3.401 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=179 \[ -\frac{3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac{e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}-\frac{3 b d^2 e n x^{r-1}}{(1-r)^2}-\frac{b d^3 n}{x}-\frac{3 b d e^2 n x^{2 r-1}}{(1-2 r)^2}-\frac{b e^3 n x^{3 r-1}}{(1-3 r)^2} \]

[Out]

-((b*d^3*n)/x) - (3*b*d^2*e*n*x^(-1 + r))/(1 - r)^2 - (3*b*d*e^2*n*x^(-1 + 2*r))/(1 - 2*r)^2 - (b*e^3*n*x^(-1
+ 3*r))/(1 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/x - (3*d^2*e*x^(-1 + r)*(a + b*Log[c*x^n]))/(1 - r) - (3*d*e^2*
x^(-1 + 2*r)*(a + b*Log[c*x^n]))/(1 - 2*r) - (e^3*x^(-1 + 3*r)*(a + b*Log[c*x^n]))/(1 - 3*r)

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Rubi [A]  time = 0.397323, antiderivative size = 150, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {270, 2334, 14} \[ -\left (\frac{3 d^2 e x^{r-1}}{1-r}+\frac{d^3}{x}+\frac{3 d e^2 x^{2 r-1}}{1-2 r}+\frac{e^3 x^{3 r-1}}{1-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-1}}{(1-r)^2}-\frac{b d^3 n}{x}-\frac{3 b d e^2 n x^{2 r-1}}{(1-2 r)^2}-\frac{b e^3 n x^{3 r-1}}{(1-3 r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^2,x]

[Out]

-((b*d^3*n)/x) - (3*b*d^2*e*n*x^(-1 + r))/(1 - r)^2 - (3*b*d*e^2*n*x^(-1 + 2*r))/(1 - 2*r)^2 - (b*e^3*n*x^(-1
+ 3*r))/(1 - 3*r)^2 - (d^3/x + (3*d^2*e*x^(-1 + r))/(1 - r) + (3*d*e^2*x^(-1 + 2*r))/(1 - 2*r) + (e^3*x^(-1 +
3*r))/(1 - 3*r))*(a + b*Log[c*x^n])

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])

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]]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\left (\frac{d^3}{x}+\frac{3 d^2 e x^{-1+r}}{1-r}+\frac{3 d e^2 x^{-1+2 r}}{1-2 r}+\frac{e^3 x^{-1+3 r}}{1-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3+\frac{3 d^2 e x^r}{-1+r}+\frac{3 d e^2 x^{2 r}}{-1+2 r}+\frac{e^3 x^{3 r}}{-1+3 r}}{x^2} \, dx\\ &=-\left (\frac{d^3}{x}+\frac{3 d^2 e x^{-1+r}}{1-r}+\frac{3 d e^2 x^{-1+2 r}}{1-2 r}+\frac{e^3 x^{-1+3 r}}{1-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d^3}{x^2}+\frac{3 d^2 e x^{-2+r}}{-1+r}+\frac{3 d e^2 x^{2 (-1+r)}}{-1+2 r}+\frac{e^3 x^{-2+3 r}}{-1+3 r}\right ) \, dx\\ &=-\frac{b d^3 n}{x}-\frac{3 b d^2 e n x^{-1+r}}{(1-r)^2}-\frac{3 b d e^2 n x^{-1+2 r}}{(1-2 r)^2}-\frac{b e^3 n x^{-1+3 r}}{(1-3 r)^2}-\left (\frac{d^3}{x}+\frac{3 d^2 e x^{-1+r}}{1-r}+\frac{3 d e^2 x^{-1+2 r}}{1-2 r}+\frac{e^3 x^{-1+3 r}}{1-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.392607, size = 181, normalized size = 1.01 \[ \frac{a \left (\frac{3 d^2 e x^r}{r-1}-d^3+\frac{3 d e^2 x^{2 r}}{2 r-1}+\frac{e^3 x^{3 r}}{3 r-1}\right )+b \log \left (c x^n\right ) \left (\frac{3 d^2 e x^r}{r-1}-d^3+\frac{3 d e^2 x^{2 r}}{2 r-1}+\frac{e^3 x^{3 r}}{3 r-1}\right )+b n \left (-\frac{3 d^2 e x^r}{(r-1)^2}-d^3-\frac{3 d e^2 x^{2 r}}{(1-2 r)^2}-\frac{e^3 x^{3 r}}{(1-3 r)^2}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^2,x]

