3.4.80 \(\int x^3 (d+e x^r)^2 (a+b \log (c x^n)) \, dx\) [380]

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

[Out]

-1/16*b*d^2*n*x^4-1/4*b*e^2*n*x^(4+2*r)/(2+r)^2-2*b*d*e*n*x^(4+r)/(4+r)^2+1/4*(d^2*x^4+2*e^2*x^(4+2*r)/(2+r)+8
*d*e*x^(4+r)/(4+r))*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.11, antiderivative size = 103, 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, 2371, 12, 14} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(b*d^2*n*x^4) - (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 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 2371

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] && IGtQ[m, 0]

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*}

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Mathematica [A]
time = 0.17, size = 115, normalized size = 1.12 \begin {gather*} \frac {1}{16} x^4 \left (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 {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}+\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}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.20, size = 1924, normalized size = 18.68

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

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,method=_RETURNVERBOSE)

[Out]

1/4*x^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)/(2+r)/(4+r)*ln(x^n)-1/1
6*x^4*(-256*e^2*(x^r)^2*a+128*I*Pi*b*d^2*csgn(I*c*x^n)^3+128*I*Pi*b*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x
^r)^2-512*d*e*x^r*a+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+128*I*Pi*b*e^
2*r*csgn(I*c*x^n)^3*(x^r)^2+320*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*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*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+2*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3-256*d^2*b*ln(c)+64*b*d^2*n-256*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^3-192*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-208*a*d^2*r^2-384*a*d
^2*r-256*ln(c)*b*e^2*(x^r)^2-16*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r-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+4*I*Pi*b*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+104*I*Pi*b*d^2*r^2*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-24*I*Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-104*
I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-192*I*Pi*b*d^2*r*csgn(I*c)*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-16*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-128*I*Pi*b*e^2*r*csgn(I*x
^n)*csgn(I*c*x^n)^2*(x^r)^2-128*I*Pi*b*d*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r-128*I*Pi*b*d*e*r^2*csgn(I*x^n)*cs
gn(I*c*x^n)^2*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+4*I*Pi*b*e^2*r^3*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)*(x^r)^2+128*b*d*e*n*r*x^r-256*ln(c)*b*d*e*r^2*x^r-4*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*c*x
^n)^2*(x^r)^2+16*I*Pi*b*d*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+128*I*Pi*b*d^2*csgn(I*c)*csgn(I*x^n)*c
sgn(I*c*x^n)-104*I*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2-128*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-4
*I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+24*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+320*
I*Pi*b*d*e*r*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)-40*I
*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^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*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-640*ln(c)*b*d*e*r*x^r-2*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*c*x^n)^2-2*I*P
i*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-24*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2+192*I*Pi*b*d^2*r*csgn(I*c*
x^n)^3-32*ln(c)*b*d*e*r^3*x^r+32*b*d*e*n*r^2*x^r+104*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3+24*I*Pi*b*d^2*r^3*csgn(I*c
*x^n)^3+128*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2+128*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-25
6*I*Pi*b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r+192*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+128*I*Pi*b*d*e
*r^2*csgn(I*c*x^n)^3*x^r-320*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-320*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*
x^n)^2*x^r+256*I*Pi*b*d*e*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*c*x^n)^2*
(x^r)^2-256*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r)/(2+r)^2/(4+r)^2

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Maxima [A]
time = 0.28, size = 148, normalized size = 1.44 \begin {gather*} -\frac {1}{16} \, b d^{2} n x^{4} + \frac {1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{2} x^{4} + \frac {b e^{2} x^{2 \, r + 4} \log \left (c x^{n}\right )}{2 \, {\left (r + 2\right )}} + \frac {2 \, b d e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e^{2} n x^{2 \, r + 4}}{4 \, {\left (r + 2\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 4}}{2 \, {\left (r + 2\right )}} - \frac {2 \, b d e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {2 \, a d e x^{r + 4}}{r + 4} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (97) = 194\).
time = 0.38, size = 440, normalized size = 4.27 \begin {gather*} \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 r^{3} + 10 \, b r^{2} + 32 \, b r + 32 \, b\right )} x^{4} 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 )} x^{4} 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 )} x^{4} 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 )} x^{4} 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 )} x^{4} 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 )} x^{4} e\right )} x^{r}}{16 \, {\left (r^{4} + 12 \, r^{3} + 52 \, r^{2} + 96 \, r + 64\right )}} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1625 vs. \(2 (97) = 194\).
time = 10.10, size = 1625, normalized size = 15.78 \begin {gather*} \begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {2 a d e \log {\left (c x^{n} \right )}}{n} - \frac {a e^{2}}{4 x^{4}} - \frac {b d^{2} n x^{4}}{16} + \frac {b d^{2} x^{4} \log {\left (c x^{n} \right )}}{4} + \frac {b d e \log {\left (c x^{n} \right )}^{2}}{n} - \frac {b e^{2} n}{16 x^{4}} - \frac {b e^{2} \log {\left (c x^{n} \right )}}{4 x^{4}} & \text {for}\: r = -4 \\\frac {a d^{2} x^{4}}{4} + a d e x^{2} + \frac {a e^{2} \log {\left (c x^{n} \right )}}{n} - \frac {b d^{2} n x^{4}}{16} + \frac {b d^{2} x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {b d e n x^{2}}{2} + b d e x^{2} \log {\left (c x^{n} \right )} + \frac {b e^{2} \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {for}\: r = -2 \\\frac {4 a d^{2} r^{4} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {48 a d^{2} r^{3} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {208 a d^{2} r^{2} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {384 a d^{2} r x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 a d^{2} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {32 a d e r^{3} x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 a d e r^{2} x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {640 a d e r x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {512 a d e x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {8 a e^{2} r^{3} x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {80 a e^{2} r^{2} x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 a e^{2} r x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 a e^{2} x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {b d^{2} n r^{4} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {12 b d^{2} n r^{3} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {52 b d^{2} n r^{2} x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {96 b d^{2} n r x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {64 b d^{2} n x^{4}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {4 b d^{2} r^{4} x^{4} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {48 b d^{2} r^{3} x^{4} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {208 b d^{2} r^{2} x^{4} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {384 b d^{2} r x^{4} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 b d^{2} x^{4} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {32 b d e n r^{2} x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {128 b d e n r x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {128 b d e n x^{4} x^{r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {32 b d e r^{3} x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 b d e r^{2} x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {640 b d e r x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {512 b d e x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {4 b e^{2} n r^{2} x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {32 b e^{2} n r x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} - \frac {64 b e^{2} n x^{4} x^{2 r}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {8 b e^{2} r^{3} x^{4} x^{2 r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {80 b e^{2} r^{2} x^{4} x^{2 r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 b e^{2} r x^{4} x^{2 r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} + \frac {256 b e^{2} x^{4} x^{2 r} \log {\left (c x^{n} \right )}}{16 r^{4} + 192 r^{3} + 832 r^{2} + 1536 r + 1024} & \text {otherwise} \end {cases} \end {gather*}

