3.4.93 \(\int x^3 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [393]

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

[Out]

-1/16*b*d^3*n*x^4-3/4*b*d*e^2*n*x^(4+2*r)/(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+1/4*
(d^3*x^4+6*d*e^2*x^(4+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*ln(c*x^n))

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Rubi [A]
time = 0.26, antiderivative size = 149, normalized size of antiderivative = 1.00, 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 = {276, 2371, 12, 14} \begin {gather*} \frac {1}{4} \left (d^3 x^4+\frac {12 d^2 e x^{r+4}}{r+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 {1}{16} b d^3 n x^4-\frac {3 b d^2 e n x^{r+4}}{(r+4)^2}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(b*d^3*n*x^4) - (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 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*} \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*}

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Mathematica [A]
time = 0.16, size = 160, normalized size = 1.07 \begin {gather*} \frac {1}{16} x^4 \left (4 b d^3 n \log (x)+d^3 \left (4 a-b n-4 b n \log (x)+4 b \log \left (c x^n\right )\right )+\frac {12 d e^2 x^{2 r} \left (-b n+2 a (2+r)+2 b (2+r) \log \left (c x^n\right )\right )}{(2+r)^2}+\frac {48 d^2 e x^r \left (-b n+a (4+r)+b (4+r) \log \left (c x^n\right )\right )}{(4+r)^2}+\frac {16 e^3 x^{3 r} \left (-b n+a (4+3 r)+b (4+3 r) \log \left (c x^n\right )\right )}{(4+3 r)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 4027, normalized size = 27.03

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

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,method=_RETURNVERBOSE)

[Out]

1/4*x^4*b*(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*e^3*(x^r)^3*a-12288*d^2*e*x^r*a-12288*d*e^2*(x^r)^2*a+2048*I*Pi*b*e^3*csgn(I*c*x^n)^
3*(x^r)^3+6144*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+2880*b*d^2*e*n*r^3*x^r-29184*ln(c)*b*d*e^2*r^2*(x^r)^2-30720*ln(c)
*b*d*e^2*r*(x^r)^2-36*a*d^3*r^6-528*a*d^3*r^5-3088*a*d^3*r^4+2048*I*Pi*b*e^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)*(x^r)^3+2304*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+6144*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+18*
I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2304*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x
^r+6528*I*Pi*b*d*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+108*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*x^n
)*csgn(I*c*x^n)*(x^r)^2+216*I*Pi*b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+1368*I*Pi*b*d*e^2*r^4*csg
n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-4096*a*d^3-13056*a*d*e^2*r^3*(x^r)^2-29184*a*d*e^2*r^2*(x^r)^2-30720*
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+9*b*d^3*n*r^6+132*b*d^3*n*r^
5+772*b*d^3*n*r^4+7104*b*d^2*e*n*r^2*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-9216*a*d^3*r^3-14848*a*d^3*r^2-12288*a*d^3*r-2048*I*Pi*b*d
^3*csgn(I*c)*csgn(I*c*x^n)^2-24*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-3968*I*Pi*b*e^3*r^2*csgn(I*
c)*csgn(I*c*x^n)^2*(x^r)^3-1632*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-36*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+1024*b*d^3*n-4
8*a*e^3*r^5*(x^r)^3-640*a*e^3*r^4*(x^r)^3-4096*ln(c)*b*e^3*(x^r)^3+1024*b*e^3*n*(x^r)^3-3264*a*e^3*r^3*(x^r)^3
