3.2.82 \(\int x^2 (a+b x)^n (c+d x^3)^3 \, dx\) [182]
Optimal. Leaf size=459 \[ \frac {a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{1+n}}{b^{12} (1+n)}-\frac {a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{2+n}}{b^{12} (2+n)}+\frac {\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{3+n}}{b^{12} (3+n)}+\frac {3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{4+n}}{b^{12} (4+n)}-\frac {15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^{5+n}}{b^{12} (5+n)}+\frac {3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^{6+n}}{b^{12} (6+n)}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{7+n}}{b^{12} (7+n)}-\frac {6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{8+n}}{b^{12} (8+n)}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{9+n}}{b^{12} (9+n)}+\frac {55 a^2 d^3 (a+b x)^{10+n}}{b^{12} (10+n)}-\frac {11 a d^3 (a+b x)^{11+n}}{b^{12} (11+n)}+\frac {d^3 (a+b x)^{12+n}}{b^{12} (12+n)} \]
[Out]
a^2*(-a^3*d+b^3*c)^3*(b*x+a)^(1+n)/b^12/(1+n)-a*(-11*a^3*d+2*b^3*c)*(-a^3*d+b^3*c)^2*(b*x+a)^(2+n)/b^12/(2+n)+
(-a^3*d+b^3*c)*(55*a^6*d^2-29*a^3*b^3*c*d+b^6*c^2)*(b*x+a)^(3+n)/b^12/(3+n)+3*a^2*d*(55*a^6*d^2-56*a^3*b^3*c*d
+10*b^6*c^2)*(b*x+a)^(4+n)/b^12/(4+n)-15*a*d*(22*a^6*d^2-14*a^3*b^3*c*d+b^6*c^2)*(b*x+a)^(5+n)/b^12/(5+n)+3*d*
(154*a^6*d^2-56*a^3*b^3*c*d+b^6*c^2)*(b*x+a)^(6+n)/b^12/(6+n)+42*a^2*d^2*(-11*a^3*d+2*b^3*c)*(b*x+a)^(7+n)/b^1
2/(7+n)-6*a*d^2*(-55*a^3*d+4*b^3*c)*(b*x+a)^(8+n)/b^12/(8+n)+3*d^2*(-55*a^3*d+b^3*c)*(b*x+a)^(9+n)/b^12/(9+n)+
55*a^2*d^3*(b*x+a)^(10+n)/b^12/(10+n)-11*a*d^3*(b*x+a)^(11+n)/b^12/(11+n)+d^3*(b*x+a)^(12+n)/b^12/(12+n)
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1634}
\begin {gather*} -\frac {6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+9}}{b^{12} (n+9)}-\frac {a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac {55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}+\frac {\left (b^3 c-a^3 d\right ) \left (55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^{12} (n+3)}-\frac {15 a d \left (22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac {3 d \left (154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{n+7}}{b^{12} (n+7)}+\frac {a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{12} (n+1)}+\frac {3 a^2 d \left (55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right ) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac {11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac {d^3 (a+b x)^{n+12}}{b^{12} (n+12)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x^3)^3,x]
[Out]
(a^2*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^12*(1 + n)) - (a*(2*b^3*c - 11*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x)
^(2 + n))/(b^12*(2 + n)) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(3 + n))/(b^12*(
3 + n)) + (3*a^2*d*(10*b^6*c^2 - 56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(4 + n))/(b^12*(4 + n)) - (15*a*d*(b^6
*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)^(5 + n))/(b^12*(5 + n)) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a
^6*d^2)*(a + b*x)^(6 + n))/(b^12*(6 + n)) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d)*(a + b*x)^(7 + n))/(b^12*(7 + n))
- (6*a*d^2*(4*b^3*c - 55*a^3*d)*(a + b*x)^(8 + n))/(b^12*(8 + n)) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^(9 +
n))/(b^12*(9 + n)) + (55*a^2*d^3*(a + b*x)^(10 + n))/(b^12*(10 + n)) - (11*a*d^3*(a + b*x)^(11 + n))/(b^12*(11
+ n)) + (d^3*(a + b*x)^(12 + n))/(b^12*(12 + n))
Rule 1634
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]
Rubi steps
