\(\int x^3 (d+e x^r)^2 (a+b \log (c x^n)) \, dx\) [380]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d^2 n x^4-\frac {b e^2 n x^{2 (2+r)}}{4 (2+r)^2}-\frac {2 b d e n x^{4+r}}{(4+r)^2}+\frac {1}{4} \left (d^2 x^4+\frac {2 e^2 x^{2 (2+r)}}{2+r}+\frac {8 d e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/16*b*d^2*n*x^4-1/4*b*e^2*n*x^(4+2*r)/(2+r)^2-2*b*d*e*n*x^(4+r)/(4+r)^2+1/4*(d^2*x^4+2*e^2*x^(4+2*r)/(2+r)+8
*d*e*x^(4+r)/(4+r))*(a+b*ln(c*x^n))

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2371, 12, 14} \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \left (d^2 x^4+\frac {8 d e x^{r+4}}{r+4}+\frac {2 e^2 x^{2 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d^2 n x^4-\frac {2 b d e n x^{r+4}}{(r+4)^2}-\frac {b e^2 n x^{2 (r+2)}}{4 (r+2)^2} \]

[In]

Int[x^3*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

-1/16*(b*d^2*n*x^4) - (b*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (2*b*d*e*n*x^(4 + r))/(4 + r)^2 + ((d^2*x^4 + (2
*e^2*x^(2*(2 + r)))/(2 + r) + (8*d*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4

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}{4} \left (d^2 x^4+\frac {2 e^2 x^{2 (2+r)}}{2+r}+\frac {8 d e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{4} x^3 \left (d^2+\frac {8 d e x^r}{4+r}+\frac {2 e^2 x^{2 r}}{2+r}\right ) \, dx \\ & = \frac {1}{4} \left (d^2 x^4+\frac {2 e^2 x^{2 (2+r)}}{2+r}+\frac {8 d e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int x^3 \left (d^2+\frac {8 d e x^r}{4+r}+\frac {2 e^2 x^{2 r}}{2+r}\right ) \, dx \\ & = \frac {1}{4} \left (d^2 x^4+\frac {2 e^2 x^{2 (2+r)}}{2+r}+\frac {8 d e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (d^2 x^3+\frac {8 d e x^{3+r}}{4+r}+\frac {2 e^2 x^{3+2 r}}{2+r}\right ) \, dx \\ & = -\frac {1}{16} b d^2 n x^4-\frac {b e^2 n x^{2 (2+r)}}{4 (2+r)^2}-\frac {2 b d e n x^{4+r}}{(4+r)^2}+\frac {1}{4} \left (d^2 x^4+\frac {2 e^2 x^{2 (2+r)}}{2+r}+\frac {8 d e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.15 \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{16} x^4 \left (b n \left (-d^2-\frac {32 d e x^r}{(4+r)^2}-\frac {4 e^2 x^{2 r}}{(2+r)^2}\right )+4 a \left (d^2+\frac {8 d e x^r}{4+r}+\frac {2 e^2 x^{2 r}}{2+r}\right )+4 b \left (d^2+\frac {8 d e x^r}{4+r}+\frac {2 e^2 x^{2 r}}{2+r}\right ) \log \left (c x^n\right )\right ) \]

[In]

Integrate[x^3*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

(x^4*(b*n*(-d^2 - (32*d*e*x^r)/(4 + r)^2 - (4*e^2*x^(2*r))/(2 + r)^2) + 4*a*(d^2 + (8*d*e*x^r)/(4 + r) + (2*e^
2*x^(2*r))/(2 + r)) + 4*b*(d^2 + (8*d*e*x^r)/(4 + r) + (2*e^2*x^(2*r))/(2 + r))*Log[c*x^n]))/16

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(587\) vs. \(2(97)=194\).

