3.4.84 \(\int \frac {(d+e x^r)^2 (a+b \log (c x^n))}{x^5} \, dx\) [384]

Optimal. Leaf size=135 \[ -\frac {b d^2 n}{16 x^4}-\frac {b e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac {2 b d e n x^{-4+r}}{(4-r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {2 d e x^{-4+r} \left (a+b \log \left (c x^n\right )\right )}{4-r} \]

[Out]

-1/16*b*d^2*n/x^4-1/4*b*e^2*n/(2-r)^2/(x^(4-2*r))-2*b*d*e*n*x^(-4+r)/(4-r)^2-1/4*d^2*(a+b*ln(c*x^n))/x^4-1/2*e
^2*(a+b*ln(c*x^n))/(2-r)/(x^(4-2*r))-2*d*e*x^(-4+r)*(a+b*ln(c*x^n))/(4-r)

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Rubi [A]
time = 0.11, antiderivative size = 135, 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 )}{4 x^4}-\frac {2 d e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {b d^2 n}{16 x^4}-\frac {2 b d e n x^{r-4}}{(4-r)^2}-\frac {b e^2 n x^{-2 (2-r)}}{4 (2-r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^5,x]

[Out]

-1/16*(b*d^2*n)/x^4 - (b*e^2*n)/(4*(2 - r)^2*x^(2*(2 - r))) - (2*b*d*e*n*x^(-4 + r))/(4 - r)^2 - (d^2*(a + b*L
og[c*x^n]))/(4*x^4) - (e^2*(a + b*Log[c*x^n]))/(2*(2 - r)*x^(2*(2 - r))) - (2*d*e*x^(-4 + r)*(a + b*Log[c*x^n]
))/(4 - 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^5} \, dx &=-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {2 e^2 x^{-2 (2-r)}}{2-r}+\frac {8 d e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2 \left (8-6 r+r^2\right )+8 d e (-2+r) x^r+2 e^2 (-4+r) x^{2 r}}{4 (2-r) (4-r) x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {2 e^2 x^{-2 (2-r)}}{2-r}+\frac {8 d e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \frac {-d^2 \left (8-6 r+r^2\right )+8 d e (-2+r) x^r+2 e^2 (-4+r) x^{2 r}}{x^5} \, dx}{4 \left (8-6 r+r^2\right )}\\ &=-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {2 e^2 x^{-2 (2-r)}}{2-r}+\frac {8 d e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \left (-\frac {d^2 (-4+r) (-2+r)}{x^5}+8 d e (-2+r) x^{-5+r}+2 e^2 (-4+r) x^{-5+2 r}\right ) \, dx}{4 \left (8-6 r+r^2\right )}\\ &=-\frac {b d^2 n}{16 x^4}-\frac {b e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac {2 b d e n x^{-4+r}}{(4-r)^2}-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {2 e^2 x^{-2 (2-r)}}{2-r}+\frac {8 d e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 115, normalized size = 0.85 \begin {gather*} \frac {-4 b d^2 n \log (x)-d^2 \left (4 a+b n-4 b n \log (x)+4 b \log \left (c x^n\right )\right )+\frac {32 d e x^r \left (-b n+a (-4+r)+b (-4+r) \log \left (c x^n\right )\right )}{(-4+r)^2}+\frac {4 e^2 x^{2 r} \left (-b n+2 a (-2+r)+2 b (-2+r) \log \left (c x^n\right )\right )}{(-2+r)^2}}{16 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^5,x]

[Out]

(-4*b*d^2*n*Log[x] - d^2*(4*a + b*n - 4*b*n*Log[x] + 4*b*Log[c*x^n]) + (32*d*e*x^r*(-(b*n) + a*(-4 + r) + b*(-
4 + r)*Log[c*x^n]))/(-4 + r)^2 + (4*e^2*x^(2*r)*(-(b*n) + 2*a*(-2 + r) + 2*b*(-2 + r)*Log[c*x^n]))/(-2 + r)^2)
/(16*x^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.22, size = 1924, normalized size = 14.25

