3.4.98 \(\int x^4 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [398]

Optimal. Leaf size=151 \[ -\frac {1}{25} b d^3 n x^5-\frac {3 b d^2 e n x^{5+r}}{(5+r)^2}-\frac {3 b d e^2 n x^{5+2 r}}{(5+2 r)^2}-\frac {b e^3 n x^{5+3 r}}{(5+3 r)^2}+\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/25*b*d^3*n*x^5-3*b*d^2*e*n*x^(5+r)/(5+r)^2-3*b*d*e^2*n*x^(5+2*r)/(5+2*r)^2-b*e^3*n*x^(5+3*r)/(5+3*r)^2+1/5*
(d^3*x^5+15*d^2*e*x^(5+r)/(5+r)+15*d*e^2*x^(5+2*r)/(5+2*r)+5*e^3*x^(5+3*r)/(5+3*r))*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.26, antiderivative size = 151, 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^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^3 n x^5-\frac {3 b d^2 e n x^{r+5}}{(r+5)^2}-\frac {3 b d e^2 n x^{2 r+5}}{(2 r+5)^2}-\frac {b e^3 n x^{3 r+5}}{(3 r+5)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

-1/25*(b*d^3*n*x^5) - (3*b*d^2*e*n*x^(5 + r))/(5 + r)^2 - (3*b*d*e^2*n*x^(5 + 2*r))/(5 + 2*r)^2 - (b*e^3*n*x^(
5 + 3*r))/(5 + 3*r)^2 + ((d^3*x^5 + (15*d^2*e*x^(5 + r))/(5 + r) + (15*d*e^2*x^(5 + 2*r))/(5 + 2*r) + (5*e^3*x
^(5 + 3*r))/(5 + 3*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 )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{5} x^4 \left (d^3+\frac {15 d^2 e x^r}{5+r}+\frac {15 d e^2 x^{2 r}}{5+2 r}+\frac {5 e^3 x^{3 r}}{5+3 r}\right ) \, dx\\ &=\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int x^4 \left (d^3+\frac {15 d^2 e x^r}{5+r}+\frac {15 d e^2 x^{2 r}}{5+2 r}+\frac {5 e^3 x^{3 r}}{5+3 r}\right ) \, dx\\ &=\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \left (d^3 x^4+\frac {15 d e^2 x^{2 (2+r)}}{5+2 r}+\frac {15 d^2 e x^{4+r}}{5+r}+\frac {5 e^3 x^{4+3 r}}{5+3 r}\right ) \, dx\\ &=-\frac {1}{25} b d^3 n x^5-\frac {3 b d^2 e n x^{5+r}}{(5+r)^2}-\frac {3 b d e^2 n x^{5+2 r}}{(5+2 r)^2}-\frac {b e^3 n x^{5+3 r}}{(5+3 r)^2}+\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 164, normalized size = 1.09 \begin {gather*} \frac {1}{25} x^5 \left (5 b d^3 n \log (x)+d^3 \left (5 a-b n-5 b n \log (x)+5 b \log \left (c x^n\right )\right )+\frac {75 d^2 e x^r \left (-b n+a (5+r)+b (5+r) \log \left (c x^n\right )\right )}{(5+r)^2}+\frac {75 d 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}+\frac {25 e^3 x^{3 r} \left (-b n+a (5+3 r)+b (5+3 r) \log \left (c x^n\right )\right )}{(5+3 r)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

(x^5*(5*b*d^3*n*Log[x] + d^3*(5*a - b*n - 5*b*n*Log[x] + 5*b*Log[c*x^n]) + (75*d^2*e*x^r*(-(b*n) + a*(5 + r) +
b*(5 + r)*Log[c*x^n]))/(5 + r)^2 + (75*d*e^2*x^(2*r)*(-(b*n) + a*(5 + 2*r) + b*(5 + 2*r)*Log[c*x^n]))/(5 + 2*
r)^2 + (25*e^3*x^(3*r)*(-(b*n) + a*(5 + 3*r) + b*(5 + 3*r)*Log[c*x^n]))/(5 + 3*r)^2))/25

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.29, size = 4031, normalized size = 26.70

method	result	size
risch	\(\text {Expression too large to display}\)	\(4031\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d+e*x^r)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/5*b*x^5*(10*e^3*r^2*(x^r)^3+45*d*e^2*r^2*(x^r)^2+75*e^3*r*(x^r)^3+6*d^3*r^3+90*d^2*e*r^2*x^r+300*d*e^2*r*(x^
r)^2+125*e^3*(x^r)^3+55*d^3*r^2+375*d^2*e*r*x^r+375*d*e^2*(x^r)^2+150*d^3*r+375*d^2*e*x^r+125*d^3)/(5+3*r)/(5+
