\(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^8} \, dx\) [404]

Optimal result

Integrand size = 23, antiderivative size = 183 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n x^{-7+r}}{(7-r)^2}-\frac {3 b d e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {b e^3 n x^{-7+3 r}}{(7-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r} \]

[Out]

-1/49*b*d^3*n/x^7-3*b*d^2*e*n*x^(-7+r)/(7-r)^2-3*b*d*e^2*n*x^(-7+2*r)/(7-2*r)^2-b*e^3*n*x^(-7+3*r)/(7-3*r)^2-1
/7*d^3*(a+b*ln(c*x^n))/x^7-3*d^2*e*x^(-7+r)*(a+b*ln(c*x^n))/(7-r)-3*d*e^2*x^(-7+2*r)*(a+b*ln(c*x^n))/(7-2*r)-e
^3*x^(-7+3*r)*(a+b*ln(c*x^n))/(7-3*r)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 183, 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} \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{3 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n x^{r-7}}{(7-r)^2}-\frac {3 b d e^2 n x^{2 r-7}}{(7-2 r)^2}-\frac {b e^3 n x^{3 r-7}}{(7-3 r)^2} \]

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-1/49*(b*d^3*n)/x^7 - (3*b*d^2*e*n*x^(-7 + r))/(7 - r)^2 - (3*b*d*e^2*n*x^(-7 + 2*r))/(7 - 2*r)^2 - (b*e^3*n*x
^(-7 + 3*r))/(7 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (3*d^2*e*x^(-7 + r)*(a + b*Log[c*x^n]))/(7 - r)
- (3*d*e^2*x^(-7 + 2*r)*(a + b*Log[c*x^n]))/(7 - 2*r) - (e^3*x^(-7 + 3*r)*(a + b*Log[c*x^n]))/(7 - 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*} \text {integral}& = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}-(b n) \int \frac {-d^3+\frac {21 d^2 e x^r}{-7+r}+\frac {21 d e^2 x^{2 r}}{-7+2 r}+\frac {7 e^3 x^{3 r}}{-7+3 r}}{7 x^8} \, dx \\ & = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}-\frac {1}{7} (b n) \int \frac {-d^3+\frac {21 d^2 e x^r}{-7+r}+\frac {21 d e^2 x^{2 r}}{-7+2 r}+\frac {7 e^3 x^{3 r}}{-7+3 r}}{x^8} \, dx \\ & = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}-\frac {1}{7} (b n) \int \left (-\frac {d^3}{x^8}+\frac {21 d^2 e x^{-8+r}}{-7+r}+\frac {21 d e^2 x^{2 (-4+r)}}{-7+2 r}+\frac {7 e^3 x^{-8+3 r}}{-7+3 r}\right ) \, dx \\ & = -\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n x^{-7+r}}{(7-r)^2}-\frac {3 b d e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {b e^3 n x^{-7+3 r}}{(7-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {b n \left (-d^3-\frac {147 d^2 e x^r}{(-7+r)^2}-\frac {147 d e^2 x^{2 r}}{(7-2 r)^2}-\frac {49 e^3 x^{3 r}}{(7-3 r)^2}\right )+7 a \left (-d^3+\frac {21 d^2 e x^r}{-7+r}+\frac {21 d e^2 x^{2 r}}{-7+2 r}+\frac {7 e^3 x^{3 r}}{-7+3 r}\right )+7 b \left (-d^3+\frac {21 d^2 e x^r}{-7+r}+\frac {21 d e^2 x^{2 r}}{-7+2 r}+\frac {7 e^3 x^{3 r}}{-7+3 r}\right ) \log \left (c x^n\right )}{49 x^7} \]

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

(b*n*(-d^3 - (147*d^2*e*x^r)/(-7 + r)^2 - (147*d*e^2*x^(2*r))/(7 - 2*r)^2 - (49*e^3*x^(3*r))/(7 - 3*r)^2) + 7*
a*(-d^3 + (21*d^2*e*x^r)/(-7 + r) + (21*d*e^2*x^(2*r))/(-7 + 2*r) + (7*e^3*x^(3*r))/(-7 + 3*r)) + 7*b*(-d^3 +
(21*d^2*e*x^r)/(-7 + r) + (21*d*e^2*x^(2*r))/(-7 + 2*r) + (7*e^3*x^(3*r))/(-7 + 3*r))*Log[c*x^n])/(49*x^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1040\) vs. \(2(179)=358\).

