3.4.85 \(\int x^4 (d+e x^r)^2 (a+b \log (c x^n)) \, dx\) [385]

Optimal. Leaf size=105 \[ -\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{5+r}}{(5+r)^2}-\frac {b e^2 n x^{5+2 r}}{(5+2 r)^2}+\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/25*b*d^2*n*x^5-2*b*d*e*n*x^(5+r)/(5+r)^2-b*e^2*n*x^(5+2*r)/(5+2*r)^2+1/5*(d^2*x^5+10*d*e*x^(5+r)/(5+r)+5*e^
2*x^(5+2*r)/(5+2*r))*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.11, antiderivative size = 105, 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}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{r+5}}{(r+5)^2}-\frac {b e^2 n x^{2 r+5}}{(2 r+5)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

-1/25*(b*d^2*n*x^5) - (2*b*d*e*n*x^(5 + r))/(5 + r)^2 - (b*e^2*n*x^(5 + 2*r))/(5 + 2*r)^2 + ((d^2*x^5 + (10*d*
e*x^(5 + r))/(5 + r) + (5*e^2*x^(5 + 2*r))/(5 + 2*r))*(a + b*Log[c*x^n]))/5

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^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{5} x^4 \left (d^2+\frac {10 d e x^r}{5+r}+\frac {5 e^2 x^{2 r}}{5+2 r}\right ) \, dx\\ &=\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int x^4 \left (d^2+\frac {10 d e x^r}{5+r}+\frac {5 e^2 x^{2 r}}{5+2 r}\right ) \, dx\\ &=\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \left (d^2 x^4+\frac {5 e^2 x^{2 (2+r)}}{5+2 r}+\frac {10 d e x^{4+r}}{5+r}\right ) \, dx\\ &=-\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{5+r}}{(5+r)^2}-\frac {b e^2 n x^{5+2 r}}{(5+2 r)^2}+\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 119, normalized size = 1.13 \begin {gather*} \frac {1}{25} x^5 \left (5 b d^2 n \log (x)+d^2 \left (5 a-b n-5 b n \log (x)+5 b \log \left (c x^n\right )\right )+\frac {50 d e x^r \left (-b n+a (5+r)+b (5+r) \log \left (c x^n\right )\right )}{(5+r)^2}+\frac {25 e^2 x^{2 r} \left (-b n+a (5+2 r)+b (5+2 r) \log \left (c x^n\right )\right )}{(5+2 r)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^r)^2*(a + b*Log[c*x^n]),x]

[Out]

(x^5*(5*b*d^2*n*Log[x] + d^2*(5*a - b*n - 5*b*n*Log[x] + 5*b*Log[c*x^n]) + (50*d*e*x^r*(-(b*n) + a*(5 + r) + b
*(5 + r)*Log[c*x^n]))/(5 + r)^2 + (25*e^2*x^(2*r)*(-(b*n) + a*(5 + 2*r) + b*(5 + 2*r)*Log[c*x^n]))/(5 + 2*r)^2
))/25

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 1930, normalized size = 18.38