-4096*d^3*b*ln(c)-7424*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2-7424*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^
2-6144*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*c*x^n)^2+2304*b*d^3*n*r^3+3712*b*d^3*n*r^2+3072*b*d^3*n*r+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-12288*ln(c)*b*d^2*e*x^r-7936*a*e^3*r
^2*(x^r)^3-9216*a*e^3*r*(x^r)^3-12288*ln(c)*b*d*e^2*(x^r)^2-3264*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)^3-48*ln(c)*b*e^3*r^5*(x^r)^3-640*ln(c)*b*e^3*r^4*(x^r)^3+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+2048*I*Pi*b*d^
3*csgn(I*c*x^n)^3-4608*a*d^2*e*r^4*x^r-16896*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+6144*I*Pi*b*d*e^2*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+6144*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+9312*I*P
i*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+14592*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)*(x^r)^2+4224*b*d*e^2*n*r^2*(x^r)^2-3968*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+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)^2-432*ln(c)*b*d^2*e*r^5*x^r+18*I*Pi*b*d
^3*r^6*csgn(I*c*x^n)^3-18*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-264*I*Pi*b*d^3*r^5*csgn(I*c)*csgn(I*c*x^n
)^2-264*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-6144*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2+2048*I*Pi*b*d
^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-18*I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*c*x^n)^2+4608*I*Pi*b*d^3*r^3*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)+7424*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-6144*I*Pi*b*d*e^2*csgn(I*
c)*csgn(I*c*x^n)^2*(x^r)^2+16896*I*Pi*b*d^2*e*r*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+6144*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-6144*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+1544*I*Pi*b*d^
3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+108*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2-24*I*Pi*b*e^3*r^5*csgn(
I*c)*csgn(I*c*x^n)^2*(x^r)^3-216*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-1368*I*Pi*b*d*e^2*r^4*csgn(I
*c)*csgn(I*c*x^n)^2*(x^r)^2-1544*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2-1544*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn
(I*c*x^n)^2+1632*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(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)*csgn(I*c*x^n)^2*(x^r)^3-4608*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-2304*I*Pi*b*d^
2*e*r^4*csgn(I*c)*csgn(I*c*x^n)^2*x^r-2304*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-14592*I*Pi*b*d*e^2
*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-14592*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2736*ln(c)*b
*d*e^2*r^4*(x^r)^2-18624*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)^2-320*I*Pi*b*e^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3-320*I*Pi*b*e^3*r^4*csgn(I*x^n)*csg
n(I*c*x^n)^2*(x^r)^3+216*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r-4608*I*Pi*b*d^3*r^3*csgn(I*c)*csgn(I*c*x^n)^2-46
08*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-108*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-18048*I
*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r-1...