\begin {align*} \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx &=\int \left (-\frac {a^2 \left (-b^3 c+a^3 d\right )^3 (a+b x)^n}{b^{11}}+\frac {a \left (-b^3 c+a^3 d\right )^2 \left (-2 b^3 c+11 a^3 d\right ) (a+b x)^{1+n}}{b^{11}}+\frac {\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{2+n}}{b^{11}}+\frac {3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{3+n}}{b^{11}}-\frac {15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^{4+n}}{b^{11}}+\frac {3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^{5+n}}{b^{11}}-\frac {42 a^2 d^2 \left (-2 b^3 c+11 a^3 d\right ) (a+b x)^{6+n}}{b^{11}}+\frac {6 a d^2 \left (-4 b^3 c+55 a^3 d\right ) (a+b x)^{7+n}}{b^{11}}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{8+n}}{b^{11}}+\frac {55 a^2 d^3 (a+b x)^{9+n}}{b^{11}}-\frac {11 a d^3 (a+b x)^{10+n}}{b^{11}}+\frac {d^3 (a+b x)^{11+n}}{b^{11}}\right ) \, dx\\ &=\frac {a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{1+n}}{b^{12} (1+n)}-\frac {a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{2+n}}{b^{12} (2+n)}+\frac {\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{3+n}}{b^{12} (3+n)}+\frac {3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{4+n}}{b^{12} (4+n)}-\frac {15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^{5+n}}{b^{12} (5+n)}+\frac {3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^{6+n}}{b^{12} (6+n)}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{7+n}}{b^{12} (7+n)}-\frac {6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{8+n}}{b^{12} (8+n)}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{9+n}}{b^{12} (9+n)}+\frac {55 a^2 d^3 (a+b x)^{10+n}}{b^{12} (10+n)}-\frac {11 a d^3 (a+b x)^{11+n}}{b^{12} (11+n)}+\frac {d^3 (a+b x)^{12+n}}{b^{12} (12+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.37, size = 402, normalized size = 0.88 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {a^2 \left (b^3 c-a^3 d\right )^3}{1+n}+\frac {a \left (b^3 c-a^3 d\right )^2 \left (-2 b^3 c+11 a^3 d\right ) (a+b x)}{2+n}+\frac {\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^2}{3+n}+\frac {3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^3}{4+n}-\frac {15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^4}{5+n}+\frac {3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^5}{6+n}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^6}{7+n}+\frac {6 a d^2 \left (-4 b^3 c+55 a^3 d\right ) (a+b x)^7}{8+n}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^8}{9+n}+\frac {55 a^2 d^3 (a+b x)^9}{10+n}-\frac {11 a d^3 (a+b x)^{10}}{11+n}+\frac {d^3 (a+b x)^{11}}{12+n}\right )}{b^{12}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x^3)^3,x]
[Out]
((a + b*x)^(1 + n)*((a^2*(b^3*c - a^3*d)^3)/(1 + n) + (a*(b^3*c - a^3*d)^2*(-2*b^3*c + 11*a^3*d)*(a + b*x))/(2
+ n) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^2)/(3 + n) + (3*a^2*d*(10*b^6*c^2 -
56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^3)/(4 + n) - (15*a*d*(b^6*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)
^4)/(5 + n) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a^6*d^2)*(a + b*x)^5)/(6 + n) + (42*a^2*d^2*(2*b^3*c - 11*a
^3*d)*(a + b*x)^6)/(7 + n) + (6*a*d^2*(-4*b^3*c + 55*a^3*d)*(a + b*x)^7)/(8 + n) + (3*d^2*(b^3*c - 55*a^3*d)*(
a + b*x)^8)/(9 + n) + (55*a^2*d^3*(a + b*x)^9)/(10 + n) - (11*a*d^3*(a + b*x)^10)/(11 + n) + (d^3*(a + b*x)^11
)/(12 + n)))/b^12
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3779\) vs.
\(2(459)=918\).
time = 0.29, size = 3780, normalized size = 8.24
| | |
| method |
result |
size |
| | |
| gosper |
\(\text {Expression too large to display}\) |
\(3780\) |
| risch |
\(\text {Expression too large to display}\) |
\(4231\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(b*x+a)^n*(d*x^3+c)^3,x,method=_RETURNVERBOSE)
[Out]
-(b*x+a)^(1+n)*(-b^11*d^3*n^11*x^11-66*b^11*d^3*n^10*x^11+11*a*b^10*d^3*n^10*x^10-1925*b^11*d^3*n^9*x^11+605*a
*b^10*d^3*n^9*x^10-3*b^11*c*d^2*n^11*x^8-32670*b^11*d^3*n^8*x^11-110*a^2*b^9*d^3*n^9*x^9+14520*a*b^10*d^3*n^8*
x^10-207*b^11*c*d^2*n^10*x^8-357423*b^11*d^3*n^7*x^11-4950*a^2*b^9*d^3*n^8*x^9+24*a*b^10*c*d^2*n^10*x^7+199650