Time = 3.29 (sec) , antiderivative size = 1041, normalized size of antiderivative = 5.69

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1041\)
risch	\(\text {Expression too large to display}\)	\(4031\)

[In]

int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/49*(823543*b*ln(c*x^n)*d^3+2470629*b*d*e^2*ln(c*x^n)*(x^r)^2+823543*e^3*(x^r)^3*a-49392*b*d^3*n*r^3+139258*
b*d^3*n*r^2-201684*b*d^3*n*r+1915998*a*d*e^2*r^2*(x^r)^2-489804*a*d*e^2*r^3*(x^r)^2+2470629*d*e^2*(x^r)^2*a+24
70629*d^2*e*x^r*a+823543*a*d^3-122451*a*e^3*r^3*(x^r)^3+521017*a*e^3*r^2*(x^r)^3-1058841*a*e^3*r*(x^r)^3-588*a
*e^3*r^5*(x^r)^3+13720*a*e^3*r^4*(x^r)^3-588*(x^r)^3*ln(c*x^n)*b*e^3*r^5+13720*(x^r)^3*ln(c*x^n)*b*e^3*r^4-122
451*(x^r)^3*ln(c*x^n)*b*e^3*r^3+521017*(x^r)^3*ln(c*x^n)*b*e^3*r^2-1058841*(x^r)^3*ln(c*x^n)*b*e^3*r+2470629*b
*d^2*e*ln(c*x^n)*x^r+36*b*d^3*n*r^6-924*b*d^3*n*r^5+9457*b*d^3*n*r^4-5292*x^r*ln(c*x^n)*b*d^2*e*r^5+98784*x^r*
ln(c*x^n)*b*d^2*e*r^4-698691*x^r*ln(c*x^n)*b*d^2*e*r^3+2369787*x^r*ln(c*x^n)*b*d^2*e*r^2-3882417*x^r*ln(c*x^n)
*b*d^2*e*r-2646*(x^r)^2*ln(c*x^n)*b*d*e^2*r^5+58653*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-489804*(x^r)^2*ln(c*x^n)*b*d
*e^2*r^3+1915998*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-3529470*(x^r)^2*ln(c*x^n)*b*d*e^2*r+117649*b*d^3*n+252*ln(c*x^n
)*b*d^3*r^6-6468*ln(c*x^n)*b*d^3*r^5+66199*ln(c*x^n)*b*d^3*r^4-345744*ln(c*x^n)*b*d^3*r^3+974806*ln(c*x^n)*b*d
^3*r^2-1411788*ln(c*x^n)*b*d^3*r+823543*e^3*b*ln(c*x^n)*(x^r)^3-345744*a*d^3*r^3+974806*a*d^3*r^2-1411788*a*d^
3*r+252*a*d^3*r^6-6468*a*d^3*r^5+66199*a*d^3*r^4-698691*a*d^2*e*r^3*x^r+352947*b*d*e^2*n*(x^r)^2+352947*b*d^2*
e*n*x^r+117649*b*e^3*n*(x^r)^3-4116*b*e^3*n*r^3*(x^r)^3+31213*b*e^3*n*r^2*(x^r)^3-100842*b*e^3*n*r*(x^r)^3+236
9787*a*d^2*e*r^2*x^r+98784*a*d^2*e*r^4*x^r-3882417*a*d^2*e*r*x^r+196*b*e^3*n*r^4*(x^r)^3-3529470*a*d*e^2*r*(x^
r)^2-2646*a*d*e^2*r^5*(x^r)^2+58653*a*d*e^2*r^4*(x^r)^2-5292*a*d^2*e*r^5*x^r+158466*b*d*e^2*n*r^2*(x^r)^2+2665
11*b*d^2*e*n*r^2*x^r-403368*b*d*e^2*n*r*(x^r)^2-504210*b*d^2*e*n*r*x^r-61740*b*d^2*e*n*r^3*x^r+1323*b*d*e^2*n*
r^4*(x^r)^2-24696*b*d*e^2*n*r^3*(x^r)^2+5292*b*d^2*e*n*r^4*x^r)/x^7/(-7+3*r)^2/(-7+2*r)^2/(-7+r)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (174) = 348\).