Time = 5.23 (sec) , antiderivative size = 588, normalized size of antiderivative = 5.71

method	result	size
parallelrisch	\(-\frac {-256 a \,d^{2} x^{4}-256 x^{4} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +4 x^{4} x^{2 r} b \,e^{2} n \,r^{2}+32 x^{4} x^{2 r} b \,e^{2} n r -8 x^{4} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}-80 x^{4} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}+128 x^{4} x^{r} b d e n r -32 x^{4} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}-256 x^{4} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-640 x^{4} x^{r} \ln \left (c \,x^{n}\right ) b d e r +32 x^{4} x^{r} b d e n \,r^{2}-4 x^{4} a \,d^{2} r^{4}-48 x^{4} a \,d^{2} r^{3}-208 x^{4} a \,d^{2} r^{2}-384 x^{4} a \,d^{2} r -32 x^{4} x^{r} a d e \,r^{3}-256 x^{4} x^{r} a d e \,r^{2}-640 x^{4} x^{r} a d e r -512 b d e \ln \left (c \,x^{n}\right ) x^{r} x^{4}+128 x^{4} x^{r} b d e n -256 x^{4} x^{2 r} a \,e^{2}-256 x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2}-208 x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-384 x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2} r -512 x^{4} x^{r} a d e +x^{4} b \,d^{2} n \,r^{4}+12 x^{4} b \,d^{2} n \,r^{3}+52 x^{4} b \,d^{2} n \,r^{2}+96 x^{4} b \,d^{2} n r -4 x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-48 x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}-8 x^{4} x^{2 r} a \,e^{2} r^{3}-80 x^{4} x^{2 r} a \,e^{2} r^{2}-256 x^{4} x^{2 r} a \,e^{2} r +64 x^{4} x^{2 r} b \,e^{2} n -256 e^{2} b \ln \left (c \,x^{n}\right ) x^{2 r} x^{4}+64 b \,d^{2} n \,x^{4}}{16 \left (r^{2}+4 r +4\right ) \left (r^{2}+8 r +16\right )}\)	\(588\)
risch	\(\text {Expression too large to display}\)	\(1924\)

[In]

int(x^3*(d+e*x^r)^2*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

-1/16*(-256*a*d^2*x^4+128*x^4*x^r*b*d*e*n*r-32*x^4*x^r*ln(c*x^n)*b*d*e*r^3-256*x^4*x^r*ln(c*x^n)*b*d*e*r^2-640
*x^4*x^r*ln(c*x^n)*b*d*e*r+32*x^4*x^r*b*d*e*n*r^2-4*x^4*a*d^2*r^4-48*x^4*a*d^2*r^3-208*x^4*a*d^2*r^2-384*x^4*a
*d^2*r-256*x^4*(x^r)^2*a*e^2-256*x^4*(x^r)^2*ln(c*x^n)*b*e^2*r-32*x^4*x^r*a*d*e*r^3-256*x^4*x^r*a*d*e*r^2-640*
x^4*x^r*a*d*e*r-512*b*d*e*ln(c*x^n)*x^r*x^4+128*x^4*x^r*b*d*e*n+4*x^4*(x^r)^2*b*e^2*n*r^2+32*x^4*(x^r)^2*b*e^2
*n*r-8*x^4*(x^r)^2*ln(c*x^n)*b*e^2*r^3-80*x^4*(x^r)^2*ln(c*x^n)*b*e^2*r^2-256*x^4*ln(c*x^n)*b*d^2-208*x^4*ln(c
*x^n)*b*d^2*r^2-384*x^4*ln(c*x^n)*b*d^2*r-512*x^4*x^r*a*d*e-8*x^4*(x^r)^2*a*e^2*r^3-80*x^4*(x^r)^2*a*e^2*r^2-2
56*x^4*(x^r)^2*a*e^2*r+64*x^4*(x^r)^2*b*e^2*n-256*e^2*b*ln(c*x^n)*(x^r)^2*x^4+x^4*b*d^2*n*r^4+12*x^4*b*d^2*n*r
^3+52*x^4*b*d^2*n*r^2+96*x^4*b*d^2*n*r-4*x^4*ln(c*x^n)*b*d^2*r^4-48*x^4*ln(c*x^n)*b*d^2*r^3+64*b*d^2*n*x^4)/(r
^2+4*r+4)/(r^2+8*r+16)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (97) = 194\).