2*r)/(5+r)*ln(x^n)-1/50*x^5*(-156250*e^3*(x^r)^3*a-5000*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+181
250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+78125*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3-468750*d^2*e*x^r*a-468750*d*e^2*(x
^r)^2*a+78125*I*Pi*b*e^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+45000*b*d^2*e*n*r^3*x^r-712500*ln(c)*b*d*
e^2*r^2*(x^r)^2-937500*ln(c)*b*d*e^2*r*(x^r)^2-360*a*d^3*r^6-6600*a*d^3*r^5-48250*a*d^3*r^4-468750*I*Pi*b*d*e^
2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+187500*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+90000*I*Pi*b*d
^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-156250*a*d^3-255000*a*d*e^2*r^3*(x^r)^2-712500*a*d*e^2*r^2*(x^r)^2-
937500*a*d*e^2*r*(x^r)^2-363750*a*d^2*e*r^3*x^r-881250*a*d^2*e*r^2*x^r-1031250*a*d^2*e*r*x^r-31875*I*Pi*b*e^3*
r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3-31875*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+515625*I*Pi*b*d
^2*e*r*csgn(I*c*x^n)^3*x^r+72*b*d^3*n*r^6+1320*b*d^3*n*r^5+9650*b*d^3*n*r^4-2700*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*
csgn(I*c*x^n)^2*x^r+440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+21375*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2
+36000*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+138750*b*d^2*e*n*r^2*x^r+150000*b*d*e^2*n*r*(x
^r)^2+187500*b*d^2*e*n*r*x^r+1350*b*d*e^2*n*r^4*(x^r)^2+18000*b*d*e^2*n*r^3*(x^r)^2+5400*b*d^2*e*n*r^4*x^r+500
0*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-180000*a*d^3*r^3-362500*a*d^3*r^2-375000*a*d^3*r+234375*I*Pi*b*d^2*e*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-360*ln(c)*b*d^3*r^6-6600*ln(c)*b*d^3*r^5-48250*ln(c)*b*d^3*r^4-180000*
ln(c)*b*d^3*r^3-362500*ln(c)*b*d^3*r^2-375000*ln(c)*b*d^3*r+31250*b*d^3*n-600*a*e^3*r^5*(x^r)^3-10000*a*e^3*r^
4*(x^r)^3-156250*ln(c)*b*e^3*(x^r)^3+31250*b*e^3*n*(x^r)^3-63750*a*e^3*r^3*(x^r)^3-156250*d^3*b*ln(c)+36000*b*
d^3*n*r^3+72500*b*d^3*n*r^2+75000*b*d^3*n*r+16250*b*e^3*n*r^2*(x^r)^3+37500*b*e^3*n*r*(x^r)^3+93750*b*d*e^2*n*
(x^r)^2+93750*b*d^2*e*n*x^r-468750*ln(c)*b*d^2*e*x^r-515625*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-90000
*I*Pi*b*d^3*r^3*csgn(I*c)*csgn(I*c*x^n)^2-193750*a*e^3*r^2*(x^r)^3-281250*a*e^3*r*(x^r)^3-78125*I*Pi*b*e^3*csg
n(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-181250*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2-181250*I*Pi*b*d^3*r^2*csgn(I*
x^n)*csgn(I*c*x^n)^2-468750*ln(c)*b*d*e^2*(x^r)^2-63750*ln(c)*b*e^3*r^3*(x^r)^3-193750*ln(c)*b*e^3*r^2*(x^r)^3
-281250*ln(c)*b*e^3*r*(x^r)^3-600*ln(c)*b*e^3*r^5*(x^r)^3-10000*ln(c)*b*e^3*r^4*(x^r)^3+200*b*e^3*n*r^4*(x^r)^
3+3000*b*e^3*n*r^3*(x^r)^3-2700*a*d*e^2*r^5*(x^r)^2-42750*a*d*e^2*r^4*(x^r)^2-5400*a*d^2*e*r^5*x^r-72000*a*d^2
*e*r^4*x^r-180*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3300*I*Pi*b*d^3*r^5*csgn(I*c)*csgn(I*c*x^n)^2-96875*
I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+356250*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-1350*I*Pi*b
*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+78125*I*Pi*b*d^3*csgn(I*c*x^n)^3+3300*I*Pi*b*d^3*r^5*csgn(I*c*x
^n)^3+31875*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+82500*b*d*e^2*n*r^2*(x^r)^2-5400*ln(c)*
