d**2*n*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 2*b*d**
2*r**4*x**2*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 12*b*d**2*r**3*x**2*log(c*x**n)/(4*r**4 + 2
4*r**3 + 52*r**2 + 48*r + 16) + 26*b*d**2*r**2*x**2*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 24*
b*d**2*r*x**2*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*b*d**2*x**2*log(c*x**n)/(4*r**4 + 24*r*
*3 + 52*r**2 + 48*r + 16) - 8*b*d*e*n*r**2*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 16*b*d*e*n*r*x
**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 8*b*d*e*n*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 1
6) + 8*b*d*e*r**3*x**2*x**r*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 32*b*d*e*r**2*x**2*x**r*log
(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 40*b*d*e*r*x**2*x**r*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r*
*2 + 48*r + 16) + 16*b*d*e*x**2*x**r*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - b*e**2*n*r**2*x**2
*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 4*b*e**2*n*r*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 +
48*r + 16) - 4*b*e**2*n*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 2*b*e**2*r**3*x**2*x**(2*r)*l
og(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 10*b*e**2*r**2*x**2*x**(2*r)*log(c*x**n)/(4*r**4 + 24*r*
*3 + 52*r**2 + 48*r + 16) + 16*b*e**2*r*x**2*x**(2*r)*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8
*b*e**2*x**2*x**(2*r)*log(c*x**n)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.45 \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} \, b d^{2} n x^{2} + \frac {1}{2} \, b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d^{2} x^{2} + \frac {b e^{2} x^{2 \, r + 2} \log \left (c x^{n}\right )}{2 \, {\left (r + 1\right )}} + \frac {2 \, b d e x^{r + 2} \log \left (c x^{n}\right )}{r + 2} - \frac {b e^{2} n x^{2 \, r + 2}}{4 \, {\left (r + 1\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 2}}{2 \, {\left (r + 1\right )}} - \frac {2 \, b d e n x^{r + 2}}{{\left (r + 2\right )}^{2}} + \frac {2 \, a d e x^{r + 2}}{r + 2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (96) = 192\).

Time = 0.31 (sec) , antiderivative size = 744, normalized size of antiderivative = 7.29 \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \, b e^{2} n r^{3} x^{2} x^{2 \, r} \log \left (x\right ) + 8 \, b d e n r^{3} x^{2} x^{r} \log \left (x\right ) + 2 \, b d^{2} n r^{4} x^{2} \log \left (x\right ) - b d^{2} n r^{4} x^{2} + 2 \, b e^{2} r^{3} x^{2} x^{2 \, r} \log \left (c\right ) + 8 \, b d e r^{3} x^{2} x^{r} \log \left (c\right ) + 2 \, b d^{2} r^{4} x^{2} \log \left (c\right ) + 10 \, b e^{2} n r^{2} x^{2} x^{2 \, r} \log \left (x\right ) + 32 \, b d e n r^{2} x^{2} x^{r} \log \left (x\right ) + 12 \, b d^{2} n r^{3} x^{2} \log \left (x\right ) - b e^{2} n r^{2} x^{2} x^{2 \, r} + 2 \, a e^{2} r^{3} x^{2} x^{2 \, r} - 8 \, b d e n r^{2} x^{2} x^{r} + 8 \, a d e r^{3} x^{2} x^{r} - 6 \, b d^{2} n r^{3} x^{2} + 2 \, a d^{2} r^{4} x^{2} + 10 \, b e^{2} r^{2} x^{2} x^{2 \, r} \log \left (c\right ) + 32 \, b d e r^{2} x^{2} x^{r} \log \left (c\right ) + 12 \, b d^{2} r^{3} x^{2} \log \left (c\right ) + 16 \, b e^{2} n r x^{2} x^{2 \, r} \log \left (x\right ) + 40 \, b d e n r x^{2} x^{r} \log \left (x\right ) + 26 \, b d^{2} n r^{2} x^{2} \log \left (x\right ) - 4 \, b e^{2} n r x^{2} x^{2 \, r} + 10 \, a e^{2} r^{2} x^{2} x^{2 \, r} - 16 \, b d e n r x^{2} x^{r} + 32 \, a d e r^{2} x^{2} x^{r} - 13 \, b d^{2} n r^{2} x^{2} + 12 \, a d^{2} r^{3} x^{2} + 16 \, b e^{2} r x^{2} x^{2 \, r} \log \left (c\right ) + 40 \, b d e r x^{2} x^{r} \log \left (c\right ) + 26 \, b d^{2} r^{2} x^{2} \log \left (c\right ) + 8 \, b e^{2} n x^{2} x^{2 \, r} \log \left (x\right ) + 16 \, b d e n x^{2} x^{r} \log \left (x\right ) + 24 \, b d^{2} n r x^{2} \log \left (x\right ) - 4 \, b e^{2} n x^{2} x^{2 \, r} + 16 \, a e^{2} r x^{2} x^{2 \, r} - 8 \, b d e n x^{2} x^{r} + 40 \, a d e r x^{2} x^{r} - 12 \, b d^{2} n r x^{2} + 26 \, a d^{2} r^{2} x^{2} + 8 \, b e^{2} x^{2} x^{2 \, r} \log \left (c\right ) + 16 \, b d e x^{2} x^{r} \log \left (c\right ) + 24 \, b d^{2} r x^{2} \log \left (c\right ) + 8 \, b d^{2} n x^{2} \log \left (x\right ) + 8 \, a e^{2} x^{2} x^{2 \, r} + 16 \, a d e x^{2} x^{r} - 4 \, b d^{2} n x^{2} + 24 \, a d^{2} r x^{2} + 8 \, b d^{2} x^{2} \log \left (c\right ) + 8 \, a d^{2} x^{2}}{4 \, {\left (r^{4} + 6 \, r^{3} + 13 \, r^{2} + 12 \, r + 4\right )}} \]