*a*b^10*d^3*n^7*x^10-6288*b^11*c*d^2*n^9*x^8-2637558*b^11*d^3*n^6*x^11+990*a^3*b^8*d^3*n^8*x^8-95700*a^2*b^9*d
^3*n^7*x^9+1464*a*b^10*c*d^2*n^9*x^7+1735503*a*b^10*d^3*n^6*x^10-3*b^11*c^2*d*n^11*x^5-110718*b^11*c*d^2*n^8*x
^8-13339535*b^11*d^3*n^5*x^11+35640*a^3*b^8*d^3*n^7*x^8-168*a^2*b^9*c*d^2*n^9*x^6-1039500*a^2*b^9*d^3*n^6*x^9+
38592*a*b^10*c*d^2*n^8*x^7+9922605*a*b^10*d^3*n^5*x^10-216*b^11*c^2*d*n^10*x^5-1251927*b^11*c*d^2*n^7*x^8-4599
5730*b^11*d^3*n^4*x^11-7920*a^4*b^7*d^3*n^7*x^7+540540*a^3*b^8*d^3*n^6*x^8-9072*a^2*b^9*c*d^2*n^8*x^6-6960030*
a^2*b^9*d^3*n^5*x^9+15*a*b^10*c^2*d*n^10*x^4+577008*a*b^10*c*d^2*n^7*x^7+37586230*a*b^10*d^3*n^4*x^10-6855*b^1
1*c^2*d*n^9*x^5-9512559*b^11*c*d^2*n^6*x^8-105258076*b^11*d^3*n^3*x^11-221760*a^4*b^7*d^3*n^6*x^7+1008*a^3*b^8
*c*d^2*n^8*x^5+4490640*a^3*b^8*d^3*n^5*x^8-206640*a^2*b^9*c*d^2*n^7*x^6-29625750*a^2*b^9*d^3*n^4*x^9+1005*a*b^
10*c^2*d*n^9*x^4+5399352*a*b^10*c*d^2*n^6*x^7+92504500*a*b^10*d^3*n^3*x^10-b^11*c^3*n^11*x^2-126180*b^11*c^2*d
*n^8*x^5-49357662*b^11*c*d^2*n^5*x^8-150917976*b^11*d^3*n^2*x^11+55440*a^5*b^6*d^3*n^6*x^6-2550240*a^4*b^7*d^3
*n^5*x^7+48384*a^3*b^8*c*d^2*n^7*x^5+22224510*a^3*b^8*d^3*n^4*x^8-60*a^2*b^9*c^2*d*n^9*x^3-2592576*a^2*b^9*c*d
^2*n^6*x^6-79604800*a^2*b^9*d^3*n^3*x^9+29250*a*b^10*c^2*d*n^8*x^4+32905656*a*b^10*c*d^2*n^5*x^7+140289336*a*b
^10*d^3*n^2*x^10-75*b^11*c^3*n^10*x^2-1491309*b^11*c^2*d*n^7*x^5-173991492*b^11*c*d^2*n^4*x^8-120543840*b^11*d
^3*n*x^11+1164240*a^5*b^6*d^3*n^5*x^6-5040*a^4*b^7*c*d^2*n^7*x^4-15523200*a^4*b^7*d^3*n^4*x^7+949536*a^3*b^8*c
*d^2*n^6*x^5+66611160*a^3*b^8*d^3*n^3*x^8-3780*a^2*b^9*c^2*d*n^8*x^3-19647432*a^2*b^9*c*d^2*n^5*x^6-128997000*
a^2*b^9*d^3*n^2*x^9+2*a*b^10*c^3*n^10*x+484650*a*b^10*c^2*d*n^7*x^4+131616048*a*b^10*c*d^2*n^4*x^7+116915040*a
*b^10*d^3*n*x^10-2492*b^11*c^3*n^9*x^2-11832048*b^11*c^2*d*n^6*x^5-405697080*b^11*c*d^2*n^3*x^8-39916800*b^11*
d^3*x^11-332640*a^6*b^5*d^3*n^5*x^5+9702000*a^5*b^6*d^3*n^4*x^6-216720*a^4*b^7*c*d^2*n^6*x^4-53610480*a^4*b^7*
d^3*n^3*x^7+180*a^3*b^8*c^2*d*n^8*x^2+9858240*a^3*b^8*c*d^2*n^5*x^5+116942760*a^3*b^8*d^3*n^2*x^8-101880*a^2*b
^9*c^2*d*n^7*x^3-92807568*a^2*b^9*c*d^2*n^4*x^6-112923360*a^2*b^9*d^3*n*x^9+146*a*b^10*c^3*n^9*x+5033295*a*b^1
0*c^2*d*n^6*x^4+339003552*a*b^10*c*d^2*n^3*x^7+39916800*a*b^10*d^3*x^10-48294*b^11*c^3*n^8*x^2-63978405*b^11*c
^2*d*n^5*x^5-590770944*b^11*c*d^2*n^2*x^8-4989600*a^6*b^5*d^3*n^4*x^5+20160*a^5*b^6*c*d^2*n^6*x^3+40748400*a^5
*b^6*d^3*n^3*x^6-3664080*a^4*b^7*c*d^2*n^5*x^4-104005440*a^4*b^7*d^3*n^2*x^7+10800*a^3*b^8*c^2*d*n^7*x^2+58735
152*a^3*b^8*c*d^2*n^4*x^5+108488160*a^3*b^8*d^3*n*x^8-2*a^2*b^9*c^3*n^9-1531080*a^2*b^9*c^2*d*n^6*x^3-27165936
0*a^2*b^9*c*d^2*n^3*x^6-39916800*a^2*b^9*d^3*x^9+4692*a*b^10*c^3*n^8*x+33993765*a*b^10*c^2*d*n^5*x^4+533548224
*a*b^10*c*d^2*n^2*x^7-604581*b^11*c^3*n^7*x^2-234340020*b^11*c^2*d*n^4*x^5-477740160*b^11*c*d^2*n*x^8+1663200*
a^7*b^4*d^3*n^4*x^4-28274400*a^6*b^5*d^3*n^3*x^5+786240*a^5*b^6*c*d^2*n^5*x^3+90034560*a^5*b^6*d^3*n^2*x^6-360
*a^4*b^7*c^2*d*n^7*x-30970800*a^4*b^7*c*d^2*n^4*x^4-103498560*a^4*b^7*d^3*n*x^7+273240*a^3*b^8*c^2*d*n^6*x^2+2
04434496*a^3*b^8*c*d^2*n^3*x^5+39916800*a^3*b^8*d^3*x^8-144*a^2*b^9*c^3*n^8-14008860*a^2*b^9*c^2*d*n^5*x^3-471
409344*a^2*b^9*c*d^2*n^2*x^6+87204*a*b^10*c^3*n^7*x+149923200*a*b^10*c^2*d*n^4*x^4+457781760*a*b^10*c*d^2*n*x^
7-5112891*b^11*c^3*n^6*x^2-565580388*b^11*c^2*d*n^3*x^5-159667200*b^11*c*d^2*x^8+16632000*a^7*b^4*d^3*n^3*x^4-
60480*a^6*b^5*c*d^2*n^5*x^2-74844000*a^6*b^5*d^3*n^2*x^5+11511360*a^5*b^6*c*d^2*n^4*x^3+97796160*a^5*b^6*d^3*n
*x^6-20880*a^4*b^7*c^2*d*n^6*x-138821760*a^4*b^7*c*d^2*n^3*x^4-39916800*a^4*b^7*d^3*x^7+3773520*a^3*b^8*c^2*d*
n^5*x^2+403349184*a^3*b^8*c*d^2*n^2*x^5-4548*a^2*b^9*c^3*n^7-79939620*a^2*b^9*c^2*d*n^4*x^3-434972160*a^2*b^9*