b*d^2*e*r^5*x^r-5000*I*Pi*b*e^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3-127500*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn
(I*c*x^n)^2*(x^r)^2+140625*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-78125*I*Pi*b*d^3*csgn(I*c)
*csgn(I*c*x^n)^2-78125*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-21375*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)
^2*(x^r)^2+2700*I*Pi*b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+356250*I*Pi*b*d*e^2*r^2*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)*(x^r)^2+440625*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+96875*I*Pi*b*e^
3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+300*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)
^3-300*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+180*I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)+24125*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-234375*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^
r-515625*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+36000*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r-2700*I*Pi*b
*d^2*e*r^5*csgn(I*c)*csgn(I*c*x^n)^2*x^r-42750*ln(c)*b*d*e^2*r^4*(x^r)^2-363750*ln(c)*b*d^2*e*r^3*x^r-881250*l
n(c)*b*d^2*e*r^2*x^r-1031250*ln(c)*b*d^2*e*r*x^r-255000*ln(c)*b*d*e^2*r^3*(x^r)^2+468750*I*Pi*b*d*e^2*r*csgn(I
*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+515625*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+21375*I*Pi
*b*d*e^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+24125*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+90000*I*Pi*b*d^3
*r^3*csgn(I*c*x^n)^3-1350*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+3300*I*Pi*b*d^3*r^5*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)-234375*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-234375*I*Pi*b*d*e^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2*(x^r)^2-356250*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-181875*I*Pi*b*d^2*e*r^3*csg
n(I*x^n)*csgn(I*c*x^n)^2*x^r+468750*I*Pi*b*d*e^...

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Maxima [A]
time = 0.29, size = 228, normalized size = 1.51 \begin {gather*} -\frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{3} x^{5} + \frac {b e^{3} x^{3 \, r + 5} \log \left (c x^{n}\right )}{3 \, r + 5} + \frac {3 \, b d e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {3 \, b d^{2} e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{3} n x^{3 \, r + 5}}{{\left (3 \, r + 5\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 5}}{3 \, r + 5} - \frac {3 \, b d e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {3 \, b d^{2} e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {3 \, a d^{2} 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)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5 + b*e^3*x^(3*r + 5)*log(c*x^n)/(3*r + 5) + 3*b*d*
e^2*x^(2*r + 5)*log(c*x^n)/(2*r + 5) + 3*b*d^2*e*x^(r + 5)*log(c*x^n)/(r + 5) - b*e^3*n*x^(3*r + 5)/(3*r + 5)^
2 + a*e^3*x^(3*r + 5)/(3*r + 5) - 3*b*d*e^2*n*x^(2*r + 5)/(2*r + 5)^2 + 3*a*d*e^2*x^(2*r + 5)/(2*r + 5) - 3*b*
d^2*e*n*x^(r + 5)/(r + 5)^2 + 3*a*d^2*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. 880 vs. \(2 (145) = 290\).
