3.380 \(\int x^3 (d+e x^r)^2 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=103 \[ \frac{1}{4} \left (d^2 x^4+\frac{8 d e x^{r+4}}{r+4}+\frac{2 e^2 x^{2 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{2 b d e n x^{r+4}}{(r+4)^2}-\frac{b e^2 n x^{2 (r+2)}}{4 (r+2)^2} \]

[Out]

-(b*d^2*n*x^4)/16 - (b*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (2*b*d*e*n*x^(4 + r))/(4 + r)^2 + ((d^2*x^4 + (2*e
^2*x^(2*(2 + r)))/(2 + r) + (8*d*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4

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Rubi [A]  time = 0.154336, antiderivative size = 103, normalized size of antiderivative = 1., 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} \[ \frac{1}{4} \left (d^2 x^4+\frac{8 d e x^{r+4}}{r+4}+\frac{2 e^2 x^{2 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{2 b d e n x^{r+4}}{(r+4)^2}-\frac{b e^2 n x^{2 (r+2)}}{4 (r+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x^4)/16 - (b*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (2*b*d*e*n*x^(4 + r))/(4 + r)^2 + ((d^2*x^4 + (2*e
^2*x^(2*(2 + r)))/(2 + r) + (8*d*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4

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^3 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{4} \left (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{1}{4} x^3 \left (d^2+\frac{8 d e x^r}{4+r}+\frac{2 e^2 x^{2 r}}{2+r}\right ) \, dx\\ &=\frac{1}{4} \left (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{1}{4} (b n) \int x^3 \left (d^2+\frac{8 d e x^r}{4+r}+\frac{2 e^2 x^{2 r}}{2+r}\right ) \, dx\\ &=\frac{1}{4} \left (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{1}{4} (b n) \int \left (d^2 x^3+\frac{8 d e x^{3+r}}{4+r}+\frac{2 e^2 x^{3+2 r}}{2+r}\right ) \, dx\\ &=-\frac{1}{16} b d^2 n 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 (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*}

Mathematica [A]  time = 0.240593, size = 118, normalized size = 1.15 \[ \frac{1}{16} x^4 \left (4 a \left (d^2+\frac{8 d e x^r}{r+4}+\frac{2 e^2 x^{2 r}}{r+2}\right )+4 b \log \left (c x^n\right ) \left (d^2+\frac{8 d e x^r}{r+4}+\frac{2 e^2 x^{2 r}}{r+2}\right )+b n \left (-d^2-\frac{32 d e x^r}{(r+4)^2}-\frac{4 e^2 x^{2 r}}{(r+2)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(b*n*(-d^2 - (32*d*e*x^r)/(4 + r)^2 - (4*e^2*x^(2*r))/(2 + r)^2) + 4*a*(d^2 + (8*d*e*x^r)/(4 + r) + (2*e^
2*x^(2*r))/(2 + r)) + 4*b*(d^2 + (8*d*e*x^r)/(4 + r) + (2*e^2*x^(2*r))/(2 + r))*Log[c*x^n]))/16

