3.4.96 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^3} \, dx\) [396]

Optimal. Leaf size=191 \[ -\frac {b d^3 n}{4 x^2}-\frac {3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac {3 b d^2 e n x^{-2+r}}{(2-r)^2}-\frac {b e^3 n x^{-2+3 r}}{(2-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {3 d^2 e x^{-2+r} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^3 x^{-2+3 r} \left (a+b \log \left (c x^n\right )\right )}{2-3 r} \]

[Out]

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

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Rubi [A]
time = 0.26, 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 )}{2 x^2}-\frac {3 d^2 e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {e^3 x^{3 r-2} \left (a+b \log \left (c x^n\right )\right )}{2-3 r}-\frac {b d^3 n}{4 x^2}-\frac {3 b d^2 e n x^{r-2}}{(2-r)^2}-\frac {3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac {b e^3 n x^{3 r-2}}{(2-3 r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(b*d^3*n)/x^2 - (3*b*d*e^2*n)/(4*(1 - r)^2*x^(2*(1 - r))) - (3*b*d^2*e*n*x^(-2 + r))/(2 - r)^2 - (b*e^3*n
*x^(-2 + 3*r))/(2 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(2*x^2) - (3*d*e^2*(a + b*Log[c*x^n]))/(2*(1 - r)*x^(2*(
1 - r))) - (3*d^2*e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r) - (e^3*x^(-2 + 3*r)*(a + b*Log[c*x^n]))/(2 - 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^3} \, dx &=-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {3 d e^2 x^{-2 (1-r)}}{1-r}+\frac {6 d^2 e x^{-2+r}}{2-r}+\frac {2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^3+\frac {6 d^2 e x^r}{-2+r}+\frac {3 d e^2 x^{2 r}}{-1+r}+\frac {2 e^3 x^{3 r}}{-2+3 r}}{2 x^3} \, dx\\ &=-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {3 d e^2 x^{-2 (1-r)}}{1-r}+\frac {6 d^2 e x^{-2+r}}{2-r}+\frac {2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \frac {-d^3+\frac {6 d^2 e x^r}{-2+r}+\frac {3 d e^2 x^{2 r}}{-1+r}+\frac {2 e^3 x^{3 r}}{-2+3 r}}{x^3} \, dx\\ &=-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {3 d e^2 x^{-2 (1-r)}}{1-r}+\frac {6 d^2 e x^{-2+r}}{2-r}+\frac {2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \left (-\frac {d^3}{x^3}+\frac {6 d^2 e x^{-3+r}}{-2+r}+\frac {2 e^3 x^{3 (-1+r)}}{-2+3 r}+\frac {3 d e^2 x^{-3+2 r}}{-1+r}\right ) \, dx\\ &=-\frac {b d^3 n}{4 x^2}-\frac {3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac {3 b d^2 e n x^{-2+r}}{(2-r)^2}-\frac {b e^3 n x^{-2+3 r}}{(2-3 r)^2}-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {3 d e^2 x^{-2 (1-r)}}{1-r}+\frac {6 d^2 e x^{-2+r}}{2-r}+\frac {2 e^3 x^{-2+3 r}}{2-3 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 {-2 b d^3 n \log (x)-d^3 \left (2 a+b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right )+\frac {12 d^2 e x^r \left (-b n+a (-2+r)+b (-2+r) \log \left (c x^n\right )\right )}{(-2+r)^2}+\frac {3 d e^2 x^{2 r} \left (-b n+2 a (-1+r)+2 b (-1+r) \log \left (c x^n\right )\right )}{(-1+r)^2}+\frac {4 e^3 x^{3 r} \left (-b n+a (-2+3 r)+b (-2+3 r) \log \left (c x^n\right )\right )}{(2-3 r)^2}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*d^3*n*Log[x] - d^3*(2*a + b*n - 2*b*n*Log[x] + 2*b*Log[c*x^n]) + (12*d^2*e*x^r*(-(b*n) + a*(-2 + r) + b*
(-2 + r)*Log[c*x^n]))/(-2 + r)^2 + (3*d*e^2*x^(2*r)*(-(b*n) + 2*a*(-1 + r) + 2*b*(-1 + r)*Log[c*x^n]))/(-1 + r
)^2 + (4*e^3*x^(3*r)*(-(b*n) + a*(-2 + 3*r) + b*(-2 + 3*r)*Log[c*x^n]))/(2 - 3*r)^2)/(4*x^2)