time = 0.37, size = 880, normalized size = 5.83 \begin {gather*} \frac {5 \, {\left (36 \, b d^{3} r^{6} + 660 \, b d^{3} r^{5} + 4825 \, b d^{3} r^{4} + 18000 \, b d^{3} r^{3} + 36250 \, b d^{3} r^{2} + 37500 \, b d^{3} r + 15625 \, b d^{3}\right )} x^{5} \log \left (c\right ) + 5 \, {\left (36 \, b d^{3} n r^{6} + 660 \, b d^{3} n r^{5} + 4825 \, b d^{3} n r^{4} + 18000 \, b d^{3} n r^{3} + 36250 \, b d^{3} n r^{2} + 37500 \, b d^{3} n r + 15625 \, b d^{3} n\right )} x^{5} \log \left (x\right ) - {\left (36 \, {\left (b d^{3} n - 5 \, a d^{3}\right )} r^{6} + 660 \, {\left (b d^{3} n - 5 \, a d^{3}\right )} r^{5} + 15625 \, b d^{3} n + 4825 \, {\left (b d^{3} n - 5 \, a d^{3}\right )} r^{4} - 78125 \, a d^{3} + 18000 \, {\left (b d^{3} n - 5 \, a d^{3}\right )} r^{3} + 36250 \, {\left (b d^{3} n - 5 \, a d^{3}\right )} r^{2} + 37500 \, {\left (b d^{3} n - 5 \, a d^{3}\right )} r\right )} x^{5} + 25 \, {\left ({\left (12 \, b r^{5} + 200 \, b r^{4} + 1275 \, b r^{3} + 3875 \, b r^{2} + 5625 \, b r + 3125 \, b\right )} x^{5} e^{3} \log \left (c\right ) + {\left (12 \, b n r^{5} + 200 \, b n r^{4} + 1275 \, b n r^{3} + 3875 \, b n r^{2} + 5625 \, b n r + 3125 \, b n\right )} x^{5} e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n - 50 \, a\right )} r^{4} - 15 \, {\left (4 \, b n - 85 \, a\right )} r^{3} - 25 \, {\left (13 \, b n - 155 \, a\right )} r^{2} - 625 \, b n - 375 \, {\left (2 \, b n - 15 \, a\right )} r + 3125 \, a\right )} x^{5} e^{3}\right )} x^{3 \, r} + 75 \, {\left ({\left (18 \, b d r^{5} + 285 \, b d r^{4} + 1700 \, b d r^{3} + 4750 \, b d r^{2} + 6250 \, b d r + 3125 \, b d\right )} x^{5} e^{2} \log \left (c\right ) + {\left (18 \, b d n r^{5} + 285 \, b d n r^{4} + 1700 \, b d n r^{3} + 4750 \, b d n r^{2} + 6250 \, b d n r + 3125 \, b d n\right )} x^{5} e^{2} \log \left (x\right ) + {\left (18 \, a d r^{5} - 3 \, {\left (3 \, b d n - 95 \, a d\right )} r^{4} - 20 \, {\left (6 \, b d n - 85 \, a d\right )} r^{3} - 625 \, b d n - 50 \, {\left (11 \, b d n - 95 \, a d\right )} r^{2} + 3125 \, a d - 250 \, {\left (4 \, b d n - 25 \, a d\right )} r\right )} x^{5} e^{2}\right )} x^{2 \, r} + 75 \, {\left ({\left (36 \, b d^{2} r^{5} + 480 \, b d^{2} r^{4} + 2425 \, b d^{2} r^{3} + 5875 \, b d^{2} r^{2} + 6875 \, b d^{2} r + 3125 \, b d^{2}\right )} x^{5} e \log \left (c\right ) + {\left (36 \, b d^{2} n r^{5} + 480 \, b d^{2} n r^{4} + 2425 \, b d^{2} n r^{3} + 5875 \, b d^{2} n r^{2} + 6875 \, b d^{2} n r + 3125 \, b d^{2} n\right )} x^{5} e \log \left (x\right ) + {\left (36 \, a d^{2} r^{5} - 12 \, {\left (3 \, b d^{2} n - 40 \, a d^{2}\right )} r^{4} - 625 \, b d^{2} n - 25 \, {\left (12 \, b d^{2} n - 97 \, a d^{2}\right )} r^{3} + 3125 \, a d^{2} - 25 \, {\left (37 \, b d^{2} n - 235 \, a d^{2}\right )} r^{2} - 625 \, {\left (2 \, b d^{2} n - 11 \, a d^{2}\right )} r\right )} x^{5} e\right )} x^{r}}{25 \, {\left (36 \, r^{6} + 660 \, r^{5} + 4825 \, r^{4} + 18000 \, r^{3} + 36250 \, r^{2} + 37500 \, r + 15625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/25*(5*(36*b*d^3*r^6 + 660*b*d^3*r^5 + 4825*b*d^3*r^4 + 18000*b*d^3*r^3 + 36250*b*d^3*r^2 + 37500*b*d^3*r + 1
5625*b*d^3)*x^5*log(c) + 5*(36*b*d^3*n*r^6 + 660*b*d^3*n*r^5 + 4825*b*d^3*n*r^4 + 18000*b*d^3*n*r^3 + 36250*b*
d^3*n*r^2 + 37500*b*d^3*n*r + 15625*b*d^3*n)*x^5*log(x) - (36*(b*d^3*n - 5*a*d^3)*r^6 + 660*(b*d^3*n - 5*a*d^3
)*r^5 + 15625*b*d^3*n + 4825*(b*d^3*n - 5*a*d^3)*r^4 - 78125*a*d^3 + 18000*(b*d^3*n - 5*a*d^3)*r^3 + 36250*(b*
d^3*n - 5*a*d^3)*r^2 + 37500*(b*d^3*n - 5*a*d^3)*r)*x^5 + 25*((12*b*r^5 + 200*b*r^4 + 1275*b*r^3 + 3875*b*r^2
+ 5625*b*r + 3125*b)*x^5*e^3*log(c) + (12*b*n*r^5 + 200*b*n*r^4 + 1275*b*n*r^3 + 3875*b*n*r^2 + 5625*b*n*r + 3
125*b*n)*x^5*e^3*log(x) + (12*a*r^5 - 4*(b*n - 50*a)*r^4 - 15*(4*b*n - 85*a)*r^3 - 25*(13*b*n - 155*a)*r^2 - 6
25*b*n - 375*(2*b*n - 15*a)*r + 3125*a)*x^5*e^3)*x^(3*r) + 75*((18*b*d*r^5 + 285*b*d*r^4 + 1700*b*d*r^3 + 4750
*b*d*r^2 + 6250*b*d*r + 3125*b*d)*x^5*e^2*log(c) + (18*b*d*n*r^5 + 285*b*d*n*r^4 + 1700*b*d*n*r^3 + 4750*b*d*n
*r^2 + 6250*b*d*n*r + 3125*b*d*n)*x^5*e^2*log(x) + (18*a*d*r^5 - 3*(3*b*d*n - 95*a*d)*r^4 - 20*(6*b*d*n - 85*a
*d)*r^3 - 625*b*d*n - 50*(11*b*d*n - 95*a*d)*r^2 + 3125*a*d - 250*(4*b*d*n - 25*a*d)*r)*x^5*e^2)*x^(2*r) + 75*
((36*b*d^2*r^5 + 480*b*d^2*r^4 + 2425*b*d^2*r^3 + 5875*b*d^2*r^2 + 6875*b*d^2*r + 3125*b*d^2)*x^5*e*log(c) + (
36*b*d^2*n*r^5 + 480*b*d^2*n*r^4 + 2425*b*d^2*n*r^3 + 5875*b*d^2*n*r^2 + 6875*b*d^2*n*r + 3125*b*d^2*n)*x^5*e*
log(x) + (36*a*d^2*r^5 - 12*(3*b*d^2*n - 40*a*d^2)*r^4 - 625*b*d^2*n - 25*(12*b*d^2*n - 97*a*d^2)*r^3 + 3125*a
*d^2 - 25*(37*b*d^2*n - 235*a*d^2)*r^2 - 625*(2*b*d^2*n - 11*a*d^2)*r)*x^5*e)*x^r)/(36*r^6 + 660*r^5 + 4825*r^
4 + 18000*r^3 + 36250*r^2 + 37500*r + 15625)

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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)**3*(a+b*ln(c*x**n)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (145) = 290\).
time = 1.88, size = 1588, normalized size = 10.52 \begin {gather*} \frac {180 \, b d^{3} n r^{6} x^{5} \log \left (x\right ) + 2700 \, b d^{2} n r^{5} x^{5} x^{r} e \log \left (x\right ) - 36 \, b d^{3} n r^{6} x^{5} + 180 \, b d^{3} r^{6} x^{5} \log \left (c\right ) + 2700 \, b d^{2} r^{5} x^{5} x^{r} e \log \left (c\right ) + 3300 \, b d^{3} n r^{5} x^{5} \log \left (x\right ) + 1350 \, b d n r^{5} x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 