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Maple [C]  time = 0.293, size = 1924, normalized size = 18.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4*(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)/(2+r)/(4+r)*ln(x^n)-1/1
6*x^4*(b*d^2*n*r^4+12*b*d^2*n*r^3-256*ln(c)*b*d^2+128*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+5
2*b*d^2*n*r^2+96*b*d^2*n*r-128*I*Pi*b*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-8*a*e^2*r^3*(x^r)^2-80*a*e^2*r^2
*(x^r)^2-256*a*d^2-512*a*d*e*x^r+320*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+16*I*Pi*b*d*e*r^3*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+24*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+192*I*Pi*b*d^2*r*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+104*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-24*I*Pi*b*d^2*r^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2-24*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)-128*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^
n)^2*(x^r)^2+256*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r+40*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2+4*I*Pi*b*e^2*r^3*csg
n(I*c*x^n)^3*(x^r)^2+320*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r+128*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(
x^r)^2+2*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+128*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r-40*I*Pi*b*e
^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-40*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+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-256*I*Pi*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)
*x^r-256*a*e^2*r*(x^r)^2-208*a*d^2*r^2-384*a*d^2*r-4*a*d^2*r^4-48*a*d^2*r^3+128*I*Pi*b*d^2*csgn(I*c*x^n)^3+64*
b*d^2*n+128*b*d*e*n*x^r-8*ln(c)*b*e^2*r^3*(x^r)^2-512*ln(c)*b*d*e*x^r-80*ln(c)*b*e^2*r^2*(x^r)^2-256*ln(c)*b*e
^2*r*(x^r)^2-256*ln(c)*b*e^2*(x^r)^2+64*b*e^2*n*(x^r)^2-208*ln(c)*b*d^2*r^2-384*ln(c)*b*d^2*r-256*a*e^2*(x^r)^
2-4*ln(c)*b*d^2*r^4-48*ln(c)*b*d^2*r^3+4*b*e^2*n*r^2*(x^r)^2-32*a*d*e*r^3*x^r-256*a*d*e*r^2*x^r-640*a*d*e*r*x^
r+32*b*e^2*n*r*(x^r)^2-4*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-4*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^2*c
sgn(I*c)*(x^r)^2-320*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+4*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)*(x^r)^2-128*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+128*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-1
28*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-128*I*Pi*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+104*I*Pi*b*
d^2*r^2*csgn(I*c*x^n)^3+192*I*Pi*b*d^2*r*csgn(I*c*x^n)^3-256*ln(c)*b*d*e*r^2*x^r-640*ln(c)*b*d*e*r*x^r-32*ln(c
)*b*d*e*r^3*x^r-192*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-192*I*Pi*b*d^2*r*csgn(I*c*x^n)^2*csgn(I*c)-128*I*
Pi*b*d*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+256*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+128*b*d*e*n*
r*x^r+32*b*d*e*n*r^2*x^r+128*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2*I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(
I*c*x^n)^2-2*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)-104*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-104*I*Pi*
b*d^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)+128*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+40*I*Pi*b*e^2
*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-320*I*Pi*b*d*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r-16*I*Pi*b*d*e*
r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-16*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+128*I*Pi*b*e^2*csgn(I*c*x^
n)^3*(x^r)^2-128*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2-128*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)+2*I*Pi*b*d^2*
r^4*csgn(I*c*x^n)^3+24*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3)/(2+r)^2/(4+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^3*(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 = 1.3785, size = 1135, normalized size = 11.02 \begin{align*} \frac{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 )} x^{4} \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 )} x^{4} \log \left (x\right ) -{\left ({\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\right )} x^{4} + 4 \,{\left (2 \,{\left (b e^{2} r^{3} + 10 \, b e^{2} r^{2} + 32 \, b e^{2} r + 32 \, b e^{2}\right )} x^{4} \log \left (c\right ) + 2 \,{\left (b e^{2} n r^{3} + 10 \, b e^{2} n r^{2} + 32 \, b e^{2} n r + 32 \, b e^{2} n\right )} x^{4} \log \left (x\right ) +{\left (2 \, a e^{2} r^{3} - 16 \, b e^{2} n + 64 \, a e^{2} -{\left (b e^{2} n - 20 \, a e^{2}\right )} r^{2} - 8 \,{\left (b e^{2} n - 8 \, a e^{2}\right )} r\right )} x^{4}\right )} x^{2 \, r} + 32 \,{\left ({\left (b d e r^{3} + 8 \, b d e r^{2} + 20 \, b d e r + 16 \, b d e\right )} x^{4} \log \left (c\right ) +{\left (b d e n r^{3} + 8 \, b d e n r^{2} + 20 \, b d e n r + 16 \, b d e n\right )} x^{4} \log \left (x\right ) +{\left (a d e r^{3} - 4 \, b d e n + 16 \, a d e -{\left (b d e n - 8 \, a d e\right )} r^{2} - 4 \,{\left (b d e n - 5 \, a d e\right )} r\right )} x^{4}\right )} x^{r}}{16 \,{\left (r^{4} + 12 \, r^{3} + 52 \, r^{2} + 96 \, r + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 138.113, size = 2162, normalized size = 20.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**4/4 + 2*a*d*e*log(x) - a*e**2/(4*x**4) + b*d**2*n*x**4*log(x)/4 - b*d**2*n*x**4/16 + b*d*
