3.381 \(\int x (d+e x^r)^2 (a+b \log (c x^n)) \, dx\)
Optimal. Leaf size=102 \[ \frac{1}{2} \left (d^2 x^2+\frac{4 d e x^{r+2}}{r+2}+\frac{e^2 x^{2 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d^2 n x^2-\frac{2 b d e n x^{r+2}}{(r+2)^2}-\frac{b e^2 n x^{2 (r+1)}}{4 (r+1)^2} \]
[Out]
-(b*d^2*n*x^2)/4 - (b*e^2*n*x^(2*(1 + r)))/(4*(1 + r)^2) - (2*b*d*e*n*x^(2 + r))/(2 + r)^2 + ((d^2*x^2 + (e^2*
x^(2*(1 + r)))/(1 + r) + (4*d*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2
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Rubi [A] time = 0.131601, antiderivative size = 102, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used =
{270, 2334, 12, 14} \[ \frac{1}{2} \left (d^2 x^2+\frac{4 d e x^{r+2}}{r+2}+\frac{e^2 x^{2 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d^2 n x^2-\frac{2 b d e n x^{r+2}}{(r+2)^2}-\frac{b e^2 n x^{2 (r+1)}}{4 (r+1)^2} \]
Antiderivative was successfully verified.
[In]
Int[x*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]
[Out]
-(b*d^2*n*x^2)/4 - (b*e^2*n*x^(2*(1 + r)))/(4*(1 + r)^2) - (2*b*d*e*n*x^(2 + r))/(2 + r)^2 + ((d^2*x^2 + (e^2*
x^(2*(1 + r)))/(1 + r) + (4*d*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2
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 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]]
Rubi steps
\begin{align*} \int x \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{2} \left (d^2 x^2+\frac{e^2 x^{2 (1+r)}}{1+r}+\frac{4 d e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{2} x \left (d^2+\frac{4 d e x^r}{2+r}+\frac{e^2 x^{2 r}}{1+r}\right ) \, dx\\ &=\frac{1}{2} \left (d^2 x^2+\frac{e^2 x^{2 (1+r)}}{1+r}+\frac{4 d e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int x \left (d^2+\frac{4 d e x^r}{2+r}+\frac{e^2 x^{2 r}}{1+r}\right ) \, dx\\ &=\frac{1}{2} \left (d^2 x^2+\frac{e^2 x^{2 (1+r)}}{1+r}+\frac{4 d e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \left (d^2 x+\frac{4 d e x^{1+r}}{2+r}+\frac{e^2 x^{1+2 r}}{1+r}\right ) \, dx\\ &=-\frac{1}{4} b d^2 n x^2-\frac{b e^2 n x^{2 (1+r)}}{4 (1+r)^2}-\frac{2 b d e n x^{2+r}}{(2+r)^2}+\frac{1}{2} \left (d^2 x^2+\frac{e^2 x^{2 (1+r)}}{1+r}+\frac{4 d e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.235334, size = 116, normalized size = 1.14 \[ \frac{1}{4} x^2 \left (2 a \left (d^2+\frac{4 d e x^r}{r+2}+\frac{e^2 x^{2 r}}{r+1}\right )+2 b \log \left (c x^n\right ) \left (d^2+\frac{4 d e x^r}{r+2}+\frac{e^2 x^{2 r}}{r+1}\right )+b n \left (-d^2-\frac{8 d e x^r}{(r+2)^2}-\frac{e^2 x^{2 r}}{(r+1)^2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]
[Out]
(x^2*(b*n*(-d^2 - (8*d*e*x^r)/(2 + r)^2 - (e^2*x^(2*r))/(1 + r)^2) + 2*a*(d^2 + (4*d*e*x^r)/(2 + r) + (e^2*x^(
2*r))/(1 + r)) + 2*b*(d^2 + (4*d*e*x^r)/(2 + r) + (e^2*x^(2*r))/(1 + r))*Log[c*x^n]))/4
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Maple [C] time = 0.296, size = 1922, normalized size = 18.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(d+e*x^r)^2*(a+b*ln(c*x^n)),x)
[Out]