c*d^2*n*x^6+1034754*a*b^10*c^3*n^6*x+422084100*a*b^10*c^2*d*n^3*x^4+159667200*a*b^10*c*d^2*x^7-29651558*b^11*c
^3*n^5*x^2-848562336*b^11*c^2*d*n^2*x^5-6652800*a^8*b^3*d^3*n^3*x^3+58212000*a^7*b^4*d^3*n^2*x^4-2177280*a^6*b
^5*c*d^2*n^4*x^2-91143360*a^6*b^5*d^3*n*x^5+360*a^5*b^6*c^2*d*n^6+77837760*a^5*b^6*c*d^2*n^3*x^3+39916800*a^5*
b^6*d^3*x^6-504720*a^4*b^7*c^2*d*n^5*x-328063680*a^4*b^7*c*d^2*n^2*x^4+30706020*a^3*b^8*c^2*d*n^4*x^2+40836096
0*a^3*b^8*c*d^2*n*x^5-82656*a^2*b^9*c^3*n^6-279934320*a^2*b^9*c^2*d*n^3*x^3-159667200*a^2*b^9*c*d^2*x^6+815627
4*a*b^10*c^3*n^5*x+717481440*a*b^10*c^2*d*n^2*x^4-117115476*b^11*c^3*n^4*x^2-703304640*b^11*c^2*d*n*x^5-399168
00*a^8*b^3*d^3*n^2*x^3+120960*a^7*b^4*c*d^2*n^4*x+83160000*a^7*b^4*d^3*n*x^4-28002240*a^6*b^5*c*d^2*n^3*x^2-39
916800*a^6*b^5*d^3*x^5+20520*a^5*b^6*c^2*d*n^5+243936000*a^5*b^6*c*d^2*n^2*x^3-6537600*a^4*b^7*c^2*d*n^4*x-376
427520*a^4*b^7*c*d^2*n*x^4+147700800*a^3*b^8*c^...
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1153 vs.
\(2 (459) = 918\).
time = 0.31, size = 1153, normalized size = 2.51 \begin {gather*} \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{3}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac {3 \, {\left ({\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{9} x^{9} + {\left (n^{8} + 28 \, n^{7} + 322 \, n^{6} + 1960 \, n^{5} + 6769 \, n^{4} + 13132 \, n^{3} + 13068 \, n^{2} + 5040 \, n\right )} a b^{8} x^{8} - 8 \, {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a^{2} b^{7} x^{7} + 56 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{3} b^{6} x^{6} - 336 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{4} b^{5} x^{5} + 1680 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{5} b^{4} x^{4} - 6720 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{6} b^{3} x^{3} + 20160 \, {\left (n^{2} + n\right )} a^{7} b^{2} x^{2} - 40320 \, a^{8} b n x + 40320 \, a^{9}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{9} + 45 \, n^{8} + 870 \, n^{7} + 9450 \, n^{6} + 63273 \, n^{5} + 269325 \, n^{4} + 723680 \, n^{3} + 1172700 \, n^{2} + 1026576 \, n + 362880\right )} b^{9}} + \frac {{\left ({\left (n^{11} + 66 \, n^{10} + 1925 \, n^{9} + 32670 \, n^{8} + 357423 \, n^{7} + 2637558 \, n^{6} + 13339535 \, n^{5} + 45995730 \, n^{4} + 105258076 \, n^{3} + 150917976 \, n^{2} + 120543840 \, n + 39916800\right )} b^{12} x^{12} + {\left (n^{11} + 55 \, n^{10} + 1320 \, n^{9} + 18150 \, n^{8} + 157773 \, n^{7} + 902055 \, n^{6} + 3416930 \, n^{5} + 8409500 \, n^{4} + 12753576 \, n^{3} + 10628640 \, n^{2} + 3628800 \, n\right )} a b^{11} x^{11} - 11 \, {\left (n^{10} + 45 \, n^{9} + 870 \, n^{8} + 9450 \, n^{7} + 63273 \, n^{6} + 269325 \, n^{5} + 723680 \, n^{4} + 1172700 \, n^{3} + 1026576 \, n^{2} + 362880 \, n\right )} a^{2} b^{10} x^{10} + 110 \, {\left (n^{9} + 36 \, n^{8} + 546 \, n^{7} + 4536 \, n^{6} + 22449 \, n^{5} + 67284 \, n^{4} + 118124 \, n^{3} + 109584 \, n^{2} + 40320 \, n\right )} a^{3} b^{9} x^{9} - 990 \, {\left (n^{8} + 28 \, n^{7} + 322 \, n^{6} + 1960 \, n^{5} + 6769 \, n^{4} + 13132 \, n^{3} + 13068 \, n^{2} + 5040 \, n\right )} a^{4} b^{8} x^{8} + 7920 \, {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a^{5} b^{7} x^{7} - 55440 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{6} b^{6} x^{6} + 332640 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{7} b^{5} x^{5} - 1663200 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{8} b^{4} x^{4} + 6652800 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{9} b^{3} x^{3} - 19958400 \, {\left (n^{2} + n\right )} a^{10} b^{2} x^{2} + 39916800 \, a^{11} b n x - 39916800 \, a^{12}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{12} + 78 \, n^{11} + 2717 \, n^{10} + 55770 \, n^{9} + 749463 \, n^{8} + 6926634 \, n^{7} + 44990231 \, n^{6} + 206070150 \, n^{5} + 657206836 \, n^{4} + 1414014888 \, n^{3} + 1931559552 \, n^{2} + 1486442880 \, n + 479001600\right )} b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^3+c)^3,x, algorithm="maxima")
