3.5.2 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^4} \, dx\) [402]

Optimal. Leaf size=191 \[ -\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \]

[Out]

-1/9*b*d^3*n/x^3-1/9*b*e^3*n/(1-r)^2/(x^(3-3*r))-3*b*d^2*e*n*x^(-3+r)/(3-r)^2-3*b*d*e^2*n*x^(-3+2*r)/(3-2*r)^2
-1/3*d^3*(a+b*ln(c*x^n))/x^3-1/3*e^3*(a+b*ln(c*x^n))/(1-r)/(x^(3-3*r))-3*d^2*e*x^(-3+r)*(a+b*ln(c*x^n))/(3-r)-
3*d*e^2*x^(-3+2*r)*(a+b*ln(c*x^n))/(3-2*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 )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac {3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-1/9*(b*d^3*n)/x^3 - (b*e^3*n)/(9*(1 - r)^2*x^(3*(1 - r))) - (3*b*d^2*e*n*x^(-3 + r))/(3 - r)^2 - (3*b*d*e^2*n
*x^(-3 + 2*r))/(3 - 2*r)^2 - (d^3*(a + b*Log[c*x^n]))/(3*x^3) - (e^3*(a + b*Log[c*x^n]))/(3*(1 - r)*x^(3*(1 -
r))) - (3*d^2*e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r) - (3*d*e^2*x^(-3 + 2*r)*(a + b*Log[c*x^n]))/(3 - 2*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^4} \, dx &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}}{3 x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \frac {-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}}{x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (-\frac {d^3}{x^4}+\frac {9 d^2 e x^{-4+r}}{-3+r}+\frac {9 d e^2 x^{2 (-2+r)}}{-3+2 r}+\frac {e^3 x^{-4+3 r}}{-1+r}\right ) \, dx\\ &=-\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 159, normalized size = 0.83 \begin {gather*} \frac {-3 b d^3 n \log (x)-d^3 \left (3 a+b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+\frac {27 d^2 e x^r \left (-b n+a (-3+r)+b (-3+r) \log \left (c x^n\right )\right )}{(-3+r)^2}+\frac {e^3 x^{3 r} \left (-b n+3 a (-1+r)+3 b (-1+r) \log \left (c x^n\right )\right )}{(-1+r)^2}+\frac {27 d e^2 x^{2 r} \left (-b n+a (-3+2 r)+b (-3+2 r) \log \left (c x^n\right )\right )}{(3-2 r)^2}}{9 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

(-3*b*d^3*n*Log[x] - d^3*(3*a + b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + (27*d^2*e*x^r*(-(b*n) + a*(-3 + r) + b*
(-3 + r)*Log[c*x^n]))/(-3 + r)^2 + (e^3*x^(3*r)*(-(b*n) + 3*a*(-1 + r) + 3*b*(-1 + r)*Log[c*x^n]))/(-1 + r)^2
+ (27*d*e^2*x^(2*r)*(-(b*n) + a*(-3 + 2*r) + b*(-3 + 2*r)*Log[c*x^n]))/(3 - 2*r)^2)/(9*x^3)