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(c*x**n)/n - a*e**2/(4*x**4) - b*d**2*n*x**4/16 + b*d**2*x**4*log(c*x**n
)/4 + b*d*e*log(c*x**n)**2/n - b*e**2*n/(16*x**4) - b*e**2*log(c*x**n)/(4*x**4), Eq(r, -4)), (a*d**2*x**4/4 +
a*d*e*x**2 + a*e**2*log(c*x**n)/n - b*d**2*n*x**4/16 + b*d**2*x**4*log(c*x**n)/4 - b*d*e*n*x**2/2 + b*d*e*x**2
*log(c*x**n) + b*e**2*log(c*x**n)**2/(2*n), Eq(r, -2)), (4*a*d**2*r**4*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1
536*r + 1024) + 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 + 192*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 + 102
4) + 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) - b*d**2*n*r**4*x**4/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) - 12*b*d**2*n*r**3*x**4/(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
) - 96*b*d**2*n*r*x**4/(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*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) +
48*b*d**2*r**3*x**4*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 208*b*d**2*r**2*x**4*log(c*
x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 384*b*d**2*r*x**4*log(c*x**n)/(16*r**4 + 192*r**3 + 83
2*r**2 + 1536*r + 1024) + 256*b*d**2*x**4*log(c*x**n)/(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) - 128*b*d*e*n*r*x**4*x**r/(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*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*d*e*r**2*x**4*x**r*log
(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 640*b*d*e*r*x**4*x**r*log(c*x**n)/(16*r**4 + 192*r*
*3 + 832*r**2 + 1536*r + 1024) + 512*b*d*e*x**4*x**r*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 102
4) - 4*b*e**2*n*r**2*x**4*x**(2*r)/(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) - 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*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 8
0*b*e**2*r**2*x**4*x**(2*r)*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*r*x**4*x*
*(2*r)*log(c*x**n)/(16*r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024) + 256*b*e**2*x**4*x**(2*r)*log(c*x**n)/(16*
r**4 + 192*r**3 + 832*r**2 + 1536*r + 1024), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (97) = 194\).
time = 2.50, size = 744, normalized size = 7.22 \begin {gather*} \frac {4 \, b d^{2} n r^{4} x^{4} \log \left (x\right ) + 32 \, b d n r^{3} x^{4} x^{r} e \log \left (x\right ) - b d^{2} n r^{4} x^{4} + 4 \, b d^{2} r^{4} x^{4} \log \left (c\right ) + 32 \, b d r^{3} x^{4} x^{r} e \log \left (c\right ) + 48 \, b d^{2} n r^{3} x^{4} \log \left (x\right ) + 8 \, b n r^{3} x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 256 \, b d n r^{2} x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 8 \, b r^{3} x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 256 \, b d r^{2} x^{4} x^{r} e \log \left (c\right ) + 208 \, b d^{2} n r^{2} x^{4} \log \left (x\right ) + 80 \, b n r^{2} x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 640 \, b d n r x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 80 \, b r^{2} x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 640 \, b d r x^{4} x^{r} e \log \left (c\right ) + 384 \, b d^{2} n r x^{4} \log \left (x\right ) + 256 \, b n r x^{4} x^{2 \, r} e^{2} \log \left (x\right ) + 512 \, b d n x^{4} x^{r} e \log \left (x\right ) - 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 \left (c\right ) + 256 \, b r x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 512 \, b d x^{4} x^{r} e \log \left (c\right ) + 256 \, b d^{2} n x^{4} \log \left (x\right ) + 256 \, b n x^{4} x^{2 \, r} e^{2} \log \left (x\right ) - 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 \left (c\right ) + 256 \, b x^{4} x^{2 \, r} e^{2} \log \left (c\right ) + 256 \, a d^{2} x^{4} + 256 \, a x^{4} x^{2 \, r} e^{2}}{16 \, {\left (r^{4} + 12 \, r^{3} + 52 \, r^{2} + 96 \, r + 64\right )}} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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