3.4.97 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^5} \, dx\) [397]

Optimal. Leaf size=191 \[ -\frac {b d^3 n}{16 x^4}-\frac {3 b d e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac {3 b d^2 e n x^{-4+r}}{(4-r)^2}-\frac {b e^3 n x^{-4+3 r}}{(4-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {3 d^2 e x^{-4+r} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^3 x^{-4+3 r} \left (a+b \log \left (c x^n\right )\right )}{4-3 r} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 191, 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, 2372, 12, 14} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {e^3 x^{3 r-4} \left (a+b \log \left (c x^n\right )\right )}{4-3 r}-\frac {b d^3 n}{16 x^4}-\frac {3 b d^2 e n x^{r-4}}{(4-r)^2}-\frac {3 b d e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac {b e^3 n x^{3 r-4}}{(4-3 r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2372

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]}, Dist[a + b*Log[c*x^n], u, 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])

Rubi steps

\begin {align*} \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac {1}{4} \left (\frac {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 {-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}}{4 x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {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 \frac {-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}}{x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {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 (-\frac {d^3}{x^5}+\frac {12 d^2 e x^{-5+r}}{-4+r}+\frac {6 d e^2 x^{-5+2 r}}{-2+r}+\frac {4 e^3 x^{-5+3 r}}{-4+3 r}\right ) \, dx\\ &=-\frac {b d^3 n}{16 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 (\frac {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 = 0.84 \begin {gather*} \frac {-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 {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 {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 {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}}{16 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.32, size = 4027, normalized size = 21.08

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*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)/x^4/(-4+3*r)/(-2+r)/
(-4+r)*ln(x^n)-1/16*(4096*e^3*(x^r)^3*a+1544*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-3968*I*Pi*b*e^3*r^2*cs
gn(I*c*x^n)^3*(x^r)^3-6144*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+12288*d^2*e*x^r*a+12288*d*e^2*(x^r)^2*a+6528*I
*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+264*I*Pi*b*d^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+6144*I*Pi*b*d^3
*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-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^4-1368*I*Pi*b*d*e^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x
^n)*(x^r)^2-2304*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+4608*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*x
^n)*csgn(I*c*x^n)*(x^r)^3+18048*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*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)^2-18048*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+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+2048*I*Pi*b*d^3*csgn(I*c)*csgn(I*c
*x^n)^2+2048*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-6528*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+
9312*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+15360*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(
I*c*x^n)*(x^r)^2+16896*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+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+36*ln(c)*b*d^3*r^6-528*ln(c)*b*d^3*r^5+3088*ln(c)*b*d^3*r^4-921
6*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-48*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)+6144*I*Pi*b*d*e^2*
csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+6144*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-14592*I*Pi*b*d*e^2*r^2
*csgn(I*c*x^n)^3*(x^r)^2+9312*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r-4608*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*c*x^n)^2
*(x^r)^3-2304*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+2048*
I*Pi*b*e^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+2048*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+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)^2-3968*I*Pi*b*e^3*r^2*csgn(I*c)
*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+4608*I*Pi*b*d^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+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+4608*a*d^2*e*r^4*x^r-4608*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^
2*(x^r)^3+108*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2-6144*I*Pi*b*d^3*r*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)^2-320*I*Pi*b*e^3*r^4*csgn(I*c
*x^n)^3*(x^r)^3-2048*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+18048*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-18048*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-2048*I*Pi*
b*e^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-6528*I*Pi*b*d*e^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+4224
*b*d*e^2*n*r^2*(x^r)^2-432*ln(c)*b*d^2*e*r^5*x^r+14592*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+1536
0*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-1632*I*Pi*b*e^3*r^3*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-9312*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r+18*I*Pi*b*d^3*r^6*cs
gn(I*x^n)*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)^3-24*I*Pi*b*e^3*r^5*csgn(I*c)*cs
gn(I*c*x^n)^2*(x^r)^3-16896*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r+24*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)*(x^r)^3+14592*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6144*I*Pi*b*d^2*e*csgn(I*c
)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+1632*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+4608*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*
(x^r)^3+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+24*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-1544*I*Pi*b*d^3*r^4*csgn(
I*c*x^n)^3-7424*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-108*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-7424*I*P
i*b*d^3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+6144*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r-18*I*Pi*b*d^3*
r^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-2048*I*...

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(r-5>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (172) = 344\).
time = 0.38, size = 843, normalized size = 4.41 \begin {gather*} -\frac {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 - 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 )} 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 )} 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 )} 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 )} 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 )} 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 )} 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 )} 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 )} 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 )} e\right )} x^{r} + 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 )} \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 )} \log \left (x\right )}{16 \, {\left (9 \, r^{6} - 132 \, r^{5} + 772 \, r^{4} - 2304 \, r^{3} + 3712 \, r^{2} - 3072 \, r + 1024\right )} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(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 - 16*((3
*b*r^5 - 40*b*r^4 + 204*b*r^3 - 496*b*r^2 + 576*b*r - 256*b)*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)*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)*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)*e^2*log(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)*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)*e^2)*x^(2*r) - 48*((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)*e*log(c) + (9*b*d^2*n*r^5 - 9
6*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)*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)*e)*x^r + 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)*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)*log(x))/((9*r^6 - 132*r^5 + 772*r^4 - 230
4*r^3 + 3712*r^2 - 3072*r + 1024)*x^4)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8856 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^r*e + d)^3*(b*log(c*x^n) + a)/x^5, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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