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

Optimal. Leaf size=127 \[ -\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{-7+r}}{(7-r)^2}-\frac {b e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r} \]

[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]
time = 0.12, antiderivative size = 127, 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^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} \end {gather*}

Antiderivative was successfully verified.

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

________________________________________________________________________________________

Mathematica [A]
time = 0.17, size = 119, normalized size = 0.94 \begin {gather*} \frac {-7 b d^2 n \log (x)-d^2 \left (7 a+b n-7 b n \log (x)+7 b \log \left (c x^n\right )\right )+\frac {98 d e x^r \left (-b n+a (-7+r)+b (-7+r) \log \left (c x^n\right )\right )}{(-7+r)^2}+\frac {49 e^2 x^{2 r} \left (-b n+a (-7+2 r)+b (-7+2 r) \log \left (c x^n\right )\right )}{(7-2 r)^2}}{49 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.22, size = 1930, normalized size = 15.20

method	result	size
risch	\(\text {Expression too large to display}\)	\(1930\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*b*(-7*e^2*(x^r)^2*r+2*d^2*r^2-28*d*e*x^r*r+49*e^2*(x^r)^2-21*d^2*r+98*d*e*x^r+49*d^2)/x^7/(-7+2*r)/(-7+r)
*ln(x^n)-1/98*(33614*e^2*(x^r)^2*a+67228*d*e*x^r*a+1274*b*d^2*n*r^2-4116*b*d^2*n*r+8918*ln(c)*b*d^2*r^2-28812*
ln(c)*b*d^2*r+56*ln(c)*b*d^2*r^4-1176*ln(c)*b*d^2*r^3+33614*d^2*b*ln(c)+392*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)*x^r+4802*b*d^2*n+33614*a*d^2+8*b*d^2*n*r^4-168*b*d^2*n*r^3+56*a*d^2*r^4-1176*a*d^2*r^3-4459*I
*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1715*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-1680
7*I*Pi*b*d^2*csgn(I*c*x^n)^3+8918*a*d^2*r^2-28812*a*d^2*r+33614*ln(c)*b*e^2*(x^r)^2-5488*I*Pi*b*d*e*r^2*csgn(I
*c*x^n)^3*x^r-9604*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-196*a*e^2*r^3*(x^r)^2+3430*a*e^2*r^2*(x^r)^2
-19208*a*e^2*r*(x^r)^2+4802*b*e^2*n*(x^r)^2+10976*a*d*e*r^2*x^r-48020*a*d*e*r*x^r-1372*b*e^2*n*r*(x^r)^2+9604*
b*d*e*n*x^r+98*b*e^2*n*r^2*(x^r)^2-784*a*d*e*r^3*x^r+3430*ln(c)*b*e^2*r^2*(x^r)^2-19208*ln(c)*b*e^2*r*(x^r)^2-
196*ln(c)*b*e^2*r^3*(x^r)^2+67228*ln(c)*b*d*e*x^r-24010*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r+5488*I*Pi*b
*d*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r-1715*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2+28*I*Pi*b*d^2*r^4*csgn(I*c)
*csgn(I*c*x^n)^2-588*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2+5488*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*
x^r+9604*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-9604*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^
2*(x^r)^2-28*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+24010*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*x^n)*csgn(
I*c*x^n)*x^r-5488*I*Pi*b*d*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-1715*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)*(x^r)^2-24010*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-33614*I*Pi*b*d*e*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)*x^r+98*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-392*I*Pi*b*d*e*r^3*csgn
(I*c)*csgn(I*c*x^n)^2*x^r-392*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+4459*I*Pi*b*d^2*r^2*csgn(I*c)*csg
n(I*c*x^n)^2+98*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+9604*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+16807*I*Pi*b*
e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+14406*I*Pi*b*d^2*r*csgn(I*c*x^n)^3+14406*I*Pi*b*d^2*r*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)-5488*b*d*e*n*r*x^r+10976*ln(c)*b*d*e*r^2*x^r-588*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^
2+4459*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+28*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-16807*I*Pi*b*e
^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+392*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-28*I*Pi*b*d^2*r^4*csgn(I
*c*x^n)^3+24010*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r+1715*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-98*I*Pi
*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-48020*ln(c)*b*d*e*r*x^r+16807*I*Pi*b*e^2*csgn(I*c)*csgn(I*c*x^n
)^2*(x^r)^2+588*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-784*ln(c)*b*d*e*r^3*x^r+784*b*d*e*n*r^2*x^r
-14406*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*c*x^n)^2-14406*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-16807*I*Pi*b*d^2*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+33614*I*Pi*b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r+33614*I*Pi*b*d*e*csgn(I*x^n
)*csgn(I*c*x^n)^2*x^r+16807*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2-16807*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2+58
8*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3-4459*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3-98*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*c*x^n)
^2*(x^r)^2-33614*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r+16807*I*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2)/(-7+2*r)^2/x^7/(-7
+r)^2

________________________________________________________________________________________

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)^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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (118) = 236\).
time = 0.35, size = 422, normalized size = 3.32 \begin {gather*} -\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 ({\left (2 \, b r^{3} - 35 \, b r^{2} + 196 \, b r - 343 \, b\right )} e^{2} \log \left (c\right ) + {\left (2 \, b n r^{3} - 35 \, b n r^{2} + 196 \, b n r - 343 \, b n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a r^{3} - {\left (b n + 35 \, a\right )} r^{2} - 49 \, b n + 14 \, {\left (b n + 14 \, a\right )} r - 343 \, a\right )} e^{2}\right )} x^{2 \, r} - 98 \, {\left ({\left (4 \, b d r^{3} - 56 \, b d r^{2} + 245 \, b d r - 343 \, b d\right )} e \log \left (c\right ) + {\left (4 \, b d n r^{3} - 56 \, b d n r^{2} + 245 \, b d n r - 343 \, b d n\right )} e \log \left (x\right ) + {\left (4 \, a d r^{3} - 49 \, b d n - 4 \, {\left (b d n + 14 \, a d\right )} r^{2} - 343 \, a d + 7 \, {\left (4 \, b d n + 35 \, a d\right )} r\right )} e\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*b*r^3 - 35*b*r^2 + 196*b*r - 343*b)*e^2*log(c) + (2*b*n*r^3 -
35*b*n*r^2 + 196*b*n*r - 343*b*n)*e^2*log(x) + (2*a*r^3 - (b*n + 35*a)*r^2 - 49*b*n + 14*(b*n + 14*a)*r - 343
*a)*e^2)*x^(2*r) - 98*((4*b*d*r^3 - 56*b*d*r^2 + 245*b*d*r - 343*b*d)*e*log(c) + (4*b*d*n*r^3 - 56*b*d*n*r^2 +
245*b*d*n*r - 343*b*d*n)*e*log(x) + (4*a*d*r^3 - 49*b*d*n - 4*(b*d*n + 14*a*d)*r^2 - 343*a*d + 7*(4*b*d*n + 3
5*a*d)*r)*e)*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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8856 deep

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________