3.4.90 \(\int \frac {(d+e x^r)^2 (a+b \log (c x^n))}{x^6} \, dx\) [390]

Optimal. Leaf size=127 \[ -\frac {b d^2 n}{25 x^5}-\frac {2 b d e n x^{-5+r}}{(5-r)^2}-\frac {b e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {2 d e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r} \]

[Out]

-1/25*b*d^2*n/x^5-2*b*d*e*n*x^(-5+r)/(5-r)^2-b*e^2*n*x^(-5+2*r)/(5-2*r)^2-1/5*d^2*(a+b*ln(c*x^n))/x^5-2*d*e*x^
(-5+r)*(a+b*ln(c*x^n))/(5-r)-e^2*x^(-5+2*r)*(a+b*ln(c*x^n))/(5-2*r)

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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 )}{5 x^5}-\frac {2 d e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {e^2 x^{2 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n x^{r-5}}{(5-r)^2}-\frac {b e^2 n x^{2 r-5}}{(5-2 r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-1/25*(b*d^2*n)/x^5 - (2*b*d*e*n*x^(-5 + r))/(5 - r)^2 - (b*e^2*n*x^(-5 + 2*r))/(5 - 2*r)^2 - (d^2*(a + b*Log[
c*x^n]))/(5*x^5) - (2*d*e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r) - (e^2*x^(-5 + 2*r)*(a + b*Log[c*x^n]))/(5 -
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^6} \, dx &=-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2+\frac {10 d e x^r}{-5+r}+\frac {5 e^2 x^{2 r}}{-5+2 r}}{5 x^6} \, dx\\ &=-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \frac {-d^2+\frac {10 d e x^r}{-5+r}+\frac {5 e^2 x^{2 r}}{-5+2 r}}{x^6} \, dx\\ &=-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \left (-\frac {d^2}{x^6}+\frac {10 d e x^{-6+r}}{-5+r}+\frac {5 e^2 x^{2 (-3+r)}}{-5+2 r}\right ) \, dx\\ &=-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n x^{-5+r}}{(5-r)^2}-\frac {b e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 119, normalized size = 0.94 \begin {gather*} \frac {-5 b d^2 n \log (x)-d^2 \left (5 a+b n-5 b n \log (x)+5 b \log \left (c x^n\right )\right )+\frac {50 d e x^r \left (-b n+a (-5+r)+b (-5+r) \log \left (c x^n\right )\right )}{(-5+r)^2}+\frac {25 e^2 x^{2 r} \left (-b n+a (-5+2 r)+b (-5+2 r) \log \left (c x^n\right )\right )}{(5-2 r)^2}}{25 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^6,x]

[Out]

(-5*b*d^2*n*Log[x] - d^2*(5*a + b*n - 5*b*n*Log[x] + 5*b*Log[c*x^n]) + (50*d*e*x^r*(-(b*n) + a*(-5 + r) + b*(-
5 + r)*Log[c*x^n]))/(-5 + r)^2 + (25*e^2*x^(2*r)*(-(b*n) + a*(-5 + 2*r) + b*(-5 + 2*r)*Log[c*x^n]))/(5 - 2*r)^
2)/(25*x^5)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.23, 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^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*b*(-5*e^2*(x^r)^2*r+2*d^2*r^2-20*d*e*x^r*r+25*e^2*(x^r)^2-15*d^2*r+50*d*e*x^r+25*d^2)/x^5/(-5+2*r)/(-5+r)
*ln(x^n)-1/50*(6250*e^2*(x^r)^2*a+12500*d*e*x^r*a+650*b*d^2*n*r^2-1500*b*d^2*n*r+3250*ln(c)*b*d^2*r^2-7500*ln(
c)*b*d^2*r+40*ln(c)*b*d^2*r^4-600*ln(c)*b*d^2*r^3+6250*d^2*b*ln(c)+1250*b*d^2*n+6250*a*d^2+8*b*d^2*n*r^4-120*b
*d^2*n*r^3+40*a*d^2*r^4-600*a*d^2*r^3-3125*I*Pi*b*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-2500*I*Pi*b*
e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+3250*a*d^2*r^2-7500*a*d^2*r+20*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*c*x^n)^
2+6250*ln(c)*b*e^2*(x^r)^2-2500*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3750*I*Pi*b*d^2*r*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)+625*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-100*a*e^2*r^3*(x^r)^2+1250*a*e
^2*r^2*(x^r)^2-5000*a*e^2*r*(x^r)^2+1250*b*e^2*n*(x^r)^2+4000*a*d*e*r^2*x^r-12500*a*d*e*r*x^r-500*b*e^2*n*r*(x
^r)^2+2500*b*d*e*n*x^r+50*b*e^2*n*r^2*(x^r)^2-400*a*d*e*r^3*x^r+1250*ln(c)*b*e^2*r^2*(x^r)^2-5000*ln(c)*b*e^2*
r*(x^r)^2-100*ln(c)*b*e^2*r^3*(x^r)^2+12500*ln(c)*b*d*e*x^r-3125*I*Pi*b*d^2*csgn(I*c*x^n)^3-3125*I*Pi*b*e^2*cs
gn(I*c*x^n)^3*(x^r)^2-1625*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3-20*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3+2500*I*Pi*b*e^2*r*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-3125*I*Pi*b*d^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1625*I*Pi*b*d^
2*r^2*csgn(I*c)*csgn(I*c*x^n)^2+50*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-200*I*Pi*b*d*e*r
^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r-20*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1625*I*Pi*b*d^2*r^2*csg
n(I*x^n)*csgn(I*c*x^n)^2+200*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-6250*I*Pi*b*d*e*r*csgn(I*c
)*csgn(I*c*x^n)^2*x^r-6250*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-625*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)*(x^r)^2+2000*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6250*I*Pi*b*d*e*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)*x^r+6250*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+50*I*Pi*b*e^2*r^3*csgn(I*c*x^
n)^3*(x^r)^2+3750*I*Pi*b*d^2*r*csgn(I*c*x^n)^3+300*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+3125*I*Pi*b*e^2*csgn(I*c)*cs
gn(I*c*x^n)^2*(x^r)^2+3125*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6250*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r-
2000*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r-2000*b*d*e*n*r*x^r+4000*ln(c)*b*d*e*r^2*x^r-625*I*Pi*b*e^2*r^2*csgn(I*
c*x^n)^3*(x^r)^2+2500*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-300*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2+3125*I
*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2+3125*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+6250*I*Pi*b*d*e*r*csgn(I*c*x^n
)^3*x^r+300*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+20*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-2
00*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+200*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-50*I*Pi*b*e^2*r^3*csg
n(I*c)*csgn(I*c*x^n)^2*(x^r)^2-1625*I*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-12500*ln(c)*b*d*e*r*x^r
+6250*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+625*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-300*I*Pi
*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-3750*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*c*x^n)^2-3750*I*Pi*b*d^2*r*csgn(I*x^
n)*csgn(I*c*x^n)^2-50*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+2000*I*Pi*b*d*e*r^2*csgn(I*c)*csgn(I*
c*x^n)^2*x^r-400*ln(c)*b*d*e*r^3*x^r+400*b*d*e*n*r^2*x^r+6250*I*Pi*b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r-2000*I*
Pi*b*d*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r)/(-5+2*r)^2/x^5/(-5+r)^2