Time = 0.30 (sec) , antiderivative size = 981, normalized size of antiderivative = 5.36 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Too large to display} \]

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^8,x, algorithm="fricas")

[Out]

-1/49*(36*(b*d^3*n + 7*a*d^3)*r^6 - 924*(b*d^3*n + 7*a*d^3)*r^5 + 117649*b*d^3*n + 9457*(b*d^3*n + 7*a*d^3)*r^
4 + 823543*a*d^3 - 49392*(b*d^3*n + 7*a*d^3)*r^3 + 139258*(b*d^3*n + 7*a*d^3)*r^2 - 201684*(b*d^3*n + 7*a*d^3)
*r - 49*(12*a*e^3*r^5 - 2401*b*e^3*n - 4*(b*e^3*n + 70*a*e^3)*r^4 - 16807*a*e^3 + 21*(4*b*e^3*n + 119*a*e^3)*r
^3 - 49*(13*b*e^3*n + 217*a*e^3)*r^2 + 1029*(2*b*e^3*n + 21*a*e^3)*r + (12*b*e^3*r^5 - 280*b*e^3*r^4 + 2499*b*
e^3*r^3 - 10633*b*e^3*r^2 + 21609*b*e^3*r - 16807*b*e^3)*log(c) + (12*b*e^3*n*r^5 - 280*b*e^3*n*r^4 + 2499*b*e
^3*n*r^3 - 10633*b*e^3*n*r^2 + 21609*b*e^3*n*r - 16807*b*e^3*n)*log(x))*x^(3*r) - 147*(18*a*d*e^2*r^5 - 2401*b
*d*e^2*n - 3*(3*b*d*e^2*n + 133*a*d*e^2)*r^4 - 16807*a*d*e^2 + 28*(6*b*d*e^2*n + 119*a*d*e^2)*r^3 - 98*(11*b*d
*e^2*n + 133*a*d*e^2)*r^2 + 686*(4*b*d*e^2*n + 35*a*d*e^2)*r + (18*b*d*e^2*r^5 - 399*b*d*e^2*r^4 + 3332*b*d*e^
2*r^3 - 13034*b*d*e^2*r^2 + 24010*b*d*e^2*r - 16807*b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 399*b*d*e^2*n*r^4 +
3332*b*d*e^2*n*r^3 - 13034*b*d*e^2*n*r^2 + 24010*b*d*e^2*n*r - 16807*b*d*e^2*n)*log(x))*x^(2*r) - 147*(36*a*d^
2*e*r^5 - 2401*b*d^2*e*n - 12*(3*b*d^2*e*n + 56*a*d^2*e)*r^4 - 16807*a*d^2*e + 7*(60*b*d^2*e*n + 679*a*d^2*e)*
r^3 - 49*(37*b*d^2*e*n + 329*a*d^2*e)*r^2 + 343*(10*b*d^2*e*n + 77*a*d^2*e)*r + (36*b*d^2*e*r^5 - 672*b*d^2*e*
r^4 + 4753*b*d^2*e*r^3 - 16121*b*d^2*e*r^2 + 26411*b*d^2*e*r - 16807*b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 672
*b*d^2*e*n*r^4 + 4753*b*d^2*e*n*r^3 - 16121*b*d^2*e*n*r^2 + 26411*b*d^2*e*n*r - 16807*b*d^2*e*n)*log(x))*x^r +
7*(36*b*d^3*r^6 - 924*b*d^3*r^5 + 9457*b*d^3*r^4 - 49392*b*d^3*r^3 + 139258*b*d^3*r^2 - 201684*b*d^3*r + 1176
49*b*d^3)*log(c) + 7*(36*b*d^3*n*r^6 - 924*b*d^3*n*r^5 + 9457*b*d^3*n*r^4 - 49392*b*d^3*n*r^3 + 139258*b*d^3*n
*r^2 - 201684*b*d^3*n*r + 117649*b*d^3*n)*log(x))/((36*r^6 - 924*r^5 + 9457*r^4 - 49392*r^3 + 139258*r^2 - 201
684*r + 117649)*x^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Timed out} \]

[In]

integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**8,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^8,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-8>0)', see `assume?` for mor
e details)Is

Giac [F]

\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}} \,d x } \]

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^8, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]

[In]

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^8,x)

[Out]

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^8, x)