36000 \, b d^{2} n r^{4} x^{5} x^{r} e \log \left (x\right ) - 660 \, b d^{3} n r^{5} x^{5} + 180 \, a d^{3} r^{6} x^{5} - 2700 \, b d^{2} n r^{4} x^{5} x^{r} e + 2700 \, a d^{2} r^{5} x^{5} x^{r} e + 3300 \, b d^{3} r^{5} x^{5} \log \left (c\right ) + 1350 \, b d r^{5} x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 36000 \, b d^{2} r^{4} x^{5} x^{r} e \log \left (c\right ) + 24125 \, b d^{3} n r^{4} x^{5} \log \left (x\right ) + 300 \, b n r^{5} x^{5} x^{3 \, r} e^{3} \log \left (x\right ) + 21375 \, b d n r^{4} x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 181875 \, b d^{2} n r^{3} x^{5} x^{r} e \log \left (x\right ) - 4825 \, b d^{3} n r^{4} x^{5} + 3300 \, a d^{3} r^{5} x^{5} - 675 \, b d n r^{4} x^{5} x^{2 \, r} e^{2} + 1350 \, a d r^{5} x^{5} x^{2 \, r} e^{2} - 22500 \, b d^{2} n r^{3} x^{5} x^{r} e + 36000 \, a d^{2} r^{4} x^{5} x^{r} e + 24125 \, b d^{3} r^{4} x^{5} \log \left (c\right ) + 300 \, b r^{5} x^{5} x^{3 \, r} e^{3} \log \left (c\right ) + 21375 \, b d r^{4} x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 181875 \, b d^{2} r^{3} x^{5} x^{r} e \log \left (c\right ) + 90000 \, b d^{3} n r^{3} x^{5} \log \left (x\right ) + 5000 \, b n r^{4} x^{5} x^{3 \, r} e^{3} \log \left (x\right ) + 127500 \, b d n r^{3} x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 440625 \, b d^{2} n r^{2} x^{5} x^{r} e \log \left (x\right ) - 18000 \, b d^{3} n r^{3} x^{5} + 24125 \, a d^{3} r^{4} x^{5} - 100 \, b n r^{4} x^{5} x^{3 \, r} e^{3} + 300 \, a r^{5} x^{5} x^{3 \, r} e^{3} - 9000 \, b d n r^{3} x^{5} x^{2 \, r} e^{2} + 21375 \, a d r^{4} x^{5} x^{2 \, r} e^{2} - 69375 \, b d^{2} n r^{2} x^{5} x^{r} e + 181875 \, a d^{2} r^{3} x^{5} x^{r} e + 90000 \, b d^{3} r^{3} x^{5} \log \left (c\right ) + 5000 \, b r^{4} x^{5} x^{3 \, r} e^{3} \log \left (c\right ) + 127500 \, b d r^{3} x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 440625 \, b d^{2} r^{2} x^{5} x^{r} e \log \left (c\right ) + 181250 \, b d^{3} n r^{2} x^{5} \log \left (x\right ) + 31875 \, b n r^{3} x^{5} x^{3 \, r} e^{3} \log \left (x\right ) + 356250 \, b d n r^{2} x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 515625 \, b d^{2} n r x^{5} x^{r} e \log \left (x\right ) - 36250 \, b d^{3} n r^{2} x^{5} + 90000 \, a d^{3} r^{3} x^{5} - 1500 \, b n r^{3} x^{5} x^{3 \, r} e^{3} + 5000 \, a r^{4} x^{5} x^{3 \, r} e^{3} - 41250 \, b d n r^{2} x^{5} x^{2 \, r} e^{2} + 127500 \, a d r^{3} x^{5} x^{2 \, r} e^{2} - 93750 \, b d^{2} n r x^{5} x^{r} e + 440625 \, a d^{2} r^{2} x^{5} x^{r} e + 181250 \, b d^{3} r^{2} x^{5} \log \left (c\right ) + 31875 \, b r^{3} x^{5} x^{3 \, r} e^{3} \log \left (c\right ) + 356250 \, b d