*2*x**4*log(c)/4 + b*d*e*n*log(x)**2 + 2*b*d*e*log(c)*log(x) - b*e**2*n*log(x)/(4*x**4) - b*e**2*n/(16*x**4) -
b*e**2*log(c)/(4*x**4), Eq(r, -4)), (a*d**2*x**4/4 + a*d*e*x**2 + a*e**2*log(x) + b*d**2*n*x**4*log(x)/4 - b*
d**2*n*x**4/16 + b*d**2*x**4*log(c)/4 + b*d*e*n*x**2*log(x) - b*d*e*n*x**2/2 + b*d*e*x**2*log(c) + b*e**2*n*lo
g(x)**2/2 + b*e**2*log(c)*log(x), Eq(r, -2)), (4*a*d**2*r**4*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 10
24) + 48*a*d**2*r**3*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 208*a*d**2*r**2*x**4/(16*r**4 + 19
2*r**3 + 832*r**2 + 1536*r + 1024) + 384*a*d**2*r*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*a
*d**2*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 32*a*d*e*r**3*x**4*x**r/(16*r**4 + 192*r**3 + 832
*r**2 + 1536*r + 1024) + 256*a*d*e*r**2*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 640*a*d*e*
r*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 512*a*d*e*x**4*x**r/(16*r**4 + 192*r**3 + 832*r*
*2 + 1536*r + 1024) + 8*a*e**2*r**3*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 80*a*e**2*
r**2*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*a*e**2*r*x**4*x**(2*r)/(16*r**4 + 192
*r**3 + 832*r**2 + 1536*r + 1024) + 256*a*e**2*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) +
4*b*d**2*n*r**4*x**4*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - b*d**2*n*r**4*x**4/(16*r**4 + 1
92*r**3 + 832*r**2 + 1536*r + 1024) + 48*b*d**2*n*r**3*x**4*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1
024) - 12*b*d**2*n*r**3*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 208*b*d**2*n*r**2*x**4*log(x)/(
16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 52*b*d**2*n*r**2*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r
+ 1024) + 384*b*d**2*n*r*x**4*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 96*b*d**2*n*r*x**4/(16
*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*d**2*n*x**4*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*
r + 1024) - 64*b*d**2*n*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 4*b*d**2*r**4*x**4*log(c)/(16*r
**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 48*b*d**2*r**3*x**4*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*
r + 1024) + 208*b*d**2*r**2*x**4*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 384*b*d**2*r*x**4*lo
g(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*d**2*x**4*log(c)/(16*r**4 + 192*r**3 + 832*r**2 +
1536*r + 1024) + 32*b*d*e*n*r**3*x**4*x**r*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*d*e
*n*r**2*x**4*x**r*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 32*b*d*e*n*r**2*x**4*x**r/(16*r**4
+ 192*r**3 + 832*r**2 + 1536*r + 1024) + 640*b*d*e*n*r*x**4*x**r*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*
r + 1024) - 128*b*d*e*n*r*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 512*b*d*e*n*x**4*x**r*lo
g(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 128*b*d*e*n*x**4*x**r/(16*r**4 + 192*r**3 + 832*r**2 +
1536*r + 1024) + 32*b*d*e*r**3*x**4*x**r*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*d*e*r*
*2*x**4*x**r*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 640*b*d*e*r*x**4*x**r*log(c)/(16*r**4 +
192*r**3 + 832*r**2 + 1536*r + 1024) + 512*b*d*e*x**4*x**r*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 10
24) + 8*b*e**2*n*r**3*x**4*x**(2*r)*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 80*b*e**2*n*r**2*
x**4*x**(2*r)*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 4*b*e**2*n*r**2*x**4*x**(2*r)/(16*r**4
+ 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*n*r*x**4*x**(2*r)*log(x)/(16*r**4 + 192*r**3 + 832*r**2 +
1536*r + 1024) - 32*b*e**2*n*r*x**4*x**(2*r)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*n*x*
*4*x**(2*r)*log(x)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 64*b*e**2*n*x**4*x**(2*r)/(16*r**4 + 192*
r**3 + 832*r**2 + 1536*r + 1024) + 8*b*e**2*r**3*x**4*x**(2*r)*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r
+ 1024) + 80*b*e**2*r**2*x**4*x**(2*r)*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*r*x
**4*x**(2*r)*log(c)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*x**4*x**(2*r)*log(c)/(16*r**4
+ 192*r**3 + 832*r**2 + 1536*r + 1024), True))