1/2*b*x^2*(e^2*(x^r)^2*r+d^2*r^2+4*d*e*x^r*r+2*e^2*(x^r)^2+3*d^2*r+4*d*e*x^r+2*d^2)/(1+r)/(2+r)*ln(x^n)-1/4*x^
2*(b*d^2*n*r^4+6*b*d^2*n*r^3-8*ln(c)*b*d^2+20*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+13*b*d^2*n*
r^2+12*b*d^2*n*r+13*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2*a*e^2*r^3*(x^r)^2-10*a*e^2*r^2*(x^r)^
2-8*a*d^2+4*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-16*a*d*e*x^r+16*I*Pi*b*d*e*r^2*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)*x^r+4*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r-8*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-
8*I*Pi*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+6*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*I*Pi*b*d^2*
r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+4*I*Pi*b*d^2*csgn(I*c*x^n)^3-I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*
(x^r)^2-I*Pi*b*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-8*I*Pi*b*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+16*I
*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r+I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-8*I*Pi*b*e^2*r*csgn(I*x^n
)*csgn(I*c*x^n)^2*(x^r)^2-16*a*e^2*r*(x^r)^2-26*a*d^2*r^2-24*a*d^2*r-2*a*d^2*r^4-12*a*d^2*r^3+4*b*d^2*n+8*b*d*
e*n*x^r-2*ln(c)*b*e^2*r^3*(x^r)^2-16*ln(c)*b*d*e*x^r-10*ln(c)*b*e^2*r^2*(x^r)^2-16*ln(c)*b*e^2*r*(x^r)^2-8*ln(
c)*b*e^2*(x^r)^2+4*b*e^2*n*(x^r)^2-26*ln(c)*b*d^2*r^2-24*ln(c)*b*d^2*r-8*a*e^2*(x^r)^2-2*ln(c)*b*d^2*r^4-12*ln
(c)*b*d^2*r^3+b*e^2*n*r^2*(x^r)^2-8*a*d*e*r^3*x^r-32*a*d*e*r^2*x^r-40*a*d*e*r*x^r+4*b*e^2*n*r*(x^r)^2-5*I*Pi*b
*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-5*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+20*I*Pi*b*d*e*
r*csgn(I*c*x^n)^3*x^r+4*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-16*I*Pi*b*d*e*r^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2*x^r-16*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)*(x^r)^2+5*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-12*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(
I*c*x^n)^2-12*I*Pi*b*d^2*r*csgn(I*c*x^n)^2*csgn(I*c)-13*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*b*d^2*
r^4*csgn(I*c*x^n)^3-32*ln(c)*b*d*e*r^2*x^r-40*ln(c)*b*d*e*r*x^r-8*ln(c)*b*d*e*r^3*x^r-4*I*Pi*b*e^2*csgn(I*x^n)
*csgn(I*c*x^n)^2*(x^r)^2-4*I*Pi*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-13*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^2*csgn
(I*c)+4*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d^2*r
^4*csgn(I*c*x^n)^2*csgn(I*c)-20*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-20*I*Pi*b*d*e*r*csgn(I*c*x^n)^2*c
sgn(I*c)*x^r+16*b*d*e*n*r*x^r+8*b*d*e*n*r^2*x^r+8*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r+5*I*Pi*b*e^2*r^2*csgn(I*c*x^n
)^3*(x^r)^2+I*Pi*b*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-6*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I*Pi*b*d^2*r
^3*csgn(I*c*x^n)^2*csgn(I*c)+8*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-4*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)