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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^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*b*(-2*e^3*r^2*(x^r)^3-9*d*e^2*r^2*(x^r)^2+6*e^3*r*(x^r)^3+3*d^3*r^3-18*d^2*e*r^2*x^r+24*d*e^2*r*(x^r)^2-4
*e^3*(x^r)^3-11*d^3*r^2+30*d^2*e*r*x^r-12*d*e^2*(x^r)^2+12*d^3*r-12*d^2*e*x^r-4*d^3)/x^2/(-2+3*r)/(-1+r)/(-2+r
)*ln(x^n)-1/4*(32*e^3*(x^r)^3*a+96*d^2*e*x^r*a+96*d*e^2*(x^r)^2*a+16*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-36
0*b*d^2*e*n*r^3*x^r+912*ln(c)*b*d*e^2*r^2*(x^r)^2-480*ln(c)*b*d*e^2*r*(x^r)^2+72*I*Pi*b*e^3*r*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)*(x^r)^3+102*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+18*a*d^3*r^6-132*a*
d^3*r^5+386*a*d^3*r^4+27*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+27*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*x^n)*cs
gn(I*c*x^n)*(x^r)^2+54*I*Pi*b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+48*I*Pi*b*d^2*e*csgn(I*c)*csgn
(I*c*x^n)^2*x^r-9*I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-240*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x
^n)^2*(x^r)^2-66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2+32*a*d^3-816*a*d*e^2*r^3*(x^r)^2+912*a*d*e^2*r^2*(
x^r)^2-480*a*d*e^2*r*(x^r)^2-1164*a*d^2*e*r^3*x^r+1128*a*d^2*e*r^2*x^r-528*a*d^2*e*r*x^r+9*b*d^3*n*r^6-66*b*d^
3*n*r^5+193*b*d^3*n*r^4-6*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+444*b*d^2*e*n*r^2*x^r-192*b*d*e^2*n
*r*(x^r)^2-240*b*d^2*e*n*r*x^r+27*b*d*e^2*n*r^4*(x^r)^2-144*b*d*e^2*n*r^3*(x^r)^2+108*b*d^2*e*n*r^4*x^r-576*a*
d^3*r^3+464*a*d^3*r^2-192*a*d^3*r-264*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-72*I*Pi*b*e^3*r*csgn(I*c)*c
sgn(I*c*x^n)^2*(x^r)^3+18*ln(c)*b*d^3*r^6-132*ln(c)*b*d^3*r^5+386*ln(c)*b*d^3*r^4-576*ln(c)*b*d^3*r^3+464*ln(c
)*b*d^3*r^2-192*ln(c)*b*d^3*r+16*b*d^3*n-12*a*e^3*r^5*(x^r)^3+80*a*e^3*r^4*(x^r)^3+32*ln(c)*b*e^3*(x^r)^3+16*b
*e^3*n*(x^r)^3-204*a*e^3*r^3*(x^r)^3+32*d^3*b*ln(c)-288*b*d^3*n*r^3+232*b*d^3*n*r^2-96*b*d^3*n*r+52*b*e^3*n*r^
2*(x^r)^3-48*b*e^3*n*r*(x^r)^3+48*b*d*e^2*n*(x^r)^2+48*b*d^2*e*n*x^r+96*ln(c)*b*d^2*e*x^r+408*I*Pi*b*d*e^2*r^3
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+248*a*e^3*r^2*(x^r)^3-144*a*e^3*r*(x^r)^3-193*I*Pi*b*d^3*r^4*csgn
(I*c*x^n)^3+96*ln(c)*b*d*e^2*(x^r)^2-204*ln(c)*b*e^3*r^3*(x^r)^3+248*ln(c)*b*e^3*r^2*(x^r)^3-144*ln(c)*b*e^3*r
*(x^r)^3-12*ln(c)*b*e^3*r^5*(x^r)^3+80*ln(c)*b*e^3*r^4*(x^r)^3+4*b*e^3*n*r^4*(x^r)^3-24*b*e^3*n*r^3*(x^r)^3-54
*a*d*e^2*r^5*(x^r)^2+342*a*d*e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r+576*a*d^2*e*r^4*x^r-16*I*Pi*b*d^3*csgn(I*c*x^
n)^3+564*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r+564*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-4
8*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+6*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)*(x^r)^3-27*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-456*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(I*x^n)*csg
n(I*c*x^n)*(x^r)^2-564*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-171*I*Pi*b*d*e^2*r^4*csgn(I*c)
*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+288*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-232*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)+40*I*Pi*b*e^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+40*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)
^2*(x^r)^3+16*I*Pi*b*e^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+16*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+2
32*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2-288*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+48*I*Pi*b*d*e^2*csgn(I*c)
*csgn(I*c*x^n)^2*(x^r)^2-96*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2+264*b*d*e^2*n*r^2*(x^r)^2-108*ln(c)*b*d^2
*e*r^5*x^r+582*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+66*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-408*
I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-264*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+6*I*Pi
*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-408*I*Pi*b*d*e^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+288*I*Pi*b*d^3*r^3*c
sgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+96*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+232*I*Pi*b*d^3*r^2*csgn
(I*x^n)*csgn(I*c*x^n)^2+9*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+193*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*c*x^n
)^2+193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-124*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3-48*I*Pi*b*d*e^2*
csgn(I*c*x^n)^3*(x^r)^2+342*ln(c)*b*d*e^2*r^4*(x^r)^2-1164*ln(c)*b*d^2*e*r^3*x^r+1128*ln(c)*b*d^2*e*r^2*x^r-52
8*ln(c)*b*d^2*e*r*x^r-816*ln(c)*b*d*e^2*r^3*(x^r)^2-582*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r-582*I*P
i*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-240*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+48*I*Pi*b*d
*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+124*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-456*I*Pi*b*d*e
^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-564*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-16*I*Pi*b*e^3*csgn(I*c)*csgn(I*x^n)*cs
gn(I*c*x^n)*(x^r)^3+102*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+72*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-6*I*Pi*
b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+240*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+2
64*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-288*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)*x^r-171*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-193*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+4
8*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+...