method	result	size
risch	\(\text {Expression too large to display}\)	\(1930\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d+e*x^r)^2*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/5*b*x^5*(5*e^2*(x^r)^2*r+2*d^2*r^2+20*d*e*x^r*r+25*e^2*(x^r)^2+15*d^2*r+50*d*e*x^r+25*d^2)/(5+2*r)/(5+r)*ln(
x^n)-1/50*x^5*(-6250*e^2*(x^r)^2*a-625*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-12500*d*e*x^r*a+3125
*I*Pi*b*d^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1625*I*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2+50*I*Pi*b*e^2*r^
3*csgn(I*c*x^n)^3*(x^r)^2+650*b*d^2*n*r^2+1500*b*d^2*n*r-3250*ln(c)*b*d^2*r^2-7500*ln(c)*b*d^2*r-40*ln(c)*b*d^
2*r^4-600*ln(c)*b*d^2*r^3-6250*d^2*b*ln(c)+1250*b*d^2*n-6250*a*d^2+8*b*d^2*n*r^4+120*b*d^2*n*r^3-40*a*d^2*r^4-
600*a*d^2*r^3+20*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3250*a*d^2*r^2-7500*a*d^2*r-2500*I*Pi*b*e^
2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6250*ln(c)*b*e^2*(x^r)^2-100*a*e^2*r^3*(x^r)^2-1250*a*e^2*r^2*(x^r)^2-
5000*a*e^2*r*(x^r)^2+1250*b*e^2*n*(x^r)^2-3125*I*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2-3125*I*Pi*b*d^2*csgn(I*x^n
)*csgn(I*c*x^n)^2-50*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-4000*a*d*e*r^2*x^r-12500*a*d*e*r*x^r+500
*b*e^2*n*r*(x^r)^2+2500*b*d*e*n*x^r+50*b*e^2*n*r^2*(x^r)^2-400*a*d*e*r^3*x^r-1250*ln(c)*b*e^2*r^2*(x^r)^2-5000
*ln(c)*b*e^2*r*(x^r)^2-100*ln(c)*b*e^2*r^3*(x^r)^2-12500*ln(c)*b*d*e*x^r+3125*I*Pi*b*d^2*csgn(I*c*x^n)^3+1625*
I*Pi*b*d^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2000*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r-200*I*Pi*b*d*e*r^3*
csgn(I*c)*csgn(I*c*x^n)^2*x^r+3750*I*Pi*b*d^2*r*csgn(I*c*x^n)^3+20*I*Pi*b*d^2*r^4*csgn(I*c*x^n)^3+200*I*Pi*b*d
*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+300*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3125*I*P
i*b*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+6250*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r-2000*I*Pi*b*d*e*r^2*
csgn(I*c)*csgn(I*c*x^n)^2*x^r-2000*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-1625*I*Pi*b*d^2*r^2*csgn(I*x
^n)*csgn(I*c*x^n)^2-3750*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*c*x^n)^2-3750*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-
200*I*Pi*b*d*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-3125*I*Pi*b*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-3125*I*Pi
*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+6250*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r-20*I*Pi*b*d^2*r^4*csgn(I*x^n)*c
sgn(I*c*x^n)^2-20*I*Pi*b*d^2*r^4*csgn(I*c)*csgn(I*c*x^n)^2-300*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2-300*I*
Pi*b*d^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2+200*I*Pi*b*d*e*r^3*csgn(I*c*x^n)^3*x^r+2000*b*d*e*n*r*x^r-4000*ln(c)*
b*d*e*r^2*x^r+625*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+50*I*Pi*b*e^2*r^3*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)*(x^r)^2+6250*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+2000*I*Pi*b*d*e*r^2*csg
n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-625*I*Pi*b*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-6250*I*Pi*b*d*e*csgn
(I*c)*csgn(I*c*x^n)^2*x^r-2500*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-50*I*Pi*b*e^2*r^3*csgn(I*x^n)*cs
gn(I*c*x^n)^2*(x^r)^2+300*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3+3125*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2+3750*I*Pi*b*d
^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-12500*ln(c)*b*d*e*r*x^r+625*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2+25
00*I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-6250*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-400*ln(c)*b*d*e*r^3*x^
r+400*b*d*e*n*r^2*x^r+1625*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3+2500*I*Pi*b*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)*(x^r)^2-6250*I*Pi*b*d*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-6250*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+62
50*I*Pi*b*d*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r)/(5+2*r)^2/(5+r)^2