method	result	size
risch	\(\text {Expression too large to display}\)	\(1924\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)^2*(a+b*ln(c*x^n))/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*b*(-2*e^2*(x^r)^2*r+d^2*r^2-8*d*e*x^r*r+8*e^2*(x^r)^2-6*d^2*r+16*d*e*x^r+8*d^2)/x^4/(-2+r)/(-4+r)*ln(x^n)
-1/16*(256*e^2*(x^r)^2*a+512*d*e*x^r*a+128*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-320*I*Pi*b
*d*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r+52*b*d^2*n*r^2-96*b*d^2*n*r+208*ln(c)*b*d^2*r^2-384*ln(c)*b*d^2*r+4*ln(c)
*b*d^2*r^4-48*ln(c)*b*d^2*r^3+128*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+256*d^2*b*ln(c)+64*b*d^2*n+25
6*a*d^2+b*d^2*n*r^4-12*b*d^2*n*r^3+4*a*d^2*r^4-48*a*d^2*r^3-40*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2+2*I*Pi*b
*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+208*a*d^2*r^2-384*a*d^2*r-24*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1
92*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*c*x^n)^2+128*I*Pi*b*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-128*I*Pi*b*e^2*r*cs
gn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+256*ln(c)*b*e^2*(x^r)^2-128*I*Pi*b*d^2*csgn(I*c*x^n)^3-8*a*e^2*r^3*(x^r)^2+80*
a*e^2*r^2*(x^r)^2-256*a*e^2*r*(x^r)^2+64*b*e^2*n*(x^r)^2+256*a*d*e*r^2*x^r-640*a*d*e*r*x^r-32*b*e^2*n*r*(x^r)^
2+128*b*d*e*n*x^r+4*b*e^2*n*r^2*(x^r)^2-32*a*d*e*r^3*x^r+80*ln(c)*b*e^2*r^2*(x^r)^2-256*ln(c)*b*e^2*r*(x^r)^2-
8*ln(c)*b*e^2*r^3*(x^r)^2+512*ln(c)*b*d*e*x^r-256*I*Pi*b*d*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+128*I*Pi*
b*d*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r-128*I*Pi*b*d^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-4*I*Pi*b*e^2*r^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-192*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2+320*I*Pi*b*d*e*r*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)*x^r-40*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+128*I*Pi*b*d*e*r^2*csg
n(I*x^n)*csgn(I*c*x^n)^2*x^r+4*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-16*I*Pi*b*d*e*r^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2*x^r+24*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+104*I*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2+2
*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*c*x^n)^2+256*I*Pi*b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r-128*b*d*e*n*r*x^r+256*l
n(c)*b*d*e*r^2*x^r-128*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r+24*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)-256*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r-128*I*Pi*b*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+4*I*Pi*b*e^2*r
^3*csgn(I*c*x^n)^3*(x^r)^2+128*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-24*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*c*x^n)^
2+104*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-128*I*Pi*b*d*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-2*
I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+16*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r+256*I*Pi*b*d*e*csgn(I
*x^n)*csgn(I*c*x^n)^2*x^r+40*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-4*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(
I*c*x^n)^2*(x^r)^2+40*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-640*ln(c)*b*d*e*r*x^r-104*I*Pi*b*d^2*
r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+320*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r-128*I*Pi*b*e^2*r*csgn(I*x^n)*csgn
(I*c*x^n)^2*(x^r)^2-320*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-16*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*c*x^n)
^2*x^r-32*ln(c)*b*d*e*r^3*x^r+32*b*d*e*n*r^2*x^r+128*I*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2+128*I*Pi*b*d^2*csgn(
I*x^n)*csgn(I*c*x^n)^2+192*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+192*I*Pi*b*d^2*r*csgn(I*c*x^n)^3-1
28*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2-104*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3-2*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3+16*I
*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r)/(-2+r)^2/x^4/(-4+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^5,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-5>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. 413 vs. \(2 (119) = 238\).
time = 0.38, size = 413, normalized size = 3.06 \begin {gather*} -\frac {{\left (b d^{2} n + 4 \, a d^{2}\right )} r^{4} + 64 \, b d^{2} n - 12 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} r^{3} + 256 \, a d^{2} + 52 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} r^{2} - 96 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} r - 4 \, {\left (2 \, {\left (b r^{3} - 10 \, b r^{2} + 32 \, b r - 32 \, b\right )} e^{2} \log \left (c\right ) + 2 \, {\left (b n r^{3} - 10 \, b n r^{2} + 32 \, b n r - 32 \, b n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a r^{3} - {\left (b n + 20 \, a\right )} r^{2} - 16 \, b n + 8 \, {\left (b n + 8 \, a\right )} r - 64 \, a\right )} e^{2}\right )} x^{2 \, r} - 32 \, {\left ({\left (b d r^{3} - 8 \, b d r^{2} + 20 \, b d r - 16 \, b d\right )} e \log \left (c\right ) + {\left (b d n r^{3} - 8 \, b d n r^{2} + 20 \, b d n r - 16 \, b d n\right )} e \log \left (x\right ) + {\left (a d r^{3} - 4 \, b d n - {\left (b d n + 8 \, a d\right )} r^{2} - 16 \, a d + 4 \, {\left (b d n + 5 \, a d\right )} r\right )} e\right )} x^{r} + 4 \, {\left (b d^{2} r^{4} - 12 \, b d^{2} r^{3} + 52 \, b d^{2} r^{2} - 96 \, b d^{2} r + 64 \, b d^{2}\right )} \log \left (c\right ) + 4 \, {\left (b d^{2} n r^{4} - 12 \, b d^{2} n r^{3} + 52 \, b d^{2} n r^{2} - 96 \, b d^{2} n r + 64 \, b d^{2} n\right )} \log \left (x\right )}{16 \, {\left (r^{4} - 12 \, r^{3} + 52 \, r^{2} - 96 \, r + 64\right )} x^{4}} \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^5,x, algorithm="fricas")

[Out]