^2*x^r-4*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+8*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+8*I
*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+6*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+4*I*Pi*b*e^2*csgn(I*c
*x^n)^3*(x^r)^2+13*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3+12*I*Pi*b*d^2*r*csgn(I*c*x^n)^3-4*I*Pi*b*d^2*csgn(I*x^n)*csg
n(I*c*x^n)^2-4*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c))/(1+r)^2/(2+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(x*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.41032, size = 1096, normalized size = 10.75 \begin{align*} \frac{2 \,{\left (b d^{2} r^{4} + 6 \, b d^{2} r^{3} + 13 \, b d^{2} r^{2} + 12 \, b d^{2} r + 4 \, b d^{2}\right )} x^{2} \log \left (c\right ) + 2 \,{\left (b d^{2} n r^{4} + 6 \, b d^{2} n r^{3} + 13 \, b d^{2} n r^{2} + 12 \, b d^{2} n r + 4 \, b d^{2} n\right )} x^{2} \log \left (x\right ) -{\left ({\left (b d^{2} n - 2 \, a d^{2}\right )} r^{4} + 4 \, b d^{2} n + 6 \,{\left (b d^{2} n - 2 \, a d^{2}\right )} r^{3} - 8 \, a d^{2} + 13 \,{\left (b d^{2} n - 2 \, a d^{2}\right )} r^{2} + 12 \,{\left (b d^{2} n - 2 \, a d^{2}\right )} r\right )} x^{2} +{\left (2 \,{\left (b e^{2} r^{3} + 5 \, b e^{2} r^{2} + 8 \, b e^{2} r + 4 \, b e^{2}\right )} x^{2} \log \left (c\right ) + 2 \,{\left (b e^{2} n r^{3} + 5 \, b e^{2} n r^{2} + 8 \, b e^{2} n r + 4 \, b e^{2} n\right )} x^{2} \log \left (x\right ) +{\left (2 \, a e^{2} r^{3} - 4 \, b e^{2} n + 8 \, a e^{2} -{\left (b e^{2} n - 10 \, a e^{2}\right )} r^{2} - 4 \,{\left (b e^{2} n - 4 \, a e^{2}\right )} r\right )} x^{2}\right )} x^{2 \, r} + 8 \,{\left ({\left (b d e r^{3} + 4 \, b d e r^{2} + 5 \, b d e r + 2 \, b d e\right )} x^{2} \log \left (c\right ) +{\left (b d e n r^{3} + 4 \, b d e n r^{2} + 5 \, b d e n r + 2 \, b d e n\right )} x^{2} \log \left (x\right ) +{\left (a d e r^{3} - b d e n + 2 \, a d e -{\left (b d e n - 4 \, a d e\right )} r^{2} -{\left (2 \, b d e n - 5 \, a d e\right )} r\right )} x^{2}\right )} x^{r}}{4 \,{\left (r^{4} + 6 \, r^{3} + 13 \, r^{2} + 12 \, r + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="fricas")
[Out]
1/4*(2*(b*d^2*r^4 + 6*b*d^2*r^3 + 13*b*d^2*r^2 + 12*b*d^2*r + 4*b*d^2)*x^2*log(c) + 2*(b*d^2*n*r^4 + 6*b*d^2*n
*r^3 + 13*b*d^2*n*r^2 + 12*b*d^2*n*r + 4*b*d^2*n)*x^2*log(x) - ((b*d^2*n - 2*a*d^2)*r^4 + 4*b*d^2*n + 6*(b*d^2
*n - 2*a*d^2)*r^3 - 8*a*d^2 + 13*(b*d^2*n - 2*a*d^2)*r^2 + 12*(b*d^2*n - 2*a*d^2)*r)*x^2 + (2*(b*e^2*r^3 + 5*b
*e^2*r^2 + 8*b*e^2*r + 4*b*e^2)*x^2*log(c) + 2*(b*e^2*n*r^3 + 5*b*e^2*n*r^2 + 8*b*e^2*n*r + 4*b*e^2*n)*x^2*log
(x) + (2*a*e^2*r^3 - 4*b*e^2*n + 8*a*e^2 - (b*e^2*n - 10*a*e^2)*r^2 - 4*(b*e^2*n - 4*a*e^2)*r)*x^2)*x^(2*r) +
8*((b*d*e*r^3 + 4*b*d*e*r^2 + 5*b*d*e*r + 2*b*d*e)*x^2*log(c) + (b*d*e*n*r^3 + 4*b*d*e*n*r^2 + 5*b*d*e*n*r + 2
*b*d*e*n)*x^2*log(x) + (a*d*e*r^3 - b*d*e*n + 2*a*d*e - (b*d*e*n - 4*a*d*e)*r^2 - (2*b*d*e*n - 5*a*d*e)*r)*x^2