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Maxima [A]
time = 0.29, size = 152, normalized size = 1.45 \begin {gather*} -\frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{2} x^{5} + \frac {b e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {2 \, b d e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {2 \, b d e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {2 \, a d e x^{r + 5}}{r + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/25*b*d^2*n*x^5 + 1/5*b*d^2*x^5*log(c*x^n) + 1/5*a*d^2*x^5 + b*e^2*x^(2*r + 5)*log(c*x^n)/(2*r + 5) + 2*b*d*
e*x^(r + 5)*log(c*x^n)/(r + 5) - b*e^2*n*x^(2*r + 5)/(2*r + 5)^2 + a*e^2*x^(2*r + 5)/(2*r + 5) - 2*b*d*e*n*x^(
r + 5)/(r + 5)^2 + 2*a*d*e*x^(r + 5)/(r + 5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (101) = 202\).
time = 0.37, size = 449, normalized size = 4.28 \begin {gather*} \frac {5 \, {\left (4 \, b d^{2} r^{4} + 60 \, b d^{2} r^{3} + 325 \, b d^{2} r^{2} + 750 \, b d^{2} r + 625 \, b d^{2}\right )} x^{5} \log \left (c\right ) + 5 \, {\left (4 \, b d^{2} n r^{4} + 60 \, b d^{2} n r^{3} + 325 \, b d^{2} n r^{2} + 750 \, b d^{2} n r + 625 \, b d^{2} n\right )} x^{5} \log \left (x\right ) - {\left (4 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{4} + 625 \, b d^{2} n + 60 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{3} - 3125 \, a d^{2} + 325 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{2} + 750 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r\right )} x^{5} + 25 \, {\left ({\left (2 \, b r^{3} + 25 \, b r^{2} + 100 \, b r + 125 \, b\right )} x^{5} e^{2} \log \left (c\right ) + {\left (2 \, b n r^{3} + 25 \, b n r^{2} + 100 \, b n r + 125 \, b n\right )} x^{5} e^{2} \log \left (x\right ) + {\left (2 \, a r^{3} - {\left (b n - 25 \, a\right )} r^{2} - 25 \, b n - 10 \, {\left (b n - 10 \, a\right )} r + 125 \, a\right )} x^{5} e^{2}\right )} x^{2 \, r} + 50 \, {\left ({\left (4 \, b d r^{3} + 40 \, b d r^{2} + 125 \, b d r + 125 \, b d\right )} x^{5} e \log \left (c\right ) + {\left (4 \, b d n r^{3} + 40 \, b d n r^{2} + 125 \, b d n r + 125 \, b d n\right )} x^{5} e \log \left (x\right ) + {\left (4 \, a d r^{3} - 25 \, b d n - 4 \, {\left (b d n - 10 \, a d\right )} r^{2} + 125 \, a d - 5 \, {\left (4 \, b d n - 25 \, a d\right )} r\right )} x^{5} e\right )} x^{r}}{25 \, {\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/25*(5*(4*b*d^2*r^4 + 60*b*d^2*r^3 + 325*b*d^2*r^2 + 750*b*d^2*r + 625*b*d^2)*x^5*log(c) + 5*(4*b*d^2*n*r^4 +
60*b*d^2*n*r^3 + 325*b*d^2*n*r^2 + 750*b*d^2*n*r + 625*b*d^2*n)*x^5*log(x) - (4*(b*d^2*n - 5*a*d^2)*r^4 + 625
*b*d^2*n + 60*(b*d^2*n - 5*a*d^2)*r^3 - 3125*a*d^2 + 325*(b*d^2*n - 5*a*d^2)*r^2 + 750*(b*d^2*n - 5*a*d^2)*r)*
x^5 + 25*((2*b*r^3 + 25*b*r^2 + 100*b*r + 125*b)*x^5*e^2*log(c) + (2*b*n*r^3 + 25*b*n*r^2 + 100*b*n*r + 125*b*
n)*x^5*e^2*log(x) + (2*a*r^3 - (b*n - 25*a)*r^2 - 25*b*n - 10*(b*n - 10*a)*r + 125*a)*x^5*e^2)*x^(2*r) + 50*((
4*b*d*r^3 + 40*b*d*r^2 + 125*b*d*r + 125*b*d)*x^5*e*log(c) + (4*b*d*n*r^3 + 40*b*d*n*r^2 + 125*b*d*n*r + 125*b
*d*n)*x^5*e*log(x) + (4*a*d*r^3 - 25*b*d*n - 4*(b*d*n - 10*a*d)*r^2 + 125*a*d - 5*(4*b*d*n - 25*a*d)*r)*x^5*e)
*x^r)/(4*r^4 + 60*r^3 + 325*r^2 + 750*r + 625)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d+e*x**r)**2*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (101) = 202\).