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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^3,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-3>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. 844 vs. \(2 (172) = 344\).
time = 0.35, size = 844, normalized size = 4.42 \begin {gather*} -\frac {9 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{6} - 66 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{5} + 16 \, b d^{3} n + 193 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{4} + 32 \, a d^{3} - 288 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{3} + 232 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{2} - 96 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r - 4 \, {\left ({\left (3 \, b r^{5} - 20 \, b r^{4} + 51 \, b r^{3} - 62 \, b r^{2} + 36 \, b r - 8 \, b\right )} e^{3} \log \left (c\right ) + {\left (3 \, b n r^{5} - 20 \, b n r^{4} + 51 \, b n r^{3} - 62 \, b n r^{2} + 36 \, b n r - 8 \, b n\right )} e^{3} \log \left (x\right ) + {\left (3 \, a r^{5} - {\left (b n + 20 \, a\right )} r^{4} + 3 \, {\left (2 \, b n + 17 \, a\right )} r^{3} - {\left (13 \, b n + 62 \, a\right )} r^{2} - 4 \, b n + 12 \, {\left (b n + 3 \, a\right )} r - 8 \, a\right )} e^{3}\right )} x^{3 \, r} - 3 \, {\left (2 \, {\left (9 \, b d r^{5} - 57 \, b d r^{4} + 136 \, b d r^{3} - 152 \, b d r^{2} + 80 \, b d r - 16 \, b d\right )} e^{2} \log \left (c\right ) + 2 \, {\left (9 \, b d n r^{5} - 57 \, b d n r^{4} + 136 \, b d n r^{3} - 152 \, b d n r^{2} + 80 \, b d n r - 16 \, b d n\right )} e^{2} \log \left (x\right ) + {\left (18 \, a d r^{5} - 3 \, {\left (3 \, b d n + 38 \, a d\right )} r^{4} + 16 \, {\left (3 \, b d n + 17 \, a d\right )} r^{3} - 16 \, b d n - 8 \, {\left (11 \, b d n + 38 \, a d\right )} r^{2} - 32 \, a d + 32 \, {\left (2 \, b d n + 5 \, a d\right )} r\right )} e^{2}\right )} x^{2 \, r} - 12 \, {\left ({\left (9 \, b d^{2} r^{5} - 48 \, b d^{2} r^{4} + 97 \, b d^{2} r^{3} - 94 \, b d^{2} r^{2} + 44 \, b d^{2} r - 8 \, b d^{2}\right )} e \log \left (c\right ) + {\left (9 \, b d^{2} n r^{5} - 48 \, b d^{2} n r^{4} + 97 \, b d^{2} n r^{3} - 94 \, b d^{2} n r^{2} + 44 \, b d^{2} n r - 8 \, b d^{2} n\right )} e \log \left (x\right ) + {\left (9 \, a d^{2} r^{5} - 3 \, {\left (3 \, b d^{2} n + 16 \, a d^{2}\right )} r^{4} - 4 \, b d^{2} n + {\left (30 \, b d^{2} n + 97 \, a d^{2}\right )} r^{3} - 8 \, a d^{2} - {\left (37 \, b d^{2} n + 94 \, a d^{2}\right )} r^{2} + 4 \, {\left (5 \, b d^{2} n + 11 \, a d^{2}\right )} r\right )} e\right )} x^{r} + 2 \, {\left (9 \, b d^{3} r^{6} - 66 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} - 288 \, b d^{3} r^{3} + 232 \, b d^{3} r^{2} - 96 \, b d^{3} r + 16 \, b d^{3}\right )} \log \left (c\right ) + 2 \, {\left (9 \, b d^{3} n r^{6} - 66 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} - 288 \, b d^{3} n r^{3} + 232 \, b d^{3} n r^{2} - 96 \, b d^{3} n r + 16 \, b d^{3} n\right )} \log \left (x\right )}{4 \, {\left (9 \, r^{6} - 66 \, r^{5} + 193 \, r^{4} - 288 \, r^{3} + 232 \, r^{2} - 96 \, r + 16\right )} x^{2}} \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^3,x, algorithm="fricas")

