\(\int \frac {(d+e x^r)^2 (a+b \log (c x^n))}{x^8} \, dx\) [391]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{-7+r}}{(7-r)^2}-\frac {b e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 127, 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} \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{r-7}}{(7-r)^2}-\frac {b e^2 n x^{2 r-7}}{(7-2 r)^2} \]

[In]

Int[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-1/49*(b*d^2*n)/x^7 - (2*b*d*e*n*x^(-7 + r))/(7 - r)^2 - (b*e^2*n*x^(-7 + 2*r))/(7 - 2*r)^2 - (d^2*(a + b*Log[
c*x^n]))/(7*x^7) - (2*d*e*x^(-7 + r)*(a + b*Log[c*x^n]))/(7 - r) - (e^2*x^(-7 + 2*r)*(a + b*Log[c*x^n]))/(7 -
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^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-(b n) \int \frac {-d^2+\frac {14 d e x^r}{-7+r}+\frac {7 e^2 x^{2 r}}{-7+2 r}}{7 x^8} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {1}{7} (b n) \int \frac {-d^2+\frac {14 d e x^r}{-7+r}+\frac {7 e^2 x^{2 r}}{-7+2 r}}{x^8} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {1}{7} (b n) \int \left (-\frac {d^2}{x^8}+\frac {14 d e x^{-8+r}}{-7+r}+\frac {7 e^2 x^{2 (-4+r)}}{-7+2 r}\right ) \, dx \\ & = -\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{-7+r}}{(7-r)^2}-\frac {b e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {b n \left (-d^2-\frac {98 d e x^r}{(-7+r)^2}-\frac {49 e^2 x^{2 r}}{(7-2 r)^2}\right )+a \left (-7 d^2+\frac {98 d e x^r}{-7+r}+\frac {49 e^2 x^{2 r}}{-7+2 r}\right )+7 b \left (-d^2+\frac {14 d e x^r}{-7+r}+\frac {7 e^2 x^{2 r}}{-7+2 r}\right ) \log \left (c x^n\right )}{49 x^7} \]

[In]

Integrate[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^8,x]

[Out]

(b*n*(-d^2 - (98*d*e*x^r)/(-7 + r)^2 - (49*e^2*x^(2*r))/(7 - 2*r)^2) + a*(-7*d^2 + (98*d*e*x^r)/(-7 + r) + (49
*e^2*x^(2*r))/(-7 + 2*r)) + 7*b*(-d^2 + (14*d*e*x^r)/(-7 + r) + (7*e^2*x^(2*r))/(-7 + 2*r))*Log[c*x^n])/(49*x^
7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(123)=246\).

Time = 3.26 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.72

method	result	size
parallelrisch	\(-\frac {16807 b \ln \left (c \,x^{n}\right ) d^{2}+4802 b d e n \,x^{r}+637 b \,d^{2} n \,r^{2}+33614 d e \,x^{r} a -2058 b \,d^{2} n r +33614 d e \,x^{r} b \ln \left (c \,x^{n}\right )+28 a \,d^{2} r^{4}-588 a \,d^{2} r^{3}-392 a d e \,r^{3} x^{r}+2401 b \,d^{2} n +16807 a \,d^{2}+28 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-588 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}+4459 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-14406 \ln \left (c \,x^{n}\right ) b \,d^{2} r +4 b \,d^{2} n \,r^{4}-84 b \,d^{2} n \,r^{3}-2744 b d e n r \,x^{r}+4459 a \,d^{2} r^{2}-14406 a \,d^{2} r +1715 a \,e^{2} r^{2} x^{2 r}-9604 a \,e^{2} r \,x^{2 r}+2401 b \,e^{2} n \,x^{2 r}-98 a \,e^{2} r^{3} x^{2 r}+16807 e^{2} x^{2 r} b \ln \left (c \,x^{n}\right )+392 b d e n \,r^{2} x^{r}+5488 a d e \,r^{2} x^{r}-24010 a d e r \,x^{r}-686 b \,e^{2} n r \,x^{2 r}-98 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}+1715 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-9604 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +49 b \,e^{2} n \,r^{2} x^{2 r}+16807 e^{2} x^{2 r} a -392 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}+5488 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-24010 x^{r} \ln \left (c \,x^{n}\right ) b d e r}{49 x^{7} \left (-7+2 r \right )^{2} \left (-7+r \right )^{2}}\)	\(473\)
risch	\(\text {Expression too large to display}\)	\(1930\)

[In]