[Out]
((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c^3/((n^3 + 6*n^2 + 11*n + 6
)*b^3) + 3*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*
a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4
*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a)^n*c^2*d/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n +
720)*b^6) + 3*((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^
9*x^9 + (n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132*n^3 + 13068*n^2 + 5040*n)*a*b^8*x^8 - 8*(n^7 + 2
1*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a^2*b^7*x^7 + 56*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 2
74*n^2 + 120*n)*a^3*b^6*x^6 - 336*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^4*b^5*x^5 + 1680*(n^4 + 6*n^3 + 11
*n^2 + 6*n)*a^5*b^4*x^4 - 6720*(n^3 + 3*n^2 + 2*n)*a^6*b^3*x^3 + 20160*(n^2 + n)*a^7*b^2*x^2 - 40320*a^8*b*n*x
+ 40320*a^9)*(b*x + a)^n*c*d^2/((n^9 + 45*n^8 + 870*n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 11
72700*n^2 + 1026576*n + 362880)*b^9) + ((n^11 + 66*n^10 + 1925*n^9 + 32670*n^8 + 357423*n^7 + 2637558*n^6 + 13
339535*n^5 + 45995730*n^4 + 105258076*n^3 + 150917976*n^2 + 120543840*n + 39916800)*b^12*x^12 + (n^11 + 55*n^1
0 + 1320*n^9 + 18150*n^8 + 157773*n^7 + 902055*n^6 + 3416930*n^5 + 8409500*n^4 + 12753576*n^3 + 10628640*n^2 +
3628800*n)*a*b^11*x^11 - 11*(n^10 + 45*n^9 + 870*n^8 + 9450*n^7 + 63273*n^6 + 269325*n^5 + 723680*n^4 + 11727
00*n^3 + 1026576*n^2 + 362880*n)*a^2*b^10*x^10 + 110*(n^9 + 36*n^8 + 546*n^7 + 4536*n^6 + 22449*n^5 + 67284*n^
4 + 118124*n^3 + 109584*n^2 + 40320*n)*a^3*b^9*x^9 - 990*(n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132
*n^3 + 13068*n^2 + 5040*n)*a^4*b^8*x^8 + 7920*(n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)
*a^5*b^7*x^7 - 55440*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^6*b^6*x^6 + 332640*(n^5 + 10*n^4 +
35*n^3 + 50*n^2 + 24*n)*a^7*b^5*x^5 - 1663200*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^8*b^4*x^4 + 6652800*(n^3 + 3*n^2
+ 2*n)*a^9*b^3*x^3 - 19958400*(n^2 + n)*a^10*b^2*x^2 + 39916800*a^11*b*n*x - 39916800*a^12)*(b*x + a)^n*d^3/((
n^12 + 78*n^11 + 2717*n^10 + 55770*n^9 + 749463*n^8 + 6926634*n^7 + 44990231*n^6 + 206070150*n^5 + 657206836*n
^4 + 1414014888*n^3 + 1931559552*n^2 + 1486442880*n + 479001600)*b^12)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3564 vs.
\(2 (459) = 918\).
time = 0.40, size = 3564, normalized size = 7.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^3+c)^3,x, algorithm="fricas")
[Out]
(2*a^3*b^9*c^3*n^9 + 144*a^3*b^9*c^3*n^8 + 4548*a^3*b^9*c^3*n^7 + 159667200*a^3*b^9*c^3 - 239500800*a^6*b^6*c^
2*d + 159667200*a^9*b^3*c*d^2 - 39916800*a^12*d^3 + (b^12*d^3*n^11 + 66*b^12*d^3*n^10 + 1925*b^12*d^3*n^9 + 32
670*b^12*d^3*n^8 + 357423*b^12*d^3*n^7 + 2637558*b^12*d^3*n^6 + 13339535*b^12*d^3*n^5 + 45995730*b^12*d^3*n^4
+ 105258076*b^12*d^3*n^3 + 150917976*b^12*d^3*n^2 + 120543840*b^12*d^3*n + 39916800*b^12*d^3)*x^12 + (a*b^11*d
^3*n^11 + 55*a*b^11*d^3*n^10 + 1320*a*b^11*d^3*n^9 + 18150*a*b^11*d^3*n^8 + 157773*a*b^11*d^3*n^7 + 902055*a*b
^11*d^3*n^6 + 3416930*a*b^11*d^3*n^5 + 8409500*a*b^11*d^3*n^4 + 12753576*a*b^11*d^3*n^3 + 10628640*a*b^11*d^3*
n^2 + 3628800*a*b^11*d^3*n)*x^11 - 11*(a^2*b^10*d^3*n^10 + 45*a^2*b^10*d^3*n^9 + 870*a^2*b^10*d^3*n^8 + 9450*a
^2*b^10*d^3*n^7 + 63273*a^2*b^10*d^3*n^6 + 269325*a^2*b^10*d^3*n^5 + 723680*a^2*b^10*d^3*n^4 + 1172700*a^2*b^1
0*d^3*n^3 + 1026576*a^2*b^10*d^3*n^2 + 362880*a^2*b^10*d^3*n)*x^10 + (3*b^12*c*d^2*n^11 + 207*b^12*c*d^2*n^10
+ 159667200*b^12*c*d^2 + 2*(3144*b^12*c*d^2 + 55*a^3*b^9*d^3)*n^9 + 18*(6151*b^12*c*d^2 + 220*a^3*b^9*d^3)*n^8