r^{2} x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 515625 \, b d^{2} r x^{5} x^{r} e \log \left (c\right ) + 187500 \, b d^{3} n r x^{5} \log \left (x\right ) + 96875 \, b n r^{2} x^{5} x^{3 \, r} e^{3} \log \left (x\right ) + 468750 \, b d n r x^{5} x^{2 \, r} e^{2} \log \left (x\right ) + 234375 \, b d^{2} n x^{5} x^{r} e \log \left (x\right ) - 37500 \, b d^{3} n r x^{5} + 181250 \, a d^{3} r^{2} x^{5} - 8125 \, b n r^{2} x^{5} x^{3 \, r} e^{3} + 31875 \, a r^{3} x^{5} x^{3 \, r} e^{3} - 75000 \, b d n r x^{5} x^{2 \, r} e^{2} + 356250 \, a d r^{2} x^{5} x^{2 \, r} e^{2} - 46875 \, b d^{2} n x^{5} x^{r} e + 515625 \, a d^{2} r x^{5} x^{r} e + 187500 \, b d^{3} r x^{5} \log \left (c\right ) + 96875 \, b r^{2} x^{5} x^{3 \, r} e^{3} \log \left (c\right ) + 468750 \, b d r x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 234375 \, b d^{2} x^{5} x^{r} e \log \left (c\right ) + 78125 \, b d^{3} n x^{5} \log \left (x\right ) + 140625 \, b n r x^{5} x^{3 \, r} e^{3} \log \left (x\right ) + 234375 \, b d n x^{5} x^{2 \, r} e^{2} \log \left (x\right ) - 15625 \, b d^{3} n x^{5} + 187500 \, a d^{3} r x^{5} - 18750 \, b n r x^{5} x^{3 \, r} e^{3} + 96875 \, a r^{2} x^{5} x^{3 \, r} e^{3} - 46875 \, b d n x^{5} x^{2 \, r} e^{2} + 468750 \, a d r x^{5} x^{2 \, r} e^{2} + 234375 \, a d^{2} x^{5} x^{r} e + 78125 \, b d^{3} x^{5} \log \left (c\right ) + 140625 \, b r x^{5} x^{3 \, r} e^{3} \log \left (c\right ) + 234375 \, b d x^{5} x^{2 \, r} e^{2} \log \left (c\right ) + 78125 \, b n x^{5} x^{3 \, r} e^{3} \log \left (x\right ) + 78125 \, a d^{3} x^{5} - 15625 \, b n x^{5} x^{3 \, r} e^{3} + 140625 \, a r x^{5} x^{3 \, r} e^{3} + 234375 \, a d x^{5} x^{2 \, r} e^{2} + 78125 \, b x^{5} x^{3 \, r} e^{3} \log \left (c\right ) + 78125 \, a x^{5} x^{3 \, r} e^{3}}{25 \, {\left (36 \, r^{6} + 660 \, r^{5} + 4825 \, r^{4} + 18000 \, r^{3} + 36250 \, r^{2} + 37500 \, r + 15625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/25*(180*b*d^3*n*r^6*x^5*log(x) + 2700*b*d^2*n*r^5*x^5*x^r*e*log(x) - 36*b*d^3*n*r^6*x^5 + 180*b*d^3*r^6*x^5*
log(c) + 2700*b*d^2*r^5*x^5*x^r*e*log(c) + 3300*b*d^3*n*r^5*x^5*log(x) + 1350*b*d*n*r^5*x^5*x^(2*r)*e^2*log(x)
+ 36000*b*d^2*n*r^4*x^5*x^r*e*log(x) - 660*b*d^3*n*r^5*x^5 + 180*a*d^3*r^6*x^5 - 2700*b*d^2*n*r^4*x^5*x^r*e +
2700*a*d^2*r^5*x^5*x^r*e + 3300*b*d^3*r^5*x^5*log(c) + 1350*b*d*r^5*x^5*x^(2*r)*e^2*log(c) + 36000*b*d^2*r^4*
x^5*x^r*e*log(c) + 24125*b*d^3*n*r^4*x^5*log(x) + 300*b*n*r^5*x^5*x^(3*r)*e^3*log(x) + 21375*b*d*n*r^4*x^5*x^(
2*r)*e^2*log(x) + 181875*b*d^2*n*r^3*x^5*x^r*e*log(x) - 4825*b*d^3*n*r^4*x^5 + 3300*a*d^3*r^5*x^5 - 675*b*d*n*
r^4*x^5*x^(2*r)*e^2 + 1350*a*d*r^5*x^5*x^(2*r)*e^2 - 22500*b*d^2*n*r^3*x^5*x^r*e + 36000*a*d^2*r^4*x^5*x^r*e +