int((d+e*x^r)^2*(a+b*ln(c*x^n))/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/49/x^7*(16807*b*ln(c*x^n)*d^2-686*b*e^2*n*r*(x^r)^2+4802*b*d*e*n*x^r+1715*a*e^2*r^2*(x^r)^2-9604*a*e^2*r*(x
^r)^2+2401*b*e^2*n*(x^r)^2-98*a*e^2*r^3*(x^r)^2+16807*e^2*(x^r)^2*a+637*b*d^2*n*r^2+33614*d*e*x^r*a+16807*e^2*
(x^r)^2*b*ln(c*x^n)-2058*b*d^2*n*r+33614*d*e*x^r*b*ln(c*x^n)+28*a*d^2*r^4-588*a*d^2*r^3-392*a*d*e*r^3*x^r+2401
*b*d^2*n+16807*a*d^2+28*ln(c*x^n)*b*d^2*r^4-588*ln(c*x^n)*b*d^2*r^3+4459*ln(c*x^n)*b*d^2*r^2-14406*ln(c*x^n)*b
*d^2*r+4*b*d^2*n*r^4-84*b*d^2*n*r^3-2744*b*d*e*n*r*x^r+4459*a*d^2*r^2-14406*a*d^2*r-98*(x^r)^2*ln(c*x^n)*b*e^2
*r^3+1715*(x^r)^2*ln(c*x^n)*b*e^2*r^2-9604*(x^r)^2*ln(c*x^n)*b*e^2*r+392*b*d*e*n*r^2*x^r+5488*a*d*e*r^2*x^r-24
010*a*d*e*r*x^r+49*b*e^2*n*r^2*(x^r)^2-392*x^r*ln(c*x^n)*b*d*e*r^3+5488*x^r*ln(c*x^n)*b*d*e*r^2-24010*x^r*ln(c
*x^n)*b*d*e*r)/(-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. 466 vs. \(2 (118) = 236\).

Time = 0.32 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.67 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {4 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{4} + 2401 \, b d^{2} n - 84 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{3} + 16807 \, a d^{2} + 637 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{2} - 2058 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r - 49 \, {\left (2 \, a e^{2} r^{3} - 49 \, b e^{2} n - 343 \, a e^{2} - {\left (b e^{2} n + 35 \, a e^{2}\right )} r^{2} + 14 \, {\left (b e^{2} n + 14 \, a e^{2}\right )} r + {\left (2 \, b e^{2} r^{3} - 35 \, b e^{2} r^{2} + 196 \, b e^{2} r - 343 \, b e^{2}\right )} \log \left (c\right ) + {\left (2 \, b e^{2} n r^{3} - 35 \, b e^{2} n r^{2} + 196 \, b e^{2} n r - 343 \, b e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 98 \, {\left (4 \, a d e r^{3} - 49 \, b d e n - 343 \, a d e - 4 \, {\left (b d e n + 14 \, a d e\right )} r^{2} + 7 \, {\left (4 \, b d e n + 35 \, a d e\right )} r + {\left (4 \, b d e r^{3} - 56 \, b d e r^{2} + 245 \, b d e r - 343 \, b d e\right )} \log \left (c\right ) + {\left (4 \, b d e n r^{3} - 56 \, b d e n r^{2} + 245 \, b d e n r - 343 \, b d e n\right )} \log \left (x\right )\right )} x^{r} + 7 \, {\left (4 \, b d^{2} r^{4} - 84 \, b d^{2} r^{3} + 637 \, b d^{2} r^{2} - 2058 \, b d^{2} r + 2401 \, b d^{2}\right )} \log \left (c\right ) + 7 \, {\left (4 \, b d^{2} n r^{4} - 84 \, b d^{2} n r^{3} + 637 \, b d^{2} n r^{2} - 2058 \, b d^{2} n r + 2401 \, b d^{2} n\right )} \log \left (x\right )}{49 \, {\left (4 \, r^{4} - 84 \, r^{3} + 637 \, r^{2} - 2058 \, r + 2401\right )} x^{7}} \]

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^8,x, algorithm="fricas")

[Out]

-1/49*(4*(b*d^2*n + 7*a*d^2)*r^4 + 2401*b*d^2*n - 84*(b*d^2*n + 7*a*d^2)*r^3 + 16807*a*d^2 + 637*(b*d^2*n + 7*
a*d^2)*r^2 - 2058*(b*d^2*n + 7*a*d^2)*r - 49*(2*a*e^2*r^3 - 49*b*e^2*n - 343*a*e^2 - (b*e^2*n + 35*a*e^2)*r^2
+ 14*(b*e^2*n + 14*a*e^2)*r + (2*b*e^2*r^3 - 35*b*e^2*r^2 + 196*b*e^2*r - 343*b*e^2)*log(c) + (2*b*e^2*n*r^3 -
35*b*e^2*n*r^2 + 196*b*e^2*n*r - 343*b*e^2*n)*log(x))*x^(2*r) - 98*(4*a*d*e*r^3 - 49*b*d*e*n - 343*a*d*e - 4*
(b*d*e*n + 14*a*d*e)*r^2 + 7*(4*b*d*e*n + 35*a*d*e)*r + (4*b*d*e*r^3 - 56*b*d*e*r^2 + 245*b*d*e*r - 343*b*d*e)
*log(c) + (4*b*d*e*n*r^3 - 56*b*d*e*n*r^2 + 245*b*d*e*n*r - 343*b*d*e*n)*log(x))*x^r + 7*(4*b*d^2*r^4 - 84*b*d
^2*r^3 + 637*b*d^2*r^2 - 2058*b*d^2*r + 2401*b*d^2)*log(c) + 7*(4*b*d^2*n*r^4 - 84*b*d^2*n*r^3 + 637*b*d^2*n*r
^2 - 2058*b*d^2*n*r + 2401*b*d^2*n)*log(x))/((4*r^4 - 84*r^3 + 637*r^2 - 2058*r + 2401)*x^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Timed out} \]

[In]

integrate((d+e*x**r)**2*(a+b*ln(c*x**n))/x**8,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d+e*x^r)^2*(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]

\[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}} \,d x } \]

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^2*(b*log(c*x^n) + a)/x^8, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]

[In]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^8,x)

[Out]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^8, x)