+ 3*(417309*b^12*c*d^2 + 20020*a^3*b^9*d^3)*n^7 + 567*(16777*b^12*c*d^2 + 880*a^3*b^9*d^3)*n^6 + 6*(8226277*b
^12*c*d^2 + 411565*a^3*b^9*d^3)*n^5 + 36*(4833097*b^12*c*d^2 + 205590*a^3*b^9*d^3)*n^4 + 40*(10142427*b^12*c*d
^2 + 324841*a^3*b^9*d^3)*n^3 + 288*(2051288*b^12*c*d^2 + 41855*a^3*b^9*d^3)*n^2 + 5760*(82941*b^12*c*d^2 + 770
*a^3*b^9*d^3)*n)*x^9 + 3*(a*b^11*c*d^2*n^11 + 61*a*b^11*c*d^2*n^10 + 1608*a*b^11*c*d^2*n^9 + 6*(4007*a*b^11*c*
d^2 - 55*a^4*b^8*d^3)*n^8 + 21*(10713*a*b^11*c*d^2 - 440*a^4*b^8*d^3)*n^7 + 21*(65289*a*b^11*c*d^2 - 5060*a^4*
b^8*d^3)*n^6 + 2*(2742001*a*b^11*c*d^2 - 323400*a^4*b^8*d^3)*n^5 + 2*(7062574*a*b^11*c*d^2 - 1116885*a^4*b^8*d
^3)*n^4 + 264*(84209*a*b^11*c*d^2 - 16415*a^4*b^8*d^3)*n^3 + 360*(52984*a*b^11*c*d^2 - 11979*a^4*b^8*d^3)*n^2
+ 1663200*(4*a*b^11*c*d^2 - a^4*b^8*d^3)*n)*x^8 - 24*(a^2*b^10*c*d^2*n^10 + 54*a^2*b^10*c*d^2*n^9 + 1230*a^2*b
^10*c*d^2*n^8 + 6*(2572*a^2*b^10*c*d^2 - 55*a^5*b^7*d^3)*n^7 + 21*(5569*a^2*b^10*c*d^2 - 330*a^5*b^7*d^3)*n^6
+ 42*(13153*a^2*b^10*c*d^2 - 1375*a^5*b^7*d^3)*n^5 + 10*(161702*a^2*b^10*c*d^2 - 24255*a^5*b^7*d^3)*n^4 + 24*(
116917*a^2*b^10*c*d^2 - 22330*a^5*b^7*d^3)*n^3 + 360*(7192*a^2*b^10*c*d^2 - 1617*a^5*b^7*d^3)*n^2 + 237600*(4*
a^2*b^10*c*d^2 - a^5*b^7*d^3)*n)*x^7 + 72*(1148*a^3*b^9*c^3 - 5*a^6*b^6*c^2*d)*n^6 + 3*(b^12*c^2*d*n^11 + 72*b
^12*c^2*d*n^10 + 79833600*b^12*c^2*d + (2285*b^12*c^2*d + 56*a^3*b^9*c*d^2)*n^9 + 12*(3505*b^12*c^2*d + 224*a^
3*b^9*c*d^2)*n^8 + 3*(165701*b^12*c^2*d + 17584*a^3*b^9*c*d^2)*n^7 + 48*(82167*b^12*c^2*d + 11410*a^3*b^9*c*d^
2 - 385*a^6*b^6*d^3)*n^6 + (21326135*b^12*c^2*d + 3263064*a^3*b^9*c*d^2 - 277200*a^6*b^6*d^3)*n^5 + 12*(650944
5*b^12*c^2*d + 946456*a^3*b^9*c*d^2 - 130900*a^6*b^6*d^3)*n^4 + 4*(47131699*b^12*c^2*d + 5602072*a^3*b^9*c*d^2
- 1039500*a^6*b^6*d^3)*n^3 + 96*(2946397*b^12*c^2*d + 236320*a^3*b^9*c*d^2 - 52745*a^6*b^6*d^3)*n^2 + 2880*(8
1401*b^12*c^2*d + 3080*a^3*b^9*c*d^2 - 770*a^6*b^6*d^3)*n)*x^6 + 6*(158683*a^3*b^9*c^3 - 3420*a^6*b^6*c^2*d)*n
^5 + 3*(a*b^11*c^2*d*n^11 + 67*a*b^11*c^2*d*n^10 + 1950*a*b^11*c^2*d*n^9 + 6*(5385*a*b^11*c^2*d - 56*a^4*b^8*c
*d^2)*n^8 + 3*(111851*a*b^11*c^2*d - 4816*a^4*b^8*c*d^2)*n^7 + 3*(755417*a*b^11*c^2*d - 81424*a^4*b^8*c*d^2)*n
^6 + 560*(17848*a*b^11*c^2*d - 3687*a^4*b^8*c*d^2 + 198*a^7*b^5*d^3)*n^5 + 4*(7034735*a*b^11*c^2*d - 2313696*a
^4*b^8*c*d^2 + 277200*a^7*b^5*d^3)*n^4 + 96*(498251*a*b^11*c^2*d - 227822*a^4*b^8*c*d^2 + 40425*a^7*b^5*d^3)*n
^3 + 576*(75857*a*b^11*c^2*d - 43568*a^4*b^8*c*d^2 + 9625*a^7*b^5*d^3)*n^2 + 2661120*(6*a*b^11*c^2*d - 4*a^4*b
^8*c*d^2 + a^7*b^5*d^3)*n)*x^5 + 72*(100058*a^3*b^9*c^3 - 6725*a^6*b^6*c^2*d)*n^4 - 15*(a^2*b^10*c^2*d*n^10 +
63*a^2*b^10*c^2*d*n^9 + 1698*a^2*b^10*c^2*d*n^8 + 6*(4253*a^2*b^10*c^2*d - 56*a^5*b^7*c*d^2)*n^7 + 3*(77827*a^
2*b^10*c^2*d - 4368*a^5*b^7*c*d^2)*n^6 + 3*(444109*a^2*b^10*c^2*d - 63952*a^5*b^7*c*d^2)*n^5 + 4*(1166393*a^2*
b^10*c^2*d - 324324*a^5*b^7*c*d^2 + 27720*a^8*b^4*d^3)*n^4 + 12*(789721*a^2*b^10*c^2*d - 338800*a^5*b^7*c*d^2
+ 55440*a^8*b^4*d^3)*n^3 + 144*(68927*a^2*b^10*c^2*d - 38948*a^5*b^7*c*d^2 + 8470*a^8*b^4*d^3)*n^2 + 665280*(6
*a^2*b^10*c^2*d - 4*a^5*b^7*c*d^2 + a^8*b^4*d^3)*n)*x^4 + 8*(4473299*a^3*b^9*c^3 - 756675*a^6*b^6*c^2*d + 1512
0*a^9*b^3*c*d^2)*n^3 + (b^12*c^3*n^11 + 75*b^12*c^3*n^10 + 159667200*b^12*c^3 + 4*(623*b^12*c^3 + 15*a^3*b^9*c
^2*d)*n^9 + 18*(2683*b^12*c^3 + 200*a^3*b^9*c^2*d)*n^8 + 3*(201527*b^12*c^3 + 30360*a^3*b^9*c^2*d)*n^7 + 9*(56
8099*b^12*c^3 + 139760*a^3*b^9*c^2*d - 2240*a^6*b^6*c*d^2)*n^6 + 2*(14825779*b^12*c^3 + 5117670*a^3*b^9*c^2*d
- 362880*a^6*b^6*c*d^2)*n^5 + 12*(9759623*b^12*c^3 + 4102800*a^3*b^9*c^2*d - 777840*a^6*b^6*c*d^2)*n^4 + 8*(38
232551*b^12*c^3 + 16529190*a^3*b^9*c^2*d - 6229440*a^6*b^6*c*d^2 + 831600*a^9*b^3*d^3)*n^3 + 576*(861864*b^12*