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Maxima [A]
time = 0.29, size = 222, normalized size = 1.49 \begin {gather*} -\frac {1}{16} \, b d^{3} n x^{4} + \frac {1}{4} \, b d^{3} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{3} x^{4} + \frac {b e^{3} x^{3 \, r + 4} \log \left (c x^{n}\right )}{3 \, r + 4} + \frac {3 \, b d e^{2} x^{2 \, r + 4} \log \left (c x^{n}\right )}{2 \, {\left (r + 2\right )}} + \frac {3 \, b d^{2} e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e^{3} n x^{3 \, r + 4}}{{\left (3 \, r + 4\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 4}}{3 \, r + 4} - \frac {3 \, b d e^{2} n x^{2 \, r + 4}}{4 \, {\left (r + 2\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 4}}{2 \, {\left (r + 2\right )}} - \frac {3 \, b d^{2} e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 4}}{r + 4} \end {gather*}

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]

-1/16*b*d^3*n*x^4 + 1/4*b*d^3*x^4*log(c*x^n) + 1/4*a*d^3*x^4 + b*e^3*x^(3*r + 4)*log(c*x^n)/(3*r + 4) + 3/2*b*
d*e^2*x^(2*r + 4)*log(c*x^n)/(r + 2) + 3*b*d^2*e*x^(r + 4)*log(c*x^n)/(r + 4) - b*e^3*n*x^(3*r + 4)/(3*r + 4)^
2 + a*e^3*x^(3*r + 4)/(3*r + 4) - 3/4*b*d*e^2*n*x^(2*r + 4)/(r + 2)^2 + 3/2*a*d*e^2*x^(2*r + 4)/(r + 2) - 3*b*
d^2*e*n*x^(r + 4)/(r + 4)^2 + 3*a*d^2*e*x^(r + 4)/(r + 4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (141) = 282\).
time = 0.36, size = 879, normalized size = 5.90 \begin {gather*} \frac {4 \, {\left (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}\right )} x^{4} \log \left (c\right ) + 4 \, {\left (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\right )} x^{4} \log \left (x\right ) - {\left (9 \, {\left (b d^{3} n - 4 \, a d^{3}\right )} r^{6} + 132 \, {\left (b d^{3} n - 4 \, a d^{3}\right )} r^{5} + 1024 \, b d^{3} n + 772 \, {\left (b d^{3} n - 4 \, a d^{3}\right )} r^{4} - 4096 \, a d^{3} + 2304 \, {\left (b d^{3} n - 4 \, a d^{3}\right )} r^{3} + 3712 \, {\left (b d^{3} n - 4 \, a d^{3}\right )} r^{2} + 3072 \, {\left (b d^{3} n - 4 \, a d^{3}\right )} r\right )} x^{4} + 16 \, {\left ({\left (3 \, b r^{5} + 40 \, b r^{4} + 204 \, b r^{3} + 496 \, b r^{2} + 576 \, b r + 256 \, b\right )} x^{4} e^{3} \log \left (c\right ) + {\left (3 \, b n r^{5} + 40 \, b n r^{4} + 204 \, b n r^{3} + 496 \, b n r^{2} + 576 \, b n r + 256 \, b n\right )} x^{4} e^{3} \log \left (x\right ) + {\left (3 \, a r^{5} - {\left (b n - 40 \, a\right )} r^{4} - 12 \, {\left (b n - 17 \, a\right )} r^{3} - 4 \, {\left (13 \, b n - 124 \, a\right )} r^{2} - 64 \, b n - 96 \, {\left (b n - 6 \, a\right )} r + 256 \, a\right )} x^{4} e^{3}\right )} x^{3 \, r} + 12 \, {\left (2 \, {\left (9 \, b d r^{5} + 114 \, b d r^{4} + 544 \, b d r^{3} + 1216 \, b d r^{2} + 1280 \, b d r + 512 \, b d\right )} x^{4} e^{2} \log \left (c\right ) + 2 \, {\left (9 \, b d n r^{5} + 114 \, b d n r^{4} + 544 \, b d n r^{3} + 1216 \, b d n r^{2} + 1280 \, b d n r + 512 \, b d n\right )} x^{4} e^{2} \log \left (x\right ) + {\left (18 \, a d r^{5} - 3 \, {\left (3 \, b d n - 76 \, a d\right )} r^{4} - 32 \, {\left (3 \, b d n - 34 \, a d\right )} r^{3} - 256 \, b d n - 32 \, {\left (11 \, b d n - 76 \, a d\right )} r^{2} + 1024 \, a d - 512 \, {\left (b d n - 5 \, a d\right )} r\right )} x^{4} e^{2}\right )} x^{2 \, r} + 48 \, {\left ({\left (9 \, b d^{2} r^{5} + 96 \, b d^{2} r^{4} + 388 \, b d^{2} r^{3} + 752 \, b d^{2} r^{2} + 704 \, b d^{2} r + 256 \, b d^{2}\right )} x^{4} e \log \left (c\right ) + {\left (9 \, b d^{2} n r^{5} + 96 \, b d^{2} n r^{4} + 388 \, b d^{2} n r^{3} + 752 \, b d^{2} n r^{2} + 704 \, b d^{2} n r + 256 \, b d^{2} n\right )} x^{4} e \log \left (x\right ) + {\left (9 \, a d^{2} r^{5} - 3 \, {\left (3 \, b d^{2} n - 32 \, a d^{2}\right )} r^{4} - 64 \, b d^{2} n - 4 \, {\left (15 \, b d^{2} n - 97 \, a d^{2}\right )} r^{3} + 256 \, a d^{2} - 4 \, {\left (37 \, b d^{2} n - 188 \, a d^{2}\right )} r^{2} - 32 \, {\left (5 \, b d^{2} n - 22 \, a d^{2}\right )} r\right )} x^{4} e\right )} x^{r}}{16 \, {\left (9 \, r^{6} + 132 \, r^{5} + 772 \, r^{4} + 2304 \, r^{3} + 3712 \, r^{2} + 3072 \, r + 1024\right )}} \end {gather*}