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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^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*b*(-2*e^3*r^2*(x^r)^3-9*d*e^2*r^2*(x^r)^2+9*e^3*r*(x^r)^3+2*d^3*r^3-18*d^2*e*r^2*x^r+36*d*e^2*r*(x^r)^2-9
*e^3*(x^r)^3-11*d^3*r^2+45*d^2*e*r*x^r-27*d*e^2*(x^r)^2+18*d^3*r-27*d^2*e*x^r-9*d^3)/x^3/(-1+r)/(-3+2*r)/(-3+r
)*ln(x^n)-1/18*(486*e^3*(x^r)^3*a+2430*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-459*I*Pi*b*e^3*r^3*csgn(I*c)*csg
n(I*c*x^n)^2*(x^r)^3-120*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-243*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)+1458*d^2*e*x^r*a+1458*d*e^2*(x^r)^2*a+132*I*Pi*b*d^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-864*I*Pi*b*d
^2*e*r^4*csgn(I*c*x^n)^3*x^r+729*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+729*I*Pi*b*d*e^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2*(x^r)^2-513*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-1080*b*d^2*e*n*r^3*x^r+6156*ln(c)*b*d*e^2*
r^2*(x^r)^2-4860*ln(c)*b*d*e^2*r*(x^r)^2+579*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2+24*a*d^3*r^6-264*a*d^3*r
^5+1158*a*d^3*r^4-12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-864*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)*x^r-1836*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2430*I*Pi*b*d*e^2*r*csgn(I*x^
n)*csgn(I*c*x^n)^2*(x^r)^2+486*a*d^3-3672*a*d*e^2*r^3*(x^r)^2+6156*a*d*e^2*r^2*(x^r)^2-4860*a*d*e^2*r*(x^r)^2-
5238*a*d^2*e*r^3*x^r+7614*a*d^2*e*r^2*x^r-5346*a*d^2*e*r*x^r-12*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)
^3+54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+8*b*d^3*n*r^6-88*b*d^3*n*r^5+386*b*d^3*n*r^4+513*I*Pi*b*d*e^2*r
^4*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+3807*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r-3078*I*Pi*b*d*e^2*r^2
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-3807*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+199
8*b*d^2*e*n*r^2*x^r-1296*b*d*e^2*n*r*(x^r)^2-1620*b*d^2*e*n*r*x^r+54*b*d*e^2*n*r^4*(x^r)^2-432*b*d*e^2*n*r^3*(
x^r)^2+216*b*d^2*e*n*r^4*x^r-2592*a*d^3*r^3+3132*a*d^3*r^2-1944*a*d^3*r-12*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3-243*
I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+243*I*Pi*b*d^3*csgn(I*c)*csgn(I*c*x^n)^2+243*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*
c*x^n)^2+24*ln(c)*b*d^3*r^6-264*ln(c)*b*d^3*r^5+1158*ln(c)*b*d^3*r^4-2592*ln(c)*b*d^3*r^3+3132*ln(c)*b*d^3*r^2
-1944*ln(c)*b*d^3*r+162*b*d^3*n-24*a*e^3*r^5*(x^r)^3+240*a*e^3*r^4*(x^r)^3+486*ln(c)*b*e^3*(x^r)^3+162*b*e^3*n
*(x^r)^3-918*a*e^3*r^3*(x^r)^3+486*d^3*b*ln(c)+2619*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r-729*I*Pi*b*e^3*r*csgn
(I*c)*csgn(I*c*x^n)^2*(x^r)^3-729*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-243*I*Pi*b*e^3*csgn(I*c)*cs
gn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-1566*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-864*b*d^3*n*r^3+1044*b
*d^3*n*r^2-648*b*d^3*n*r+234*b*e^3*n*r^2*(x^r)^3-324*b*e^3*n*r*(x^r)^3+486*b*d*e^2*n*(x^r)^2+486*b*d^2*e*n*x^r
+1458*ln(c)*b*d^2*e*x^r+1674*a*e^3*r^2*(x^r)^3-1458*a*e^3*r*(x^r)^3+837*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^
n)^2*(x^r)^3-3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-3807*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+108*I*Pi*
b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+1458*ln(c)*b*d*e^2*(x^r)^2-918*ln(c)*b*e^3*r^3*(x^r)^3+1674*ln(c)*b*e^3*r^2*(x
^r)^3-1458*ln(c)*b*e^3*r*(x^r)^3-24*ln(c)*b*e^3*r^5*(x^r)^3+240*ln(c)*b*e^3*r^4*(x^r)^3+8*b*e^3*n*r^4*(x^r)^3-
72*b*e^3*n*r^3*(x^r)^3-108*a*d*e^2*r^5*(x^r)^2+1026*a*d*e^2*r^4*(x^r)^2-216*a*d^2*e*r^5*x^r+1728*a*d^2*e*r^4*x
^r+1566*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2+1566*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+12*I*Pi*b*d^3
*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-579*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+729*I*Pi*b*d^2*e*csgn(
I*x^n)*csgn(I*c*x^n)^2*x^r+837*I*Pi*b*e^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+12*I*Pi*b*e^3*r^5*csgn(I*c)*cs
gn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-2673*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r+1188*b*d*e^2*n*r^2*(x^r)^2-2
16*ln(c)*b*d^2*e*r^5*x^r+1836*I*Pi*b*d*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+132*I*Pi*b*d^3*r^5*
csgn(I*c*x^n)^3+864*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*c*x^n)^2*x^r+12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3+2
673*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+108*I*Pi*b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)*x^r+2619*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-459*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*
c*x^n)^2*(x^r)^3+459*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-243*I*Pi*b*d^3*csgn(I*c*x^n)^3
-729*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-729*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r-837*I*Pi*b*e^3*r^2*csgn(I*c*x^n
)^3*(x^r)^3+579*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+3078*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(
x^r)^2+1026*ln(c)*b*d*e^2*r^4*(x^r)^2-5238*ln(c)*b*d^2*e*r^3*x^r+7614*ln(c)*b*d^2*e*r^2*x^r-5346*ln(c)*b*d^2*e
*r*x^r-3672*ln(c)*b*d*e^2*r^3*(x^r)^2-1296*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-972*I*Pi*b*d^3*r*csgn(I*
c)*csgn(I*c*x^n)^2-972*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2+3807*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n
)^2*x^r+972*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-2430*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-729*I*Pi*b*d*e^
2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-2673*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-729*I*Pi*b*d
^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-26...