[In]

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

[Out]

1/4*(2*b*e^2*n*r^3*x^2*x^(2*r)*log(x) + 8*b*d*e*n*r^3*x^2*x^r*log(x) + 2*b*d^2*n*r^4*x^2*log(x) - b*d^2*n*r^4*
x^2 + 2*b*e^2*r^3*x^2*x^(2*r)*log(c) + 8*b*d*e*r^3*x^2*x^r*log(c) + 2*b*d^2*r^4*x^2*log(c) + 10*b*e^2*n*r^2*x^
2*x^(2*r)*log(x) + 32*b*d*e*n*r^2*x^2*x^r*log(x) + 12*b*d^2*n*r^3*x^2*log(x) - b*e^2*n*r^2*x^2*x^(2*r) + 2*a*e
^2*r^3*x^2*x^(2*r) - 8*b*d*e*n*r^2*x^2*x^r + 8*a*d*e*r^3*x^2*x^r - 6*b*d^2*n*r^3*x^2 + 2*a*d^2*r^4*x^2 + 10*b*
e^2*r^2*x^2*x^(2*r)*log(c) + 32*b*d*e*r^2*x^2*x^r*log(c) + 12*b*d^2*r^3*x^2*log(c) + 16*b*e^2*n*r*x^2*x^(2*r)*
log(x) + 40*b*d*e*n*r*x^2*x^r*log(x) + 26*b*d^2*n*r^2*x^2*log(x) - 4*b*e^2*n*r*x^2*x^(2*r) + 10*a*e^2*r^2*x^2*
x^(2*r) - 16*b*d*e*n*r*x^2*x^r + 32*a*d*e*r^2*x^2*x^r - 13*b*d^2*n*r^2*x^2 + 12*a*d^2*r^3*x^2 + 16*b*e^2*r*x^2
*x^(2*r)*log(c) + 40*b*d*e*r*x^2*x^r*log(c) + 26*b*d^2*r^2*x^2*log(c) + 8*b*e^2*n*x^2*x^(2*r)*log(x) + 16*b*d*
e*n*x^2*x^r*log(x) + 24*b*d^2*n*r*x^2*log(x) - 4*b*e^2*n*x^2*x^(2*r) + 16*a*e^2*r*x^2*x^(2*r) - 8*b*d*e*n*x^2*
x^r + 40*a*d*e*r*x^2*x^r - 12*b*d^2*n*r*x^2 + 26*a*d^2*r^2*x^2 + 8*b*e^2*x^2*x^(2*r)*log(c) + 16*b*d*e*x^2*x^r
*log(c) + 24*b*d^2*r*x^2*log(c) + 8*b*d^2*n*x^2*log(x) + 8*a*e^2*x^2*x^(2*r) + 16*a*d*e*x^2*x^r - 4*b*d^2*n*x^
2 + 24*a*d^2*r*x^2 + 8*b*d^2*x^2*log(c) + 8*a*d^2*x^2)/(r^4 + 6*r^3 + 13*r^2 + 12*r + 4)

Mupad [F(-1)]

Timed out. \[ \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

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

[Out]

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