________________________________________________________________________________________

Giac [B]  time = 1.31919, size = 1004, normalized size = 9.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*b*d^2*n*r^4*x^4*log(x) + 32*b*d*n*r^3*x^4*x^r*e*log(x) - b*d^2*n*r^4*x^4 + 4*b*d^2*r^4*x^4*log(c) + 32
*b*d*r^3*x^4*x^r*e*log(c) + 48*b*d^2*n*r^3*x^4*log(x) + 8*b*n*r^3*x^4*x^(2*r)*e^2*log(x) + 256*b*d*n*r^2*x^4*x
^r*e*log(x) - 12*b*d^2*n*r^3*x^4 + 4*a*d^2*r^4*x^4 - 32*b*d*n*r^2*x^4*x^r*e + 32*a*d*r^3*x^4*x^r*e + 48*b*d^2*
r^3*x^4*log(c) + 8*b*r^3*x^4*x^(2*r)*e^2*log(c) + 256*b*d*r^2*x^4*x^r*e*log(c) + 208*b*d^2*n*r^2*x^4*log(x) +
80*b*n*r^2*x^4*x^(2*r)*e^2*log(x) + 640*b*d*n*r*x^4*x^r*e*log(x) - 52*b*d^2*n*r^2*x^4 + 48*a*d^2*r^3*x^4 - 4*b
*n*r^2*x^4*x^(2*r)*e^2 + 8*a*r^3*x^4*x^(2*r)*e^2 - 128*b*d*n*r*x^4*x^r*e + 256*a*d*r^2*x^4*x^r*e + 208*b*d^2*r
^2*x^4*log(c) + 80*b*r^2*x^4*x^(2*r)*e^2*log(c) + 640*b*d*r*x^4*x^r*e*log(c) + 384*b*d^2*n*r*x^4*log(x) + 256*
b*n*r*x^4*x^(2*r)*e^2*log(x) + 512*b*d*n*x^4*x^r*e*log(x) - 96*b*d^2*n*r*x^4 + 208*a*d^2*r^2*x^4 - 32*b*n*r*x^
4*x^(2*r)*e^2 + 80*a*r^2*x^4*x^(2*r)*e^2 - 128*b*d*n*x^4*x^r*e + 640*a*d*r*x^4*x^r*e + 384*b*d^2*r*x^4*log(c)
+ 256*b*r*x^4*x^(2*r)*e^2*log(c) + 512*b*d*x^4*x^r*e*log(c) + 256*b*d^2*n*x^4*log(x) + 256*b*n*x^4*x^(2*r)*e^2
*log(x) - 64*b*d^2*n*x^4 + 384*a*d^2*r*x^4 - 64*b*n*x^4*x^(2*r)*e^2 + 256*a*r*x^4*x^(2*r)*e^2 + 512*a*d*x^4*x^
r*e + 256*b*d^2*x^4*log(c) + 256*b*x^4*x^(2*r)*e^2*log(c) + 256*a*d^2*x^4 + 256*a*x^4*x^(2*r)*e^2)/(r^4 + 12*r
^3 + 52*r^2 + 96*r + 64)