[Out]

-1/4*(9*(b*d^3*n + 2*a*d^3)*r^6 - 66*(b*d^3*n + 2*a*d^3)*r^5 + 16*b*d^3*n + 193*(b*d^3*n + 2*a*d^3)*r^4 + 32*a
*d^3 - 288*(b*d^3*n + 2*a*d^3)*r^3 + 232*(b*d^3*n + 2*a*d^3)*r^2 - 96*(b*d^3*n + 2*a*d^3)*r - 4*((3*b*r^5 - 20
*b*r^4 + 51*b*r^3 - 62*b*r^2 + 36*b*r - 8*b)*e^3*log(c) + (3*b*n*r^5 - 20*b*n*r^4 + 51*b*n*r^3 - 62*b*n*r^2 +
36*b*n*r - 8*b*n)*e^3*log(x) + (3*a*r^5 - (b*n + 20*a)*r^4 + 3*(2*b*n + 17*a)*r^3 - (13*b*n + 62*a)*r^2 - 4*b*
n + 12*(b*n + 3*a)*r - 8*a)*e^3)*x^(3*r) - 3*(2*(9*b*d*r^5 - 57*b*d*r^4 + 136*b*d*r^3 - 152*b*d*r^2 + 80*b*d*r
- 16*b*d)*e^2*log(c) + 2*(9*b*d*n*r^5 - 57*b*d*n*r^4 + 136*b*d*n*r^3 - 152*b*d*n*r^2 + 80*b*d*n*r - 16*b*d*n)
*e^2*log(x) + (18*a*d*r^5 - 3*(3*b*d*n + 38*a*d)*r^4 + 16*(3*b*d*n + 17*a*d)*r^3 - 16*b*d*n - 8*(11*b*d*n + 38
*a*d)*r^2 - 32*a*d + 32*(2*b*d*n + 5*a*d)*r)*e^2)*x^(2*r) - 12*((9*b*d^2*r^5 - 48*b*d^2*r^4 + 97*b*d^2*r^3 - 9
4*b*d^2*r^2 + 44*b*d^2*r - 8*b*d^2)*e*log(c) + (9*b*d^2*n*r^5 - 48*b*d^2*n*r^4 + 97*b*d^2*n*r^3 - 94*b*d^2*n*r
^2 + 44*b*d^2*n*r - 8*b*d^2*n)*e*log(x) + (9*a*d^2*r^5 - 3*(3*b*d^2*n + 16*a*d^2)*r^4 - 4*b*d^2*n + (30*b*d^2*
n + 97*a*d^2)*r^3 - 8*a*d^2 - (37*b*d^2*n + 94*a*d^2)*r^2 + 4*(5*b*d^2*n + 11*a*d^2)*r)*e)*x^r + 2*(9*b*d^3*r^
6 - 66*b*d^3*r^5 + 193*b*d^3*r^4 - 288*b*d^3*r^3 + 232*b*d^3*r^2 - 96*b*d^3*r + 16*b*d^3)*log(c) + 2*(9*b*d^3*
n*r^6 - 66*b*d^3*n*r^5 + 193*b*d^3*n*r^4 - 288*b*d^3*n*r^3 + 232*b*d^3*n*r^2 - 96*b*d^3*n*r + 16*b*d^3*n)*log(
x))/((9*r^6 - 66*r^5 + 193*r^4 - 288*r^3 + 232*r^2 - 96*r + 16)*x^2)

