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

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

[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)

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Rubi [A]  time = 0.18, antiderivative size = 109, normalized size of antiderivative = 0.86, 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 = {270, 2334, 12, 14} \[ -\frac {1}{7} \left (\frac {d^2}{x^7}+\frac {14 d e x^{r-7}}{7-r}+\frac {7 e^2 x^{2 r-7}}{7-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{r-7}}{(7-r)^2}-\frac {b e^2 n x^{2 r-7}}{(7-2 r)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 270

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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*}

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Mathematica [A]  time = 0.30, size = 127, normalized size = 1.00 \[ \frac {a \left (-7 d^2+\frac {98 d e x^r}{r-7}+\frac {49 e^2 x^{2 r}}{2 r-7}\right )+7 b \log \left (c x^n\right ) \left (-d^2+\frac {14 d e x^r}{r-7}+\frac {7 e^2 x^{2 r}}{2 r-7}\right )+b n \left (-d^2-\frac {98 d e x^r}{(r-7)^2}-\frac {49 e^2 x^{2 r}}{(7-2 r)^2}\right )}{49 x^7} \]

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.48, size = 466, normalized size = 3.67 \[ -\frac {4 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{4} + 2401 \, b d^{2} n - 84 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{3} + 16807 \, a d^{2} + 637 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{2} - 2058 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r - 49 \, {\left (2 \, a e^{2} r^{3} - 49 \, b e^{2} n - 343 \, a e^{2} - {\left (b e^{2} n + 35 \, a e^{2}\right )} r^{2} + 14 \, {\left (b e^{2} n + 14 \, a e^{2}\right )} r + {\left (2 \, b e^{2} r^{3} - 35 \, b e^{2} r^{2} + 196 \, b e^{2} r - 343 \, b e^{2}\right )} \log \relax (c) + {\left (2 \, b e^{2} n r^{3} - 35 \, b e^{2} n r^{2} + 196 \, b e^{2} n r - 343 \, b e^{2} n\right )} \log \relax (x)\right )} x^{2 \, r} - 98 \, {\left (4 \, a d e r^{3} - 49 \, b d e n - 343 \, a d e - 4 \, {\left (b d e n + 14 \, a d e\right )} r^{2} + 7 \, {\left (4 \, b d e n + 35 \, a d e\right )} r + {\left (4 \, b d e r^{3} - 56 \, b d e r^{2} + 245 \, b d e r - 343 \, b d e\right )} \log \relax (c) + {\left (4 \, b d e n r^{3} - 56 \, b d e n r^{2} + 245 \, b d e n r - 343 \, b d e n\right )} \log \relax (x)\right )} x^{r} + 7 \, {\left (4 \, b d^{2} r^{4} - 84 \, b d^{2} r^{3} + 637 \, b d^{2} r^{2} - 2058 \, b d^{2} r + 2401 \, b d^{2}\right )} \log \relax (c) + 7 \, {\left (4 \, b d^{2} n r^{4} - 84 \, b d^{2} n r^{3} + 637 \, b d^{2} n r^{2} - 2058 \, b d^{2} n r + 2401 \, b d^{2} n\right )} \log \relax (x)}{49 \, {\left (4 \, r^{4} - 84 \, r^{3} + 637 \, r^{2} - 2058 \, r + 2401\right )} x^{7}} \]

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*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}}\,{d x} \]

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

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maple [C]  time = 0.34, size = 1930, normalized size = 15.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 r-8 equal to -1?

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

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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