Time = 0.31 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.74 \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {4 \, {\left (b d^{2} r^{4} + 12 \, b d^{2} r^{3} + 52 \, b d^{2} r^{2} + 96 \, b d^{2} r + 64 \, b d^{2}\right )} x^{4} \log \left (c\right ) + 4 \, {\left (b d^{2} n r^{4} + 12 \, b d^{2} n r^{3} + 52 \, b d^{2} n r^{2} + 96 \, b d^{2} n r + 64 \, b d^{2} n\right )} x^{4} \log \left (x\right ) - {\left ({\left (b d^{2} n - 4 \, a d^{2}\right )} r^{4} + 64 \, b d^{2} n + 12 \, {\left (b d^{2} n - 4 \, a d^{2}\right )} r^{3} - 256 \, a d^{2} + 52 \, {\left (b d^{2} n - 4 \, a d^{2}\right )} r^{2} + 96 \, {\left (b d^{2} n - 4 \, a d^{2}\right )} r\right )} x^{4} + 4 \, {\left (2 \, {\left (b e^{2} r^{3} + 10 \, b e^{2} r^{2} + 32 \, b e^{2} r + 32 \, b e^{2}\right )} x^{4} \log \left (c\right ) + 2 \, {\left (b e^{2} n r^{3} + 10 \, b e^{2} n r^{2} + 32 \, b e^{2} n r + 32 \, b e^{2} n\right )} x^{4} \log \left (x\right ) + {\left (2 \, a e^{2} r^{3} - 16 \, b e^{2} n + 64 \, a e^{2} - {\left (b e^{2} n - 20 \, a e^{2}\right )} r^{2} - 8 \, {\left (b e^{2} n - 8 \, a e^{2}\right )} r\right )} x^{4}\right )} x^{2 \, r} + 32 \, {\left ({\left (b d e r^{3} + 8 \, b d e r^{2} + 20 \, b d e r + 16 \, b d e\right )} x^{4} \log \left (c\right ) + {\left (b d e n r^{3} + 8 \, b d e n r^{2} + 20 \, b d e n r + 16 \, b d e n\right )} x^{4} \log \left (x\right ) + {\left (a d e r^{3} - 4 \, b d e n + 16 \, a d e - {\left (b d e n - 8 \, a d e\right )} r^{2} - 4 \, {\left (b d e n - 5 \, a d e\right )} r\right )} x^{4}\right )} x^{r}}{16 \, {\left (r^{4} + 12 \, r^{3} + 52 \, r^{2} + 96 \, r + 64\right )}} \]

[In]

integrate(x^3*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/16*(4*(b*d^2*r^4 + 12*b*d^2*r^3 + 52*b*d^2*r^2 + 96*b*d^2*r + 64*b*d^2)*x^4*log(c) + 4*(b*d^2*n*r^4 + 12*b*d
^2*n*r^3 + 52*b*d^2*n*r^2 + 96*b*d^2*n*r + 64*b*d^2*n)*x^4*log(x) - ((b*d^2*n - 4*a*d^2)*r^4 + 64*b*d^2*n + 12
*(b*d^2*n - 4*a*d^2)*r^3 - 256*a*d^2 + 52*(b*d^2*n - 4*a*d^2)*r^2 + 96*(b*d^2*n - 4*a*d^2)*r)*x^4 + 4*(2*(b*e^
2*r^3 + 10*b*e^2*r^2 + 32*b*e^2*r + 32*b*e^2)*x^4*log(c) + 2*(b*e^2*n*r^3 + 10*b*e^2*n*r^2 + 32*b*e^2*n*r + 32
*b*e^2*n)*x^4*log(x) + (2*a*e^2*r^3 - 16*b*e^2*n + 64*a*e^2 - (b*e^2*n - 20*a*e^2)*r^2 - 8*(b*e^2*n - 8*a*e^2)
*r)*x^4)*x^(2*r) + 32*((b*d*e*r^3 + 8*b*d*e*r^2 + 20*b*d*e*r + 16*b*d*e)*x^4*log(c) + (b*d*e*n*r^3 + 8*b*d*e*n
*r^2 + 20*b*d*e*n*r + 16*b*d*e*n)*x^4*log(x) + (a*d*e*r^3 - 4*b*d*e*n + 16*a*d*e - (b*d*e*n - 8*a*d*e)*r^2 - 4
*(b*d*e*n - 5*a*d*e)*r)*x^4)*x^r)/(r^4 + 12*r^3 + 52*r^2 + 96*r + 64)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1625 vs. \(2 (97) = 194\).