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Sympy [A]
time = 75.88, size = 338, normalized size = 1.77 \begin {gather*} - \frac {a d^{3}}{2 x^{2}} + 3 a d^{2} e \left (\begin {cases} \frac {x^{r}}{r x^{2} - 2 x^{2}} & \text {for}\: r \neq 2 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{2} - 2 x^{2}} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{3 r}}{3 r x^{2} - 2 x^{2}} & \text {for}\: r \neq \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n}{4 x^{2}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{2 x^{2}} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r x^{2} - 2 x^{2}} & \text {for}\: r \neq 2 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r - 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 2 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r - 2}}{r - 2} & \text {for}\: r \neq 2 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - 3 b d e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r}}{2 r x^{2} - 2 x^{2}} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r - 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d e^{2} \left (\begin {cases} \frac {x^{2 r - 2}}{2 r - 2} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{3} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{3 r}}{3 r x^{2} - 2 x^{2}} & \text {for}\: r \neq \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 r - 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {2}{3} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r - 2}}{3 r - 2} & \text {for}\: r \neq \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \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**3,x)

[Out]

-a*d**3/(2*x**2) + 3*a*d**2*e*Piecewise((x**r/(r*x**2 - 2*x**2), Ne(r, 2)), (log(x), True)) + 3*a*d*e**2*Piece
wise((x**(2*r)/(2*r*x**2 - 2*x**2), Ne(r, 1)), (log(x), True)) + a*e**3*Piecewise((x**(3*r)/(3*r*x**2 - 2*x**2
), Ne(r, 2/3)), (log(x), True)) - b*d**3*n/(4*x**2) - b*d**3*log(c*x**n)/(2*x**2) - 3*b*d**2*e*n*Piecewise((Pi
ecewise((x**r/(r*x**2 - 2*x**2), Ne(r, 2)), (log(x), True))/(r - 2), (r > -oo) & (r < oo) & Ne(r, 2)), (log(x)
**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r - 2)/(r - 2), Ne(r, 2)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*
n*Piecewise((Piecewise((x**(2*r)/(2*r*x**2 - 2*x**2), Ne(r, 1)), (log(x), True))/(2*r - 2), (r > -oo) & (r < o
o) & Ne(r, 1)), (log(x)**2/2, True)) + 3*b*d*e**2*Piecewise((x**(2*r - 2)/(2*r - 2), Ne(r, 1)), (log(x), True)
)*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r)/(3*r*x**2 - 2*x**2), Ne(r, 2/3)), (log(x), True))/(3*r
- 2), (r > -oo) & (r < oo) & Ne(r, 2/3)), (log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r - 2)/(3*r - 2), Ne
(r, 2/3)), (log(x), True))*log(c*x**n)

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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^3,x, algorithm="giac")

[Out]

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

[Out]

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

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