[Out]

(b*n*(-d^3 - (3*d^2*e*x^r)/(-1 + r)^2 - (3*d*e^2*x^(2*r))/(1 - 2*r)^2 - (e^3*x^(3*r))/(1 - 3*r)^2) + a*(-d^3 +
(3*d^2*e*x^r)/(-1 + r) + (3*d*e^2*x^(2*r))/(-1 + 2*r) + (e^3*x^(3*r))/(-1 + 3*r)) + b*(-d^3 + (3*d^2*e*x^r)/(
-1 + r) + (3*d*e^2*x^(2*r))/(-1 + 2*r) + (e^3*x^(3*r))/(-1 + 3*r))*Log[c*x^n])/x

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Maple [C]  time = 0.35, size = 4031, normalized size = 22.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^2,x)

[Out]

-b*(-2*e^3*r^2*(x^r)^3-9*d*e^2*r^2*(x^r)^2+3*e^3*r*(x^r)^3+6*d^3*r^3-18*d^2*e*r^2*x^r+12*d*e^2*r*(x^r)^2-e^3*(
x^r)^3-11*d^3*r^2+15*d^2*e*r*x^r-3*d*e^2*(x^r)^2+6*d^3*r-3*d^2*e*x^r-d^3)/x/(-1+3*r)/(-1+2*r)/(-1+r)*ln(x^n)-1
/2*(2*a*d^3+291*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+2*a*e^3*(x^r)^3+2*ln(c)*b*d^3-30*I*Pi
*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-30*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+72*b*d^3*n*
r^6-264*b*d^3*n*r^5+386*b*d^3*n*r^4+72*a*d^3*r^6-264*a*d^3*r^5+386*a*d^3*r^4+9*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x
^r)^3+51*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3-24*a*e^3*r^5*(x^r)^3+80*a*e^3*r^4*(x^r)^3+6*a*d*e^2*(x^r)^2+6*
a*d^2*e*x^r+2*b*e^3*n*(x^r)^3-102*a*e^3*r^3*(x^r)^3+62*a*e^3*r^2*(x^r)^3-18*a*e^3*r*(x^r)^3+2*ln(c)*b*e^3*(x^r
)^3-288*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2
*(x^r)^3+108*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3+I*Pi*b*d^3*csgn(I*
c*x^n)^2*csgn(I*c)+I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-288*b*d^3*n*r^3+116*b*d^3*n*r^2-24*b*d^3*n*r+72*ln(c
)*b*d^3*r^6-264*ln(c)*b*d^3*r^5+386*ln(c)*b*d^3*r^4-288*ln(c)*b*d^3*r^3+116*ln(c)*b*d^3*r^2-24*ln(c)*b*d^3*r-2
88*a*d^3*r^3+116*a*d^3*r^2-24*a*d^3*r-60*a*d*e^2*r*(x^r)^2-582*a*d^2*e*r^3*x^r+282*a*d^2*e*r^2*x^r-66*a*d^2*e*
r*x^r+51*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+2*b*d^3*n-I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)+80*ln(c)*b*e^3*r^4*(x^r)^3-102*ln(c)*b*e^3*r^3*(x^r)^3+62*ln(c)*b*e^3*r^2*(x^r)^3-18*ln(c)*b*e^
3*r*(x^r)^3+6*ln(c)*b*d^2*e*x^r+6*ln(c)*b*d*e^2*(x^r)^2+26*b*e^3*n*r^2*(x^r)^3-12*b*e^3*n*r*(x^r)^3+6*b*d*e^2*
n*(x^r)^2+6*b*d^2*e*n*x^r-408*a*d*e^2*r^3*(x^r)^2+228*a*d*e^2*r^2*(x^r)^2+8*b*e^3*n*r^4*(x^r)^3-24*b*e^3*n*r^3
*(x^r)^3-108*a*d*e^2*r^5*(x^r)^2+342*a*d*e^2*r^4*(x^r)^2-216*a*d^2*e*r^5*x^r+576*a*d^2*e*r^4*x^r-24*ln(c)*b*e^
3*r^5*(x^r)^3+132*b*d*e^2*n*r^2*(x^r)^2+222*b*d^2*e*n*r^2*x^r-193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)-I*Pi*b*d^3*csgn(I*c*x^n)^3-9*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+291*I*Pi*b*d^2*e*r^3*csgn(I
*c*x^n)^3*x^r-9*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-171*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)*(x^r)^2-114*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-141*I*Pi*b*d^2*e*r^2*csgn
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-3*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-31*I*Pi*b*e^3*r^2*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-36*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-114*I*Pi*b*d*
e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-141*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+144*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-48
*b*d*e^2*n*r*(x^r)^2-60*b*d^2*e*n*r*x^r+54*b*d*e^2*n*r^4*(x^r)^2-144*b*d*e^2*n*r^3*(x^r)^2+216*b*d^2*e*n*r^4*x
^r-360*b*d^2*e*n*r^3*x^r-108*ln(c)*b*d*e^2*r^5*(x^r)^2+342*ln(c)*b*d*e^2*r^4*(x^r)^2-216*ln(c)*b*d^2*e*r^5*x^r
+576*ln(c)*b*d^2*e*r^4*x^r-I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+3*I*Pi*b*d*e^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2*(x^r)^2+3*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+3*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^
n)^2*x^r-33*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-58*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)+I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+36*I*Pi*b*d^3*r