Time = 7.10 (sec) , antiderivative size = 1625, normalized size of antiderivative = 15.78 \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x**3*(d+e*x**r)**2*(a+b*ln(c*x**n)),x)

[Out]

Piecewise((a*d**2*x**4/4 + 2*a*d*e*log(c*x**n)/n - a*e**2/(4*x**4) - b*d**2*n*x**4/16 + b*d**2*x**4*log(c*x**n
)/4 + b*d*e*log(c*x**n)**2/n - b*e**2*n/(16*x**4) - b*e**2*log(c*x**n)/(4*x**4), Eq(r, -4)), (a*d**2*x**4/4 +
a*d*e*x**2 + a*e**2*log(c*x**n)/n - b*d**2*n*x**4/16 + b*d**2*x**4*log(c*x**n)/4 - b*d*e*n*x**2/2 + b*d*e*x**2
*log(c*x**n) + b*e**2*log(c*x**n)**2/(2*n), Eq(r, -2)), (4*a*d**2*r**4*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1
536*r + 1024) + 48*a*d**2*r**3*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 208*a*d**2*r**2*x**4/(16
*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 384*a*d**2*r*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 102
4) + 256*a*d**2*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 32*a*d*e*r**3*x**4*x**r/(16*r**4 + 192*
r**3 + 832*r**2 + 1536*r + 1024) + 256*a*d*e*r**2*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) +
640*a*d*e*r*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 512*a*d*e*x**4*x**r/(16*r**4 + 192*r**
3 + 832*r**2 + 1536*r + 1024) + 8*a*e**2*r**3*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) +
80*a*e**2*r**2*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*a*e**2*r*x**4*x**(2*r)/(16*
r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*a*e**2*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r
+ 1024) - b*d**2*n*r**4*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 12*b*d**2*n*r**3*x**4/(16*r**4
+ 192*r**3 + 832*r**2 + 1536*r + 1024) - 52*b*d**2*n*r**2*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024
) - 96*b*d**2*n*r*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 64*b*d**2*n*x**4/(16*r**4 + 192*r**3
+ 832*r**2 + 1536*r + 1024) + 4*b*d**2*r**4*x**4*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) +
48*b*d**2*r**3*x**4*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 208*b*d**2*r**2*x**4*log(c*
x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 384*b*d**2*r*x**4*log(c*x**n)/(16*r**4 + 192*r**3 + 83
2*r**2 + 1536*r + 1024) + 256*b*d**2*x**4*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 32*b*d
*e*n*r**2*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 128*b*d*e*n*r*x**4*x**r/(16*r**4 + 192*r
**3 + 832*r**2 + 1536*r + 1024) - 128*b*d*e*n*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 32*b
*d*e*r**3*x**4*x**r*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*d*e*r**2*x**4*x**r*log
(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 640*b*d*e*r*x**4*x**r*log(c*x**n)/(16*r**4 + 192*r*
*3 + 832*r**2 + 1536*r + 1024) + 512*b*d*e*x**4*x**r*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 102
4) - 4*b*e**2*n*r**2*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 32*b*e**2*n*r*x**4*x**(2*
r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 64*b*e**2*n*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2
+ 1536*r + 1024) + 8*b*e**2*r**3*x**4*x**(2*r)*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 8
0*b*e**2*r**2*x**4*x**(2*r)*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*r*x**4*x*
*(2*r)*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*x**4*x**(2*r)*log(c*x**n)/(16*
r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.44 \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b d^{2} n x^{4} + \frac {1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{2} x^{4} + \frac {b e^{2} x^{2 \, r + 4} \log \left (c x^{n}\right )}{2 \, {\left (r + 2\right )}} + \frac {2 \, b d e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e^{2} n x^{2 \, r + 4}}{4 \, {\left (r + 2\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 4}}{2 \, {\left (r + 2\right )}} - \frac {2 \, b d e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {2 \, a d e x^{r + 4}}{r + 4} \]