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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^4,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-4>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. \(2 (173) = 346\).
time = 0.35, size = 843, normalized size = 4.41 \begin {gather*} -\frac {4 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{6} - 44 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{5} + 81 \, b d^{3} n + 193 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{4} + 243 \, a d^{3} - 432 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{3} + 522 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{2} - 324 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r - {\left (3 \, {\left (4 \, b r^{5} - 40 \, b r^{4} + 153 \, b r^{3} - 279 \, b r^{2} + 243 \, b r - 81 \, b\right )} e^{3} \log \left (c\right ) + 3 \, {\left (4 \, b n r^{5} - 40 \, b n r^{4} + 153 \, b n r^{3} - 279 \, b n r^{2} + 243 \, b n r - 81 \, b n\right )} e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n + 30 \, a\right )} r^{4} + 9 \, {\left (4 \, b n + 51 \, a\right )} r^{3} - 9 \, {\left (13 \, b n + 93 \, a\right )} r^{2} - 81 \, b n + 81 \, {\left (2 \, b n + 9 \, a\right )} r - 243 \, a\right )} e^{3}\right )} x^{3 \, r} - 27 \, {\left ({\left (2 \, b d r^{5} - 19 \, b d r^{4} + 68 \, b d r^{3} - 114 \, b d r^{2} + 90 \, b d r - 27 \, b d\right )} e^{2} \log \left (c\right ) + {\left (2 \, b d n r^{5} - 19 \, b d n r^{4} + 68 \, b d n r^{3} - 114 \, b d n r^{2} + 90 \, b d n r - 27 \, b d n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a d r^{5} - {\left (b d n + 19 \, a d\right )} r^{4} + 4 \, {\left (2 \, b d n + 17 \, a d\right )} r^{3} - 9 \, b d n - 2 \, {\left (11 \, b d n + 57 \, a d\right )} r^{2} - 27 \, a d + 6 \, {\left (4 \, b d n + 15 \, a d\right )} r\right )} e^{2}\right )} x^{2 \, r} - 27 \, {\left ({\left (4 \, b d^{2} r^{5} - 32 \, b d^{2} r^{4} + 97 \, b d^{2} r^{3} - 141 \, b d^{2} r^{2} + 99 \, b d^{2} r - 27 \, b d^{2}\right )} e \log \left (c\right ) + {\left (4 \, b d^{2} n r^{5} - 32 \, b d^{2} n r^{4} + 97 \, b d^{2} n r^{3} - 141 \, b d^{2} n r^{2} + 99 \, b d^{2} n r - 27 \, b d^{2} n\right )} e \log \left (x\right ) + {\left (4 \, a d^{2} r^{5} - 4 \, {\left (b d^{2} n + 8 \, a d^{2}\right )} r^{4} - 9 \, b d^{2} n + {\left (20 \, b d^{2} n + 97 \, a d^{2}\right )} r^{3} - 27 \, a d^{2} - {\left (37 \, b d^{2} n + 141 \, a d^{2}\right )} r^{2} + 3 \, {\left (10 \, b d^{2} n + 33 \, a d^{2}\right )} r\right )} e\right )} x^{r} + 3 \, {\left (4 \, b d^{3} r^{6} - 44 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} - 432 \, b d^{3} r^{3} + 522 \, b d^{3} r^{2} - 324 \, b d^{3} r + 81 \, b d^{3}\right )} \log \left (c\right ) + 3 \, {\left (4 \, b d^{3} n r^{6} - 44 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} - 432 \, b d^{3} n r^{3} + 522 \, b d^{3} n r^{2} - 324 \, b d^{3} n r + 81 \, b d^{3} n\right )} \log \left (x\right )}{9 \, {\left (4 \, r^{6} - 44 \, r^{5} + 193 \, r^{4} - 432 \, r^{3} + 522 \, r^{2} - 324 \, r + 81\right )} x^{3}} \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^4,x, algorithm="fricas")

[Out]