-1/16*((b*d^2*n + 4*a*d^2)*r^4 + 64*b*d^2*n - 12*(b*d^2*n + 4*a*d^2)*r^3 + 256*a*d^2 + 52*(b*d^2*n + 4*a*d^2)*
r^2 - 96*(b*d^2*n + 4*a*d^2)*r - 4*(2*(b*r^3 - 10*b*r^2 + 32*b*r - 32*b)*e^2*log(c) + 2*(b*n*r^3 - 10*b*n*r^2
+ 32*b*n*r - 32*b*n)*e^2*log(x) + (2*a*r^3 - (b*n + 20*a)*r^2 - 16*b*n + 8*(b*n + 8*a)*r - 64*a)*e^2)*x^(2*r)
- 32*((b*d*r^3 - 8*b*d*r^2 + 20*b*d*r - 16*b*d)*e*log(c) + (b*d*n*r^3 - 8*b*d*n*r^2 + 20*b*d*n*r - 16*b*d*n)*e
*log(x) + (a*d*r^3 - 4*b*d*n - (b*d*n + 8*a*d)*r^2 - 16*a*d + 4*(b*d*n + 5*a*d)*r)*e)*x^r + 4*(b*d^2*r^4 - 12*
b*d^2*r^3 + 52*b*d^2*r^2 - 96*b*d^2*r + 64*b*d^2)*log(c) + 4*(b*d^2*n*r^4 - 12*b*d^2*n*r^3 + 52*b*d^2*n*r^2 -
96*b*d^2*n*r + 64*b*d^2*n)*log(x))/((r^4 - 12*r^3 + 52*r^2 - 96*r + 64)*x^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2127 vs. \(2 (119) = 238\).
time = 8.59, size = 2127, normalized size = 15.76 \begin {gather*} \text {Too large to display} \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**5,x)

[Out]

Piecewise((-a*d**2/(4*x**4) - a*d*e/x**2 + a*e**2*log(x) + b*d**2*(-n/(16*x**4) - log(c*x**n)/(4*x**4)) + 2*b*
d*e*(-n/(4*x**2) - log(c*x**n)/(2*x**2)) - b*e**2*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n)
, True)), Eq(r, 2)), (-a*d**2/(4*x**4) + 2*a*d*e*log(c*x**n)/n + a*e**2*x**4/4 - b*d**2*n/(16*x**4) - b*d**2*l
og(c*x**n)/(4*x**4) + b*d*e*log(c*x**n)**2/n - b*e**2*n*x**4/16 + b*e**2*x**4*log(c*x**n)/4, Eq(r, 4)), (-4*a*
d**2*r**4/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 48*a*d**2*r**3/(16*r**4*x
**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 208*a*d**2*r**2/(16*r**4*x**4 - 192*r**3*x**4
+ 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 384*a*d**2*r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 153
6*r*x**4 + 1024*x**4) - 256*a*d**2/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) +
32*a*d*e*r**3*x**r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 256*a*d*e*r**2*x
**r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 640*a*d*e*r*x**r/(16*r**4*x**4
- 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 512*a*d*e*x**r/(16*r**4*x**4 - 192*r**3*x**4 + 83
2*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 8*a*e**2*r**3*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4
- 1536*r*x**4 + 1024*x**4) - 80*a*e**2*r**2*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x*
*4 + 1024*x**4) + 256*a*e**2*r*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**
4) - 256*a*e**2*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - b*d**2*n*r
**4/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 12*b*d**2*n*r**3/(16*r**4*x**4
- 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 52*b*d**2*n*r**2/(16*r**4*x**4 - 192*r**3*x**4 +
832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 96*b*d**2*n*r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*
r*x**4 + 1024*x**4) - 64*b*d**2*n/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 4
*b*d**2*r**4*log(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 48*b*d**2*
r**3*log(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 208*b*d**2*r**2*lo
g(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 384*b*d**2*r*log(c*x**n)/
(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 256*b*d**2*log(c*x**n)/(16*r**4*x**
4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 32*b*d*e*n*r**2*x**r/(16*r**4*x**4 - 192*r**3*x
**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 128*b*d*e*n*r*x**r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x
**4 - 1536*r*x**4 + 1024*x**4) - 128*b*d*e*n*x**r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4
+ 1024*x**4) + 32*b*d*e*r**3*x**r*log(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 10
24*x**4) - 256*b*d*e*r**2*x**r*log(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*
x**4) + 640*b*d*e*r*x**r*log(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4)
- 512*b*d*e*x**r*log(c*x**n)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 4*b*e*
*2*n*r**2*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 32*b*e**2*n*r*x*
*(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 64*b*e**2*n*x**(2*r)/(16*r**
4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 8*b*e**2*r**3*x**(2*r)*log(c*x**n)/(16*r**
4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 80*b*e**2*r**2*x**(2*r)*log(c*x**n)/(16*r*
*4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 256*b*e**2*r*x**(2*r)*log(c*x**n)/(16*r**
4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 256*b*e**2*x**(2*r)*log(c*x**n)/(16*r**4*x
**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4), True))

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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^5,x, algorithm="giac")

[Out]

integrate((x^r*e + d)^2*(b*log(c*x^n) + a)/x^5, 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^5} \,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^5,x)

[Out]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^5, x)

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