c^3 + 298435*a^3*b^9*c^2*d - 163940*a^6*b^6*c*d^2 + 34650*a^9*b^3*d^3)*n^2 + 5760*(76781*b^12*c^3 + 13860*a^3*
b^9*c^2*d - 9240*a^6*b^6*c*d^2 + 2310*a^9*b^3*d...
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 75191 vs.
\(2 (439) = 878\).
time = 87.44, size = 75191, normalized size = 163.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(b*x+a)**n*(d*x**3+c)**3,x)
[Out]
Piecewise((a**n*(c**3*x**3/3 + c**2*d*x**6/2 + c*d**2*x**9/3 + d**3*x**12/12), Eq(b, 0)), (27720*a**11*d**3*lo
g(a/b + x)/(27720*a**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 914
7600*a**7*b**16*x**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800
*a**3*b**20*x**8 + 1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 83711*a**11*d**3/(277
20*a**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16
*x**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**
8 + 1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 304920*a**10*b*d**3*x*log(a/b + x)/(
27720*a**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b*
*16*x**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*
x**8 + 1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 893101*a**10*b*d**3*x/(27720*a**1
1*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 +
12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 152
4600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 1524600*a**9*b**2*d**3*x**2*log(a/b + x)/(2
7720*a**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**
16*x**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x
**8 + 1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 4313045*a**9*b**2*d**3*x**2/(27720
*a**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x
**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8
+ 1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) - 168*a**8*b**3*c*d**2/(27720*a**11*b**1
2 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 + 12806
640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 1524600*a
**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 4573800*a**8*b**3*d**3*x**3*log(a/b + x)/(27720*a
**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**
4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 +
1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 12430935*a**8*b**3*d**3*x**3/(27720*a**1
1*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 +
12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 152
4600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) - 1848*a**7*b**4*c*d**2*x/(27720*a**11*b**12
+ 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 + 1280664
0*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 1524600*a**
2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 9147600*a**7*b**4*d**3*x**4*log(a/b + x)/(27720*a**
11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4
+ 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 15
24600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 23718420*a**7*b**4*d**3*x**4/(27720*a**11*
b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 + 1
2806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 15246
00*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) - 9240*a**6*b**5*c*d**2*x**2/(27720*a**11*b**12
+ 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 + 128066
40*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 1524600*a*
*2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 12806640*a**6*b**5*d**3*x**5*log(a/b + x)/(27720*a
**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**
4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 +
1524600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 31376268*a**6*b**5*d**3*x**5/(27720*a**1
1*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 +
12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 152
4600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 2...