24125*b*d^3*r^4*x^5*log(c) + 300*b*r^5*x^5*x^(3*r)*e^3*log(c) + 21375*b*d*r^4*x^5*x^(2*r)*e^2*log(c) + 181875
*b*d^2*r^3*x^5*x^r*e*log(c) + 90000*b*d^3*n*r^3*x^5*log(x) + 5000*b*n*r^4*x^5*x^(3*r)*e^3*log(x) + 127500*b*d*
n*r^3*x^5*x^(2*r)*e^2*log(x) + 440625*b*d^2*n*r^2*x^5*x^r*e*log(x) - 18000*b*d^3*n*r^3*x^5 + 24125*a*d^3*r^4*x
^5 - 100*b*n*r^4*x^5*x^(3*r)*e^3 + 300*a*r^5*x^5*x^(3*r)*e^3 - 9000*b*d*n*r^3*x^5*x^(2*r)*e^2 + 21375*a*d*r^4*
x^5*x^(2*r)*e^2 - 69375*b*d^2*n*r^2*x^5*x^r*e + 181875*a*d^2*r^3*x^5*x^r*e + 90000*b*d^3*r^3*x^5*log(c) + 5000
*b*r^4*x^5*x^(3*r)*e^3*log(c) + 127500*b*d*r^3*x^5*x^(2*r)*e^2*log(c) + 440625*b*d^2*r^2*x^5*x^r*e*log(c) + 18
1250*b*d^3*n*r^2*x^5*log(x) + 31875*b*n*r^3*x^5*x^(3*r)*e^3*log(x) + 356250*b*d*n*r^2*x^5*x^(2*r)*e^2*log(x) +
515625*b*d^2*n*r*x^5*x^r*e*log(x) - 36250*b*d^3*n*r^2*x^5 + 90000*a*d^3*r^3*x^5 - 1500*b*n*r^3*x^5*x^(3*r)*e^
3 + 5000*a*r^4*x^5*x^(3*r)*e^3 - 41250*b*d*n*r^2*x^5*x^(2*r)*e^2 + 127500*a*d*r^3*x^5*x^(2*r)*e^2 - 93750*b*d^
2*n*r*x^5*x^r*e + 440625*a*d^2*r^2*x^5*x^r*e + 181250*b*d^3*r^2*x^5*log(c) + 31875*b*r^3*x^5*x^(3*r)*e^3*log(c
) + 356250*b*d*r^2*x^5*x^(2*r)*e^2*log(c) + 515625*b*d^2*r*x^5*x^r*e*log(c) + 187500*b*d^3*n*r*x^5*log(x) + 96
875*b*n*r^2*x^5*x^(3*r)*e^3*log(x) + 468750*b*d*n*r*x^5*x^(2*r)*e^2*log(x) + 234375*b*d^2*n*x^5*x^r*e*log(x) -
37500*b*d^3*n*r*x^5 + 181250*a*d^3*r^2*x^5 - 8125*b*n*r^2*x^5*x^(3*r)*e^3 + 31875*a*r^3*x^5*x^(3*r)*e^3 - 750
00*b*d*n*r*x^5*x^(2*r)*e^2 + 356250*a*d*r^2*x^5*x^(2*r)*e^2 - 46875*b*d^2*n*x^5*x^r*e + 515625*a*d^2*r*x^5*x^r
*e + 187500*b*d^3*r*x^5*log(c) + 96875*b*r^2*x^5*x^(3*r)*e^3*log(c) + 468750*b*d*r*x^5*x^(2*r)*e^2*log(c) + 23
4375*b*d^2*x^5*x^r*e*log(c) + 78125*b*d^3*n*x^5*log(x) + 140625*b*n*r*x^5*x^(3*r)*e^3*log(x) + 234375*b*d*n*x^
5*x^(2*r)*e^2*log(x) - 15625*b*d^3*n*x^5 + 187500*a*d^3*r*x^5 - 18750*b*n*r*x^5*x^(3*r)*e^3 + 96875*a*r^2*x^5*
x^(3*r)*e^3 - 46875*b*d*n*x^5*x^(2*r)*e^2 + 468750*a*d*r*x^5*x^(2*r)*e^2 + 234375*a*d^2*x^5*x^r*e + 78125*b*d^
3*x^5*log(c) + 140625*b*r*x^5*x^(3*r)*e^3*log(c) + 234375*b*d*x^5*x^(2*r)*e^2*log(c) + 78125*b*n*x^5*x^(3*r)*e
^3*log(x) + 78125*a*d^3*x^5 - 15625*b*n*x^5*x^(3*r)*e^3 + 140625*a*r*x^5*x^(3*r)*e^3 + 234375*a*d*x^5*x^(2*r)*
e^2 + 78125*b*x^5*x^(3*r)*e^3*log(c) + 78125*a*x^5*x^(3*r)*e^3)/(36*r^6 + 660*r^5 + 4825*r^4 + 18000*r^3 + 362
50*r^2 + 37500*r + 15625)

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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 )}^3\,\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)^3*(a + b*log(c*x^n)),x)

[Out]

int(x^4*(d + e*x^r)^3*(a + b*log(c*x^n)), x)

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