^6*csgn(I*x^n)*csgn(I*c*x^n)^2-31*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3-193*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3-58
2*ln(c)*b*d^2*e*r^3*x^r+282*ln(c)*b*d^2*e*r^2*x^r-66*ln(c)*b*d^2*e*r*x^r-408*ln(c)*b*d*e^2*r^3*(x^r)^2+228*ln(
c)*b*d*e^2*r^2*(x^r)^2-60*ln(c)*b*d*e^2*r*(x^r)^2+3*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+31*I*Pi*b*e^3*r
^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+31*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+33*I*Pi*b*d^2*e*r*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+204*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-204*I*
Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-108*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r-108*I*Pi
*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+288*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+288*I*Pi*b*d
^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-291*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-291*I*Pi*b*d^2*e*r
^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-132*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-132*I*Pi*b*d^3*r^5*csgn(I*c*x^
n)^2*csgn(I*c)-12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2
-40*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+171*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^
2*(x^r)^2+171*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+40*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2
*(x^r)^3+40*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-204*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x
^r)^2-12*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r-3*I*Pi*b*d*e^2*csgn(I*c*x^n
)^3*(x^r)^2+30*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+58*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I
*c*x^n)^2+58*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)+36*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-171*I*Pi*b*d
*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-288*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+30*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x
^r)^2+33*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+204*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+144*I*Pi*b*d^3*r^3*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-54*I*Pi*b*d*e^
2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+12*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+54*I*Pi*b*d*e^2*r^
5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+12*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-12*I*Pi*b*d^3*r*csgn(I*x^n)*csgn
(I*c*x^n)^2-144*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-144*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)+193*I*
Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+193*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-58*I*Pi*b*d^3*r^2*csgn(I
*c*x^n)^3-I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3-36*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+108*I*Pi*b*d^2*e*r^5*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*x^r+132*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3+141*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n
)^2*x^r+141*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-3*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*
(x^r)^2-51*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-51*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r
)^3-33*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+132*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+114
*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+114*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-4
0*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-54*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+9*I*Pi*b*e^3*
r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3)/(-1+3*r)^2/x/(-1+2*r)^2/(-1+r)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.40963, size = 2201, normalized size = 12.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,x, algorithm="fricas")

[Out]