-1/9*(4*(b*d^3*n + 3*a*d^3)*r^6 - 44*(b*d^3*n + 3*a*d^3)*r^5 + 81*b*d^3*n + 193*(b*d^3*n + 3*a*d^3)*r^4 + 243*
a*d^3 - 432*(b*d^3*n + 3*a*d^3)*r^3 + 522*(b*d^3*n + 3*a*d^3)*r^2 - 324*(b*d^3*n + 3*a*d^3)*r - (3*(4*b*r^5 -
40*b*r^4 + 153*b*r^3 - 279*b*r^2 + 243*b*r - 81*b)*e^3*log(c) + 3*(4*b*n*r^5 - 40*b*n*r^4 + 153*b*n*r^3 - 279*
b*n*r^2 + 243*b*n*r - 81*b*n)*e^3*log(x) + (12*a*r^5 - 4*(b*n + 30*a)*r^4 + 9*(4*b*n + 51*a)*r^3 - 9*(13*b*n +
93*a)*r^2 - 81*b*n + 81*(2*b*n + 9*a)*r - 243*a)*e^3)*x^(3*r) - 27*((2*b*d*r^5 - 19*b*d*r^4 + 68*b*d*r^3 - 11
4*b*d*r^2 + 90*b*d*r - 27*b*d)*e^2*log(c) + (2*b*d*n*r^5 - 19*b*d*n*r^4 + 68*b*d*n*r^3 - 114*b*d*n*r^2 + 90*b*
d*n*r - 27*b*d*n)*e^2*log(x) + (2*a*d*r^5 - (b*d*n + 19*a*d)*r^4 + 4*(2*b*d*n + 17*a*d)*r^3 - 9*b*d*n - 2*(11*
b*d*n + 57*a*d)*r^2 - 27*a*d + 6*(4*b*d*n + 15*a*d)*r)*e^2)*x^(2*r) - 27*((4*b*d^2*r^5 - 32*b*d^2*r^4 + 97*b*d
^2*r^3 - 141*b*d^2*r^2 + 99*b*d^2*r - 27*b*d^2)*e*log(c) + (4*b*d^2*n*r^5 - 32*b*d^2*n*r^4 + 97*b*d^2*n*r^3 -
141*b*d^2*n*r^2 + 99*b*d^2*n*r - 27*b*d^2*n)*e*log(x) + (4*a*d^2*r^5 - 4*(b*d^2*n + 8*a*d^2)*r^4 - 9*b*d^2*n +
(20*b*d^2*n + 97*a*d^2)*r^3 - 27*a*d^2 - (37*b*d^2*n + 141*a*d^2)*r^2 + 3*(10*b*d^2*n + 33*a*d^2)*r)*e)*x^r +
3*(4*b*d^3*r^6 - 44*b*d^3*r^5 + 193*b*d^3*r^4 - 432*b*d^3*r^3 + 522*b*d^3*r^2 - 324*b*d^3*r + 81*b*d^3)*log(c
) + 3*(4*b*d^3*n*r^6 - 44*b*d^3*n*r^5 + 193*b*d^3*n*r^4 - 432*b*d^3*n*r^3 + 522*b*d^3*n*r^2 - 324*b*d^3*n*r +
81*b*d^3*n)*log(x))/((4*r^6 - 44*r^5 + 193*r^4 - 432*r^3 + 522*r^2 - 324*r + 81)*x^3)

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Sympy [A]
time = 88.39, size = 338, normalized size = 1.77 \begin {gather*} - \frac {a d^{3}}{3 x^{3}} + 3 a d^{2} e \left (\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{3 r}}{3 r x^{3} - 3 x^{3}} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n}{9 x^{3}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 3 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r - 3}}{r - 3} & \text {for}\: r \neq 3 \\\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^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {3}{2} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d e^{2} \left (\begin {cases} \frac {x^{2 r - 3}}{2 r - 3} & \text {for}\: r \neq \frac {3}{2} \\\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^{3} - 3 x^{3}} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r - 3}}{3 r - 3} & \text {for}\: r \neq 1 \\\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**4,x)

[Out]

-a*d**3/(3*x**3) + 3*a*d**2*e*Piecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)), (log(x), True)) + 3*a*d*e**2*Piece
wise((x**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (log(x), True)) + a*e**3*Piecewise((x**(3*r)/(3*r*x**3 - 3*x*
*3), Ne(r, 1)), (log(x), True)) - b*d**3*n/(9*x**3) - b*d**3*log(c*x**n)/(3*x**3) - 3*b*d**2*e*n*Piecewise((Pi
ecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)), (log(x), True))/(r - 3), (r > -oo) & (r < oo) & Ne(r, 3)), (log(x)
**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r - 3)/(r - 3), Ne(r, 3)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*
n*Piecewise((Piecewise((x**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (log(x), True))/(2*r - 3), (r > -oo) & (r <
oo) & Ne(r, 3/2)), (log(x)**2/2, True)) + 3*b*d*e**2*Piecewise((x**(2*r - 3)/(2*r - 3), Ne(r, 3/2)), (log(x),
True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r)/(3*r*x**3 - 3*x**3), Ne(r, 1)), (log(x), True))/
(3*r - 3), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r - 3)/(3*r - 3),
Ne(r, 1)), (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^4,x, algorithm="giac")

[Out]

integrate((x^r*e + d)^3*(b*log(c*x^n) + a)/x^4, 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^4} \,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^4,x)

[Out]

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^4, x)

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