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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^6,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-6>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (118) = 236\).
time = 0.36, size = 422, normalized size = 3.32 \begin {gather*} -\frac {4 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r^{4} + 625 \, b d^{2} n - 60 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r^{3} + 3125 \, a d^{2} + 325 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r^{2} - 750 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r - 25 \, {\left ({\left (2 \, b r^{3} - 25 \, b r^{2} + 100 \, b r - 125 \, b\right )} e^{2} \log \left (c\right ) + {\left (2 \, b n r^{3} - 25 \, b n r^{2} + 100 \, b n r - 125 \, b n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a r^{3} - {\left (b n + 25 \, a\right )} r^{2} - 25 \, b n + 10 \, {\left (b n + 10 \, a\right )} r - 125 \, a\right )} e^{2}\right )} x^{2 \, r} - 50 \, {\left ({\left (4 \, b d r^{3} - 40 \, b d r^{2} + 125 \, b d r - 125 \, b d\right )} e \log \left (c\right ) + {\left (4 \, b d n r^{3} - 40 \, b d n r^{2} + 125 \, b d n r - 125 \, b d n\right )} e \log \left (x\right ) + {\left (4 \, a d r^{3} - 25 \, b d n - 4 \, {\left (b d n + 10 \, a d\right )} r^{2} - 125 \, a d + 5 \, {\left (4 \, b d n + 25 \, a d\right )} r\right )} e\right )} x^{r} + 5 \, {\left (4 \, b d^{2} r^{4} - 60 \, b d^{2} r^{3} + 325 \, b d^{2} r^{2} - 750 \, b d^{2} r + 625 \, b d^{2}\right )} \log \left (c\right ) + 5 \, {\left (4 \, b d^{2} n r^{4} - 60 \, b d^{2} n r^{3} + 325 \, b d^{2} n r^{2} - 750 \, b d^{2} n r + 625 \, b d^{2} n\right )} \log \left (x\right )}{25 \, {\left (4 \, r^{4} - 60 \, r^{3} + 325 \, r^{2} - 750 \, r + 625\right )} x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/25*(4*(b*d^2*n + 5*a*d^2)*r^4 + 625*b*d^2*n - 60*(b*d^2*n + 5*a*d^2)*r^3 + 3125*a*d^2 + 325*(b*d^2*n + 5*a*
d^2)*r^2 - 750*(b*d^2*n + 5*a*d^2)*r - 25*((2*b*r^3 - 25*b*r^2 + 100*b*r - 125*b)*e^2*log(c) + (2*b*n*r^3 - 25
*b*n*r^2 + 100*b*n*r - 125*b*n)*e^2*log(x) + (2*a*r^3 - (b*n + 25*a)*r^2 - 25*b*n + 10*(b*n + 10*a)*r - 125*a)
*e^2)*x^(2*r) - 50*((4*b*d*r^3 - 40*b*d*r^2 + 125*b*d*r - 125*b*d)*e*log(c) + (4*b*d*n*r^3 - 40*b*d*n*r^2 + 12
5*b*d*n*r - 125*b*d*n)*e*log(x) + (4*a*d*r^3 - 25*b*d*n - 4*(b*d*n + 10*a*d)*r^2 - 125*a*d + 5*(4*b*d*n + 25*a
*d)*r)*e)*x^r + 5*(4*b*d^2*r^4 - 60*b*d^2*r^3 + 325*b*d^2*r^2 - 750*b*d^2*r + 625*b*d^2)*log(c) + 5*(4*b*d^2*n
*r^4 - 60*b*d^2*n*r^3 + 325*b*d^2*n*r^2 - 750*b*d^2*n*r + 625*b*d^2*n)*log(x))/((4*r^4 - 60*r^3 + 325*r^2 - 75
0*r + 625)*x^5)

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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**6,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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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^6,x, algorithm="giac")

[Out]

integrate((x^r*e + d)^2*(b*log(c*x^n) + a)/x^6, x)

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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^6} \,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^6,x)

[Out]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^6, x)

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