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

Optimal result

Integrand size = 23, antiderivative size = 191 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \]

[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)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 191, 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, 2372, 12, 14} \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac {3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \]

[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*} \text {integral}& = -\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}-(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 {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}-\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 {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}-\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 {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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {b n \left (-d^3-\frac {27 d^2 e x^r}{(-3+r)^2}-\frac {27 d e^2 x^{2 r}}{(3-2 r)^2}-\frac {e^3 x^{3 r}}{(-1+r)^2}\right )+3 a \left (-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}\right )+3 b \left (-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}\right ) \log \left (c x^n\right )}{9 x^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1038\) vs. \(2(183)=366\).

Time = 3.47 (sec) , antiderivative size = 1039, normalized size of antiderivative = 5.44

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1039\)
risch	\(\text {Expression too large to display}\)	\(4027\)

[In]

int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/9*(243*b*ln(c*x^n)*d^3+729*b*d*e^2*ln(c*x^n)*(x^r)^2+243*e^3*(x^r)^3*a-432*b*d^3*n*r^3+522*b*d^3*n*r^2-324*
b*d^3*n*r+3078*a*d*e^2*r^2*(x^r)^2-1836*a*d*e^2*r^3*(x^r)^2+729*d*e^2*(x^r)^2*a+729*d^2*e*x^r*a+243*a*d^3-459*
a*e^3*r^3*(x^r)^3+837*a*e^3*r^2*(x^r)^3-729*a*e^3*r*(x^r)^3-12*a*e^3*r^5*(x^r)^3+120*a*e^3*r^4*(x^r)^3-12*(x^r
)^3*ln(c*x^n)*b*e^3*r^5+120*(x^r)^3*ln(c*x^n)*b*e^3*r^4-459*(x^r)^3*ln(c*x^n)*b*e^3*r^3+837*(x^r)^3*ln(c*x^n)*
b*e^3*r^2-729*(x^r)^3*ln(c*x^n)*b*e^3*r+729*b*d^2*e*ln(c*x^n)*x^r+4*b*d^3*n*r^6-44*b*d^3*n*r^5+193*b*d^3*n*r^4
-108*x^r*ln(c*x^n)*b*d^2*e*r^5+864*x^r*ln(c*x^n)*b*d^2*e*r^4-2619*x^r*ln(c*x^n)*b*d^2*e*r^3+3807*x^r*ln(c*x^n)
*b*d^2*e*r^2-2673*x^r*ln(c*x^n)*b*d^2*e*r-54*(x^r)^2*ln(c*x^n)*b*d*e^2*r^5+513*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-1
836*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3+3078*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-2430*(x^r)^2*ln(c*x^n)*b*d*e^2*r+81*b*d^3
*n+12*ln(c*x^n)*b*d^3*r^6-132*ln(c*x^n)*b*d^3*r^5+579*ln(c*x^n)*b*d^3*r^4-1296*ln(c*x^n)*b*d^3*r^3+1566*ln(c*x
^n)*b*d^3*r^2-972*ln(c*x^n)*b*d^3*r+243*e^3*b*ln(c*x^n)*(x^r)^3-1296*a*d^3*r^3+1566*a*d^3*r^2-972*a*d^3*r+12*a
*d^3*r^6-132*a*d^3*r^5+579*a*d^3*r^4-2619*a*d^2*e*r^3*x^r+243*b*d*e^2*n*(x^r)^2+243*b*d^2*e*n*x^r+81*b*e^3*n*(
x^r)^3-36*b*e^3*n*r^3*(x^r)^3+117*b*e^3*n*r^2*(x^r)^3-162*b*e^3*n*r*(x^r)^3+3807*a*d^2*e*r^2*x^r+864*a*d^2*e*r
^4*x^r-2673*a*d^2*e*r*x^r+4*b*e^3*n*r^4*(x^r)^3-2430*a*d*e^2*r*(x^r)^2-54*a*d*e^2*r^5*(x^r)^2+513*a*d*e^2*r^4*
(x^r)^2-108*a*d^2*e*r^5*x^r+594*b*d*e^2*n*r^2*(x^r)^2+999*b*d^2*e*n*r^2*x^r-648*b*d*e^2*n*r*(x^r)^2-810*b*d^2*
e*n*r*x^r-540*b*d^2*e*n*r^3*x^r+27*b*d*e^2*n*r^4*(x^r)^2-216*b*d*e^2*n*r^3*(x^r)^2+108*b*d^2*e*n*r^4*x^r)/x^3/
(-1+r)^2/(-3+2*r)^2/(-3+r)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (175) = 350\).