)*x^r)/(r^4 + 6*r^3 + 13*r^2 + 12*r + 4)
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Sympy [A] time = 21.608, size = 2159, normalized size = 21.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(d+e*x**r)**2*(a+b*ln(c*x**n)),x)
[Out]
Piecewise((a*d**2*x**2/2 + 2*a*d*e*log(x) - a*e**2/(2*x**2) + b*d**2*n*x**2*log(x)/2 - b*d**2*n*x**2/4 + b*d**
2*x**2*log(c)/2 + b*d*e*n*log(x)**2 + 2*b*d*e*log(c)*log(x) - b*e**2*n*log(x)/(2*x**2) - b*e**2*n/(4*x**2) - b
*e**2*log(c)/(2*x**2), Eq(r, -2)), (a*d**2*x**2/2 + 2*a*d*e*x + a*e**2*log(x) + b*d**2*n*x**2*log(x)/2 - b*d**
2*n*x**2/4 + b*d**2*x**2*log(c)/2 + 2*b*d*e*n*x*log(x) - 2*b*d*e*n*x + 2*b*d*e*x*log(c) + b*e**2*n*log(x)**2/2
+ b*e**2*log(c)*log(x), Eq(r, -1)), (2*a*d**2*r**4*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 12*a*d**2*
r**3*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 26*a*d**2*r**2*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r +
16) + 24*a*d**2*r*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*a*d**2*x**2/(4*r**4 + 24*r**3 + 52*r**2 +
48*r + 16) + 8*a*d*e*r**3*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 32*a*d*e*r**2*x**2*x**r/(4*r**4
+ 24*r**3 + 52*r**2 + 48*r + 16) + 40*a*d*e*r*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 16*a*d*e*x
**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 2*a*e**2*r**3*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 +
48*r + 16) + 10*a*e**2*r**2*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 16*a*e**2*r*x**2*x**(2*r)
/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*a*e**2*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) +
2*b*d**2*n*r**4*x**2*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - b*d**2*n*r**4*x**2/(4*r**4 + 24*r**3 +
52*r**2 + 48*r + 16) + 12*b*d**2*n*r**3*x**2*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 6*b*d**2*n*r**3
*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 26*b*d**2*n*r**2*x**2*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48
*r + 16) - 13*b*d**2*n*r**2*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 24*b*d**2*n*r*x**2*log(x)/(4*r**4
+ 24*r**3 + 52*r**2 + 48*r + 16) - 12*b*d**2*n*r*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*b*d**2*n*x*
*2*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 4*b*d**2*n*x**2/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16)
+ 2*b*d**2*r**4*x**2*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 12*b*d**2*r**3*x**2*log(c)/(4*r**4 + 24
*r**3 + 52*r**2 + 48*r + 16) + 26*b*d**2*r**2*x**2*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 24*b*d**2
*r*x**2*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*b*d**2*x**2*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 4
8*r + 16) + 8*b*d*e*n*r**3*x**2*x**r*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 32*b*d*e*n*r**2*x**2*x*
*r*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 8*b*d*e*n*r**2*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48