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*r^5 + 40*b*r^4 + 204*b*r^3 + 496*b*r^2 + 576*b*r + 256*b)*x^4*
e^3*log(c) + (3*b*n*r^5 + 40*b*n*r^4 + 204*b*n*r^3 + 496*b*n*r^2 + 576*b*n*r + 256*b*n)*x^4*e^3*log(x) + (3*a*
r^5 - (b*n - 40*a)*r^4 - 12*(b*n - 17*a)*r^3 - 4*(13*b*n - 124*a)*r^2 - 64*b*n - 96*(b*n - 6*a)*r + 256*a)*x^4
*e^3)*x^(3*r) + 12*(2*(9*b*d*r^5 + 114*b*d*r^4 + 544*b*d*r^3 + 1216*b*d*r^2 + 1280*b*d*r + 512*b*d)*x^4*e^2*lo
g(c) + 2*(9*b*d*n*r^5 + 114*b*d*n*r^4 + 544*b*d*n*r^3 + 1216*b*d*n*r^2 + 1280*b*d*n*r + 512*b*d*n)*x^4*e^2*log
(x) + (18*a*d*r^5 - 3*(3*b*d*n - 76*a*d)*r^4 - 32*(3*b*d*n - 34*a*d)*r^3 - 256*b*d*n - 32*(11*b*d*n - 76*a*d)*
r^2 + 1024*a*d - 512*(b*d*n - 5*a*d)*r)*x^4*e^2)*x^(2*r) + 48*((9*b*d^2*r^5 + 96*b*d^2*r^4 + 388*b*d^2*r^3 + 7
52*b*d^2*r^2 + 704*b*d^2*r + 256*b*d^2)*x^4*e*log(c) + (9*b*d^2*n*r^5 + 96*b*d^2*n*r^4 + 388*b*d^2*n*r^3 + 752
*b*d^2*n*r^2 + 704*b*d^2*n*r + 256*b*d^2*n)*x^4*e*log(x) + (9*a*d^2*r^5 - 3*(3*b*d^2*n - 32*a*d^2)*r^4 - 64*b*
d^2*n - 4*(15*b*d^2*n - 97*a*d^2)*r^3 + 256*a*d^2 - 4*(37*b*d^2*n - 188*a*d^2)*r^2 - 32*(5*b*d^2*n - 22*a*d^2)
*r)*x^4*e)*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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (141) = 282\).
time = 2.48, size = 1588, normalized size = 10.66 \begin {gather*} \frac {36 \, b d^{3} n r^{6} x^{4} \log \left (x\right ) + 432 \, b d^{2} n r^{5} x^{4} x^{r} e \log \left (x\right ) - 9 \, b d^{3} n r^{6} x^{4} + 36 \, b d^{3} r^{6} x^{4} \log \left (c\right ) + 432 \, b d^{2} r^{5} x^{4} x^{r} e \log \left (c\right ) + 528 \, b d^{3} n r^{5} x^{4} \log \left (x\right ) + 216 \, b d n r^{5} x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 4608 \, b d^{2} n r^{4} x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 216 \, b d r^{5} x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 4608 \, b d^{2} r^{4} x^{4} x^{r} e \log \left (c\right ) + 3088 \, b d^{3} n r^{4} x^{4} \log \left (x\right ) + 48 \, b n r^{5} x^{4} x^{3 \, r} e^{3} \log \left (x\right ) + 2736 \, b d n r^{4} x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 18624 \, b d^{2} n r^{3} x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 48 \, b r^{5} x^{4} x^{3 \, r} e^{3} \log \left (c\right ) + 2736 \, b d r^{4} x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 18624 \, b d^{2} r^{3} x^{4} x^{r} e \log \left (c\right ) + 9216 \, b d^{3} n r^{3} x^{4} \log \left (x\right ) + 640 \, b n r^{4} x^{4} x^{3 \, r} e^{3} \log \left (x\right ) + 13056 \, b d n r^{3} x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 36096 \, b d^{2} n r^{2} x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 640 \, b r^{4} x^{4} x^{3 \, r} e^{3} \log \left (c\right ) + 13056 \, b d r^{3} x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 36096 \, b d^{2} r^{2} x^{4} x^{r} e \log \left (c\right ) + 14848 \, b d^{3} n r^{2} x^{4} \log \left (x\right ) + 3264 \, b n r^{3} x^{4} x^{3 \, r} e^{3} \log \left (x\right ) + 29184 \, b d n r^{2} x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 33792 \, b d^{2} n r x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 3264 \, b r^{3} x^{4} x^{3 \, r} e^{3} \log \left (c\right ) + 29184 \, b d r^{2} x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 33792 \, b d^{2} r x^{4} x^{r} e \log \left (c\right ) + 12288 \, b d^{3} n r x^{4} \log \left (x\right ) + 7936 \, b n r^{2} x^{4} x^{3 \, r} e^{3} \log \left (x\right ) + 30720 \, b d n r x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 12288 \, b d^{2} n x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 7936 \, b r^{2} x^{4} x^{3 \, r} e^{3} \log \left (c\right ) + 30720 \, b d r x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 12288 \, b d^{2} x^{4} x^{r} e \log \left (c\right ) + 4096 \, b d^{3} n x^{4} \log \left (x\right ) + 9216 \, b n r x^{4} x^{3 \, r} e^{3} \log \left (x\right ) + 12288 \, b d n x^{4} x^{2 \, r} e^{2} \log \left (x\right ) - 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 \left (c\right ) + 9216 \, b r x^{4} x^{3 \, r} e^{3} \log \left (c\right ) + 12288 \, b d x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 4096 \, b n x^{4} x^{3 \, r} e^{3} \log \left (x\right ) + 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 \left (c\right ) + 4096 \, a x^{4} x^{3 \, r} e^{3}}{16 \, {\left (9 \, r^{6} + 132 \, r^{5} + 772 \, r^{4} + 2304 \, r^{3} + 3712 \, r^{2} + 3072 \, r + 1024\right )}} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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