Time = 0.34 (sec) , antiderivative size = 980, normalized size of antiderivative = 5.13 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\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 (12 \, a e^{3} r^{5} - 81 \, b e^{3} n - 4 \, {\left (b e^{3} n + 30 \, a e^{3}\right )} r^{4} - 243 \, a e^{3} + 9 \, {\left (4 \, b e^{3} n + 51 \, a e^{3}\right )} r^{3} - 9 \, {\left (13 \, b e^{3} n + 93 \, a e^{3}\right )} r^{2} + 81 \, {\left (2 \, b e^{3} n + 9 \, a e^{3}\right )} r + 3 \, {\left (4 \, b e^{3} r^{5} - 40 \, b e^{3} r^{4} + 153 \, b e^{3} r^{3} - 279 \, b e^{3} r^{2} + 243 \, b e^{3} r - 81 \, b e^{3}\right )} \log \left (c\right ) + 3 \, {\left (4 \, b e^{3} n r^{5} - 40 \, b e^{3} n r^{4} + 153 \, b e^{3} n r^{3} - 279 \, b e^{3} n r^{2} + 243 \, b e^{3} n r - 81 \, b e^{3} n\right )} \log \left (x\right )\right )} x^{3 \, r} - 27 \, {\left (2 \, a d e^{2} r^{5} - 9 \, b d e^{2} n - {\left (b d e^{2} n + 19 \, a d e^{2}\right )} r^{4} - 27 \, a d e^{2} + 4 \, {\left (2 \, b d e^{2} n + 17 \, a d e^{2}\right )} r^{3} - 2 \, {\left (11 \, b d e^{2} n + 57 \, a d e^{2}\right )} r^{2} + 6 \, {\left (4 \, b d e^{2} n + 15 \, a d e^{2}\right )} r + {\left (2 \, b d e^{2} r^{5} - 19 \, b d e^{2} r^{4} + 68 \, b d e^{2} r^{3} - 114 \, b d e^{2} r^{2} + 90 \, b d e^{2} r - 27 \, b d e^{2}\right )} \log \left (c\right ) + {\left (2 \, b d e^{2} n r^{5} - 19 \, b d e^{2} n r^{4} + 68 \, b d e^{2} n r^{3} - 114 \, b d e^{2} n r^{2} + 90 \, b d e^{2} n r - 27 \, b d e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 27 \, {\left (4 \, a d^{2} e r^{5} - 9 \, b d^{2} e n - 4 \, {\left (b d^{2} e n + 8 \, a d^{2} e\right )} r^{4} - 27 \, a d^{2} e + {\left (20 \, b d^{2} e n + 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n + 141 \, a d^{2} e\right )} r^{2} + 3 \, {\left (10 \, b d^{2} e n + 33 \, a d^{2} e\right )} r + {\left (4 \, b d^{2} e r^{5} - 32 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} - 141 \, b d^{2} e r^{2} + 99 \, b d^{2} e r - 27 \, b d^{2} e\right )} \log \left (c\right ) + {\left (4 \, b d^{2} e n r^{5} - 32 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} - 141 \, b d^{2} e n r^{2} + 99 \, b d^{2} e n r - 27 \, b d^{2} e n\right )} \log \left (x\right )\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}} \]

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

Sympy [A] (verification not implemented)

Time = 49.68 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.82 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=- \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 \\\frac {x^{r} \log {\left (x \right )}}{x^{3}} & \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} \\\frac {x^{2 r} \log {\left (x \right )}}{x^{3}} & \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 \\\frac {x^{3 r} \log {\left (x \right )}}{x^{3}} & \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 - 3}}{r - 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 - 3}}{2 r - 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}}{3 r - 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 )} \]

[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)), (x**r*log(x)/x**3, True)) + 3*a*d*
e**2*Piecewise((x**(2*r)/(2*r*x**3 - 3*x**3), Ne(r, 3/2)), (x**(2*r)*log(x)/x**3, True)) + a*e**3*Piecewise((x
**(3*r)/(3*r*x**3 - 3*x**3), Ne(r, 1)), (x**(3*r)*log(x)/x**3, 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((Piecewise((x**(r - 3)/(r - 3), Ne(r, 3)), (log(x), True))/(r - 3), (r > -o
o) & (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 - 3)/(2*r - 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)/(3*r - 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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]

[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

Giac [F]

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

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]

[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)