[In]

integrate(x^3*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/16*b*d^2*n*x^4 + 1/4*b*d^2*x^4*log(c*x^n) + 1/4*a*d^2*x^4 + 1/2*b*e^2*x^(2*r + 4)*log(c*x^n)/(r + 2) + 2*b*
d*e*x^(r + 4)*log(c*x^n)/(r + 4) - 1/4*b*e^2*n*x^(2*r + 4)/(r + 2)^2 + 1/2*a*e^2*x^(2*r + 4)/(r + 2) - 2*b*d*e
*n*x^(r + 4)/(r + 4)^2 + 2*a*d*e*x^(r + 4)/(r + 4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (97) = 194\).

Time = 0.38 (sec) , antiderivative size = 744, normalized size of antiderivative = 7.22 \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {8 \, b e^{2} n r^{3} x^{4} x^{2 \, r} \log \left (x\right ) + 32 \, b d e n r^{3} x^{4} x^{r} \log \left (x\right ) + 4 \, b d^{2} n r^{4} x^{4} \log \left (x\right ) - b d^{2} n r^{4} x^{4} + 8 \, b e^{2} r^{3} x^{4} x^{2 \, r} \log \left (c\right ) + 32 \, b d e r^{3} x^{4} x^{r} \log \left (c\right ) + 4 \, b d^{2} r^{4} x^{4} \log \left (c\right ) + 80 \, b e^{2} n r^{2} x^{4} x^{2 \, r} \log \left (x\right ) + 256 \, b d e n r^{2} x^{4} x^{r} \log \left (x\right ) + 48 \, b d^{2} n r^{3} x^{4} \log \left (x\right ) - 4 \, b e^{2} n r^{2} x^{4} x^{2 \, r} + 8 \, a e^{2} r^{3} x^{4} x^{2 \, r} - 32 \, b d e n r^{2} x^{4} x^{r} + 32 \, a d e r^{3} x^{4} x^{r} - 12 \, b d^{2} n r^{3} x^{4} + 4 \, a d^{2} r^{4} x^{4} + 80 \, b e^{2} r^{2} x^{4} x^{2 \, r} \log \left (c\right ) + 256 \, b d e r^{2} x^{4} x^{r} \log \left (c\right ) + 48 \, b d^{2} r^{3} x^{4} \log \left (c\right ) + 256 \, b e^{2} n r x^{4} x^{2 \, r} \log \left (x\right ) + 640 \, b d e n r x^{4} x^{r} \log \left (x\right ) + 208 \, b d^{2} n r^{2} x^{4} \log \left (x\right ) - 32 \, b e^{2} n r x^{4} x^{2 \, r} + 80 \, a e^{2} r^{2} x^{4} x^{2 \, r} - 128 \, b d e n r x^{4} x^{r} + 256 \, a d e r^{2} x^{4} x^{r} - 52 \, b d^{2} n r^{2} x^{4} + 48 \, a d^{2} r^{3} x^{4} + 256 \, b e^{2} r x^{4} x^{2 \, r} \log \left (c\right ) + 640 \, b d e r x^{4} x^{r} \log \left (c\right ) + 208 \, b d^{2} r^{2} x^{4} \log \left (c\right ) + 256 \, b e^{2} n x^{4} x^{2 \, r} \log \left (x\right ) + 512 \, b d e n x^{4} x^{r} \log \left (x\right ) + 384 \, b d^{2} n r x^{4} \log \left (x\right ) - 64 \, b e^{2} n x^{4} x^{2 \, r} + 256 \, a e^{2} r x^{4} x^{2 \, r} - 128 \, b d e n x^{4} x^{r} + 640 \, a d e r x^{4} x^{r} - 96 \, b d^{2} n r x^{4} + 208 \, a d^{2} r^{2} x^{4} + 256 \, b e^{2} x^{4} x^{2 \, r} \log \left (c\right ) + 512 \, b d e x^{4} x^{r} \log \left (c\right ) + 384 \, b d^{2} r x^{4} \log \left (c\right ) + 256 \, b d^{2} n x^{4} \log \left (x\right ) + 256 \, a e^{2} x^{4} x^{2 \, r} + 512 \, a d e x^{4} x^{r} - 64 \, b d^{2} n x^{4} + 384 \, a d^{2} r x^{4} + 256 \, b d^{2} x^{4} \log \left (c\right ) + 256 \, a d^{2} x^{4}}{16 \, {\left (r^{4} + 12 \, r^{3} + 52 \, r^{2} + 96 \, r + 64\right )}} \]