*r + 16) + 40*b*d*e*n*r*x**2*x**r*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - 16*b*d*e*n*r*x**2*x**r/(4*
r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 16*b*d*e*n*x**2*x**r*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) -
8*b*d*e*n*x**2*x**r/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*b*d*e*r**3*x**2*x**r*log(c)/(4*r**4 + 24*r**
3 + 52*r**2 + 48*r + 16) + 32*b*d*e*r**2*x**2*x**r*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 40*b*d*e*
r*x**2*x**r*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 16*b*d*e*x**2*x**r*log(c)/(4*r**4 + 24*r**3 + 52
*r**2 + 48*r + 16) + 2*b*e**2*n*r**3*x**2*x**(2*r)*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 10*b*e**2
*n*r**2*x**2*x**(2*r)*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) - b*e**2*n*r**2*x**2*x**(2*r)/(4*r**4 +
24*r**3 + 52*r**2 + 48*r + 16) + 16*b*e**2*n*r*x**2*x**(2*r)*log(x)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) -
4*b*e**2*n*r*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 8*b*e**2*n*x**2*x**(2*r)*log(x)/(4*r**4
+ 24*r**3 + 52*r**2 + 48*r + 16) - 4*b*e**2*n*x**2*x**(2*r)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 2*b*e*
*2*r**3*x**2*x**(2*r)*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 10*b*e**2*r**2*x**2*x**(2*r)*log(c)/(4
*r**4 + 24*r**3 + 52*r**2 + 48*r + 16) + 16*b*e**2*r*x**2*x**(2*r)*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r +
16) + 8*b*e**2*x**2*x**(2*r)*log(c)/(4*r**4 + 24*r**3 + 52*r**2 + 48*r + 16), True))
________________________________________________________________________________________
Giac [B] time = 1.32063, size = 1004, normalized size = 9.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="giac")
[Out]
1/4*(2*b*d^2*n*r^4*x^2*log(x) + 8*b*d*n*r^3*x^2*x^r*e*log(x) - b*d^2*n*r^4*x^2 + 2*b*d^2*r^4*x^2*log(c) + 8*b*
d*r^3*x^2*x^r*e*log(c) + 12*b*d^2*n*r^3*x^2*log(x) + 2*b*n*r^3*x^2*x^(2*r)*e^2*log(x) + 32*b*d*n*r^2*x^2*x^r*e
*log(x) - 6*b*d^2*n*r^3*x^2 + 2*a*d^2*r^4*x^2 - 8*b*d*n*r^2*x^2*x^r*e + 8*a*d*r^3*x^2*x^r*e + 12*b*d^2*r^3*x^2
*log(c) + 2*b*r^3*x^2*x^(2*r)*e^2*log(c) + 32*b*d*r^2*x^2*x^r*e*log(c) + 26*b*d^2*n*r^2*x^2*log(x) + 10*b*n*r^
2*x^2*x^(2*r)*e^2*log(x) + 40*b*d*n*r*x^2*x^r*e*log(x) - 13*b*d^2*n*r^2*x^2 + 12*a*d^2*r^3*x^2 - b*n*r^2*x^2*x
^(2*r)*e^2 + 2*a*r^3*x^2*x^(2*r)*e^2 - 16*b*d*n*r*x^2*x^r*e + 32*a*d*r^2*x^2*x^r*e + 26*b*d^2*r^2*x^2*log(c) +
10*b*r^2*x^2*x^(2*r)*e^2*log(c) + 40*b*d*r*x^2*x^r*e*log(c) + 24*b*d^2*n*r*x^2*log(x) + 16*b*n*r*x^2*x^(2*r)*
e^2*log(x) + 16*b*d*n*x^2*x^r*e*log(x) - 12*b*d^2*n*r*x^2 + 26*a*d^2*r^2*x^2 - 4*b*n*r*x^2*x^(2*r)*e^2 + 10*a*
r^2*x^2*x^(2*r)*e^2 - 8*b*d*n*x^2*x^r*e + 40*a*d*r*x^2*x^r*e + 24*b*d^2*r*x^2*log(c) + 16*b*r*x^2*x^(2*r)*e^2*
log(c) + 16*b*d*x^2*x^r*e*log(c) + 8*b*d^2*n*x^2*log(x) + 8*b*n*x^2*x^(2*r)*e^2*log(x) - 4*b*d^2*n*x^2 + 24*a*
d^2*r*x^2 - 4*b*n*x^2*x^(2*r)*e^2 + 16*a*r*x^2*x^(2*r)*e^2 + 16*a*d*x^2*x^r*e + 8*b*d^2*x^2*log(c) + 8*b*x^2*x
^(2*r)*e^2*log(c) + 8*a*d^2*x^2 + 8*a*x^2*x^(2*r)*e^2)/(r^4 + 6*r^3 + 13*r^2 + 12*r + 4)