time = 3.09, size = 746, normalized size = 7.10 \begin {gather*} \frac {20 \, b d^{2} n r^{4} x^{5} \log \left (x\right ) + 200 \, b d n r^{3} x^{5} x^{r} e \log \left (x\right ) - 4 \, b d^{2} n r^{4} x^{5} + 20 \, b d^{2} r^{4} x^{5} \log \left (c\right ) + 200 \, b d r^{3} x^{5} x^{r} e \log \left (c\right ) + 300 \, b d^{2} n r^{3} x^{5} \log \left (x\right ) + 50 \, b n r^{3} x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 2000 \, b d n r^{2} x^{5} x^{r} e \log \left (x\right ) - 60 \, b d^{2} n r^{3} x^{5} + 20 \, a d^{2} r^{4} x^{5} - 200 \, b d n r^{2} x^{5} x^{r} e + 200 \, a d r^{3} x^{5} x^{r} e + 300 \, b d^{2} r^{3} x^{5} \log \left (c\right ) + 50 \, b r^{3} x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 2000 \, b d r^{2} x^{5} x^{r} e \log \left (c\right ) + 1625 \, b d^{2} n r^{2} x^{5} \log \left (x\right ) + 625 \, b n r^{2} x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 6250 \, b d n r x^{5} x^{r} e \log \left (x\right ) - 325 \, b d^{2} n r^{2} x^{5} + 300 \, a d^{2} r^{3} x^{5} - 25 \, b n r^{2} x^{5} x^{2 \, r} e^{2} + 50 \, a r^{3} x^{5} x^{2 \, r} e^{2} - 1000 \, b d n r x^{5} x^{r} e + 2000 \, a d r^{2} x^{5} x^{r} e + 1625 \, b d^{2} r^{2} x^{5} \log \left (c\right ) + 625 \, b r^{2} x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 6250 \, b d r x^{5} x^{r} e \log \left (c\right ) + 3750 \, b d^{2} n r x^{5} \log \left (x\right ) + 2500 \, b n r x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 6250 \, b d n x^{5} x^{r} e \log \left (x\right ) - 750 \, b d^{2} n r x^{5} + 1625 \, a d^{2} r^{2} x^{5} - 250 \, b n r x^{5} x^{2 \, r} e^{2} + 625 \, a r^{2} x^{5} x^{2 \, r} e^{2} - 1250 \, b d n x^{5} x^{r} e + 6250 \, a d r x^{5} x^{r} e + 3750 \, b d^{2} r x^{5} \log \left (c\right ) + 2500 \, b r x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 6250 \, b d x^{5} x^{r} e \log \left (c\right ) + 3125 \, b d^{2} n x^{5} \log \left (x\right ) + 3125 \, b n x^{5} x^{2 \, r} e^{2} \log \left (x\right ) - 625 \, b d^{2} n x^{5} + 3750 \, a d^{2} r x^{5} - 625 \, b n x^{5} x^{2 \, r} e^{2} + 2500 \, a r x^{5} x^{2 \, r} e^{2} + 6250 \, a d x^{5} x^{r} e + 3125 \, b d^{2} x^{5} \log \left (c\right ) + 3125 \, b x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 3125 \, a d^{2} x^{5} + 3125 \, a x^{5} x^{2 \, r} e^{2}}{25 \, {\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/25*(20*b*d^2*n*r^4*x^5*log(x) + 200*b*d*n*r^3*x^5*x^r*e*log(x) - 4*b*d^2*n*r^4*x^5 + 20*b*d^2*r^4*x^5*log(c)
+ 200*b*d*r^3*x^5*x^r*e*log(c) + 300*b*d^2*n*r^3*x^5*log(x) + 50*b*n*r^3*x^5*x^(2*r)*e^2*log(x) + 2000*b*d*n*
r^2*x^5*x^r*e*log(x) - 60*b*d^2*n*r^3*x^5 + 20*a*d^2*r^4*x^5 - 200*b*d*n*r^2*x^5*x^r*e + 200*a*d*r^3*x^5*x^r*e
+ 300*b*d^2*r^3*x^5*log(c) + 50*b*r^3*x^5*x^(2*r)*e^2*log(c) + 2000*b*d*r^2*x^5*x^r*e*log(c) + 1625*b*d^2*n*r
^2*x^5*log(x) + 625*b*n*r^2*x^5*x^(2*r)*e^2*log(x) + 6250*b*d*n*r*x^5*x^r*e*log(x) - 325*b*d^2*n*r^2*x^5 + 300
*a*d^2*r^3*x^5 - 25*b*n*r^2*x^5*x^(2*r)*e^2 + 50*a*r^3*x^5*x^(2*r)*e^2 - 1000*b*d*n*r*x^5*x^r*e + 2000*a*d*r^2
*x^5*x^r*e + 1625*b*d^2*r^2*x^5*log(c) + 625*b*r^2*x^5*x^(2*r)*e^2*log(c) + 6250*b*d*r*x^5*x^r*e*log(c) + 3750
*b*d^2*n*r*x^5*log(x) + 2500*b*n*r*x^5*x^(2*r)*e^2*log(x) + 6250*b*d*n*x^5*x^r*e*log(x) - 750*b*d^2*n*r*x^5 +
1625*a*d^2*r^2*x^5 - 250*b*n*r*x^5*x^(2*r)*e^2 + 625*a*r^2*x^5*x^(2*r)*e^2 - 1250*b*d*n*x^5*x^r*e + 6250*a*d*r
*x^5*x^r*e + 3750*b*d^2*r*x^5*log(c) + 2500*b*r*x^5*x^(2*r)*e^2*log(c) + 6250*b*d*x^5*x^r*e*log(c) + 3125*b*d^
2*n*x^5*log(x) + 3125*b*n*x^5*x^(2*r)*e^2*log(x) - 625*b*d^2*n*x^5 + 3750*a*d^2*r*x^5 - 625*b*n*x^5*x^(2*r)*e^
2 + 2500*a*r*x^5*x^(2*r)*e^2 + 6250*a*d*x^5*x^r*e + 3125*b*d^2*x^5*log(c) + 3125*b*x^5*x^(2*r)*e^2*log(c) + 31
25*a*d^2*x^5 + 3125*a*x^5*x^(2*r)*e^2)/(4*r^4 + 60*r^3 + 325*r^2 + 750*r + 625)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\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^4*(d + e*x^r)^2*(a + b*log(c*x^n)),x)

[Out]

int(x^4*(d + e*x^r)^2*(a + b*log(c*x^n)), x)

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