[In]

integrate(x^3*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/16*(8*b*e^2*n*r^3*x^4*x^(2*r)*log(x) + 32*b*d*e*n*r^3*x^4*x^r*log(x) + 4*b*d^2*n*r^4*x^4*log(x) - b*d^2*n*r^
4*x^4 + 8*b*e^2*r^3*x^4*x^(2*r)*log(c) + 32*b*d*e*r^3*x^4*x^r*log(c) + 4*b*d^2*r^4*x^4*log(c) + 80*b*e^2*n*r^2
*x^4*x^(2*r)*log(x) + 256*b*d*e*n*r^2*x^4*x^r*log(x) + 48*b*d^2*n*r^3*x^4*log(x) - 4*b*e^2*n*r^2*x^4*x^(2*r) +
8*a*e^2*r^3*x^4*x^(2*r) - 32*b*d*e*n*r^2*x^4*x^r + 32*a*d*e*r^3*x^4*x^r - 12*b*d^2*n*r^3*x^4 + 4*a*d^2*r^4*x^
4 + 80*b*e^2*r^2*x^4*x^(2*r)*log(c) + 256*b*d*e*r^2*x^4*x^r*log(c) + 48*b*d^2*r^3*x^4*log(c) + 256*b*e^2*n*r*x
^4*x^(2*r)*log(x) + 640*b*d*e*n*r*x^4*x^r*log(x) + 208*b*d^2*n*r^2*x^4*log(x) - 32*b*e^2*n*r*x^4*x^(2*r) + 80*
a*e^2*r^2*x^4*x^(2*r) - 128*b*d*e*n*r*x^4*x^r + 256*a*d*e*r^2*x^4*x^r - 52*b*d^2*n*r^2*x^4 + 48*a*d^2*r^3*x^4
+ 256*b*e^2*r*x^4*x^(2*r)*log(c) + 640*b*d*e*r*x^4*x^r*log(c) + 208*b*d^2*r^2*x^4*log(c) + 256*b*e^2*n*x^4*x^(
2*r)*log(x) + 512*b*d*e*n*x^4*x^r*log(x) + 384*b*d^2*n*r*x^4*log(x) - 64*b*e^2*n*x^4*x^(2*r) + 256*a*e^2*r*x^4
*x^(2*r) - 128*b*d*e*n*x^4*x^r + 640*a*d*e*r*x^4*x^r - 96*b*d^2*n*r*x^4 + 208*a*d^2*r^2*x^4 + 256*b*e^2*x^4*x^
(2*r)*log(c) + 512*b*d*e*x^4*x^r*log(c) + 384*b*d^2*r*x^4*log(c) + 256*b*d^2*n*x^4*log(x) + 256*a*e^2*x^4*x^(2
*r) + 512*a*d*e*x^4*x^r - 64*b*d^2*n*x^4 + 384*a*d^2*r*x^4 + 256*b*d^2*x^4*log(c) + 256*a*d^2*x^4)/(r^4 + 12*r
^3 + 52*r^2 + 96*r + 64)

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^3\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

int(x^3*(d + e*x^r)^2*(a + b*log(c*x^n)),x)

[Out]

int(x^3*(d + e*x^r)^2*(a + b*log(c*x^n)), x)