-(36*(b*d^3*n + a*d^3)*r^6 - 132*(b*d^3*n + a*d^3)*r^5 + b*d^3*n + 193*(b*d^3*n + a*d^3)*r^4 + a*d^3 - 144*(b*
d^3*n + a*d^3)*r^3 + 58*(b*d^3*n + a*d^3)*r^2 - 12*(b*d^3*n + a*d^3)*r - (12*a*e^3*r^5 - b*e^3*n - 4*(b*e^3*n
+ 10*a*e^3)*r^4 - a*e^3 + 3*(4*b*e^3*n + 17*a*e^3)*r^3 - (13*b*e^3*n + 31*a*e^3)*r^2 + 3*(2*b*e^3*n + 3*a*e^3)
*r + (12*b*e^3*r^5 - 40*b*e^3*r^4 + 51*b*e^3*r^3 - 31*b*e^3*r^2 + 9*b*e^3*r - b*e^3)*log(c) + (12*b*e^3*n*r^5
- 40*b*e^3*n*r^4 + 51*b*e^3*n*r^3 - 31*b*e^3*n*r^2 + 9*b*e^3*n*r - b*e^3*n)*log(x))*x^(3*r) - 3*(18*a*d*e^2*r^
5 - b*d*e^2*n - 3*(3*b*d*e^2*n + 19*a*d*e^2)*r^4 - a*d*e^2 + 4*(6*b*d*e^2*n + 17*a*d*e^2)*r^3 - 2*(11*b*d*e^2*
n + 19*a*d*e^2)*r^2 + 2*(4*b*d*e^2*n + 5*a*d*e^2)*r + (18*b*d*e^2*r^5 - 57*b*d*e^2*r^4 + 68*b*d*e^2*r^3 - 38*b
*d*e^2*r^2 + 10*b*d*e^2*r - b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 57*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 - 38*b*d
*e^2*n*r^2 + 10*b*d*e^2*n*r - b*d*e^2*n)*log(x))*x^(2*r) - 3*(36*a*d^2*e*r^5 - b*d^2*e*n - 12*(3*b*d^2*e*n + 8
*a*d^2*e)*r^4 - a*d^2*e + (60*b*d^2*e*n + 97*a*d^2*e)*r^3 - (37*b*d^2*e*n + 47*a*d^2*e)*r^2 + (10*b*d^2*e*n +
11*a*d^2*e)*r + (36*b*d^2*e*r^5 - 96*b*d^2*e*r^4 + 97*b*d^2*e*r^3 - 47*b*d^2*e*r^2 + 11*b*d^2*e*r - b*d^2*e)*l
og(c) + (36*b*d^2*e*n*r^5 - 96*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 - 47*b*d^2*e*n*r^2 + 11*b*d^2*e*n*r - b*d^2*e*
n)*log(x))*x^r + (36*b*d^3*r^6 - 132*b*d^3*r^5 + 193*b*d^3*r^4 - 144*b*d^3*r^3 + 58*b*d^3*r^2 - 12*b*d^3*r + b
*d^3)*log(c) + (36*b*d^3*n*r^6 - 132*b*d^3*n*r^5 + 193*b*d^3*n*r^4 - 144*b*d^3*n*r^3 + 58*b*d^3*n*r^2 - 12*b*d
^3*n*r + b*d^3*n)*log(x))/((36*r^6 - 132*r^5 + 193*r^4 - 144*r^3 + 58*r^2 - 12*r + 1)*x)

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Sympy [A]  time = 94.6739, size = 314, normalized size = 1.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**2,x)

[Out]

-a*d**3/x + 3*a*d**2*e*Piecewise((x**r/(r*x - x), Ne(r, 1)), (log(x), True)) + 3*a*d*e**2*Piecewise((x**(2*r)/
(2*r*x - x), Ne(r, 1/2)), (log(x), True)) + a*e**3*Piecewise((x**(3*r)/(3*r*x - x), Ne(r, 1/3)), (log(x), True
)) - b*d**3*n/x - b*d**3*log(c*x**n)/x - 3*b*d**2*e*n*Piecewise((Piecewise((x**r/(r*x - x), Ne(r, 1)), (log(x)
, True))/(r - 1), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r - 1)/(r
- 1), Ne(r - 2, -1)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n*Piecewise((Piecewise((x**(2*r)/(2*r*x - x),
Ne(r, 1/2)), (log(x), True))/(2*r - 1), (r > -oo) & (r < oo) & Ne(r, 1/2)), (log(x)**2/2, True)) + 3*b*d*e**2*
Piecewise((x**(2*r - 1)/(2*r - 1), Ne(2*r - 2, -1)), (log(x), True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewi
se((x**(3*r)/(3*r*x - x), Ne(r, 1/3)), (log(x), True))/(3*r - 1), (r > -oo) & (r < oo) & Ne(r, 1/3)), (log(x)*
*2/2, True)) + b*e**3*Piecewise((x**(3*r - 1)/(3*r - 1), Ne(3*r - 2, -1)), (log(x), True))*log(c*x**n)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{r} + d\right )}^{3}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^2, x)