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^3+c)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Polynomial exponent overflow. Error: Bad Argument Value
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Mupad [B]
time = 7.14, size = 2500, normalized size = 5.45 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(c + d*x^3)^3*(a + b*x)^n,x)
[Out]
(2*a^3*(a + b*x)^n*(79833600*b^9*c^3 - 19958400*a^9*d^3 + 101378880*b^9*c^3*n + 56231712*b^9*c^3*n^2 + 1789319
6*b^9*c^3*n^3 + 3602088*b^9*c^3*n^4 + 476049*b^9*c^3*n^5 + 41328*b^9*c^3*n^6 + 2274*b^9*c^3*n^7 + 72*b^9*c^3*n
^8 + b^9*c^3*n^9 - 119750400*a^3*b^6*c^2*d + 79833600*a^6*b^3*c*d^2 - 78222240*a^3*b^6*c^2*d*n + 21893760*a^6*
b^3*c*d^2*n - 21141720*a^3*b^6*c^2*d*n^2 + 1995840*a^6*b^3*c*d^2*n^2 - 3026700*a^3*b^6*c^2*d*n^3 + 60480*a^6*b
^3*c*d^2*n^3 - 242100*a^3*b^6*c^2*d*n^4 - 10260*a^3*b^6*c^2*d*n^5 - 180*a^3*b^6*c^2*d*n^6))/(b^12*(1486442880*
n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*n^6 + 6926634*n^7 + 749463*n^8
+ 55770*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600)) + (d^3*x^12*(a + b*x)^n*(120543840*n + 150917976*n^2 +
105258076*n^3 + 45995730*n^4 + 13339535*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11
+ 39916800))/(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*n^6 +
6926634*n^7 + 749463*n^8 + 55770*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600) + (x^3*(a + b*x)^n*(3*n + n^2
+ 2)*(79833600*b^9*c^3 + 6652800*a^9*d^3*n + 101378880*b^9*c^3*n + 56231712*b^9*c^3*n^2 + 17893196*b^9*c^3*n^3
+ 3602088*b^9*c^3*n^4 + 476049*b^9*c^3*n^5 + 41328*b^9*c^3*n^6 + 2274*b^9*c^3*n^7 + 72*b^9*c^3*n^8 + b^9*c^3*
n^9 + 39916800*a^3*b^6*c^2*d*n - 26611200*a^6*b^3*c*d^2*n + 26074080*a^3*b^6*c^2*d*n^2 - 7297920*a^6*b^3*c*d^2
*n^2 + 7047240*a^3*b^6*c^2*d*n^3 - 665280*a^6*b^3*c*d^2*n^3 + 1008900*a^3*b^6*c^2*d*n^4 - 20160*a^6*b^3*c*d^2*
n^4 + 80700*a^3*b^6*c^2*d*n^5 + 3420*a^3*b^6*c^2*d*n^6 + 60*a^3*b^6*c^2*d*n^7))/(b^9*(1486442880*n + 193155955
2*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9 +
2717*n^10 + 78*n^11 + n^12 + 479001600)) + (3*d*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 12
0)*(665280*b^6*c^2 - 18480*a^6*d^2*n + 434568*b^6*c^2*n + 117454*b^6*c^2*n^2 + 16815*b^6*c^2*n^3 + 1345*b^6*c^
2*n^4 + 57*b^6*c^2*n^5 + b^6*c^2*n^6 + 73920*a^3*b^3*c*d*n + 20272*a^3*b^3*c*d*n^2 + 1848*a^3*b^3*c*d*n^3 + 56
*a^3*b^3*c*d*n^4))/(b^6*(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 4499
0231*n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600)) - (2*a^2*n*x*(a + b
*x)^n*(79833600*b^9*c^3 - 19958400*a^9*d^3 + 101378880*b^9*c^3*n + 56231712*b^9*c^3*n^2 + 17893196*b^9*c^3*n^3
+ 3602088*b^9*c^3*n^4 + 476049*b^9*c^3*n^5 + 41328*b^9*c^3*n^6 + 2274*b^9*c^3*n^7 + 72*b^9*c^3*n^8 + b^9*c^3*
n^9 - 119750400*a^3*b^6*c^2*d + 79833600*a^6*b^3*c*d^2 - 78222240*a^3*b^6*c^2*d*n + 21893760*a^6*b^3*c*d^2*n -
21141720*a^3*b^6*c^2*d*n^2 + 1995840*a^6*b^3*c*d^2*n^2 - 3026700*a^3*b^6*c^2*d*n^3 + 60480*a^6*b^3*c*d^2*n^3
- 242100*a^3*b^6*c^2*d*n^4 - 10260*a^3*b^6*c^2*d*n^5 - 180*a^3*b^6*c^2*d*n^6))/(b^11*(1486442880*n + 193155955
2*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9 +
2717*n^10 + 78*n^11 + n^12 + 479001600)) + (d^2*x^9*(a + b*x)^n*(3960*b^3*c + 99*b^3*c*n^2 + 3*b^3*c*n^3 + 11
0*a^3*d*n + 1086*b^3*c*n)*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 +
40320))/(b^3*(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*n^6 +
6926634*n^7 + 749463*n^8 + 55770*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600)) + (a*d^3*n*x^11*(a + b*x)^n*(
10628640*n + 12753576*n^2 + 8409500*n^3 + 3416930*n^4 + 902055*n^5 + 157773*n^6 + 18150*n^7 + 1320*n^8 + 55*n^
9 + n^10 + 3628800))/(b*(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 4499
0231*n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600)) - (11*a^2*d^3*n*x^1
0*(a + b*x)^n*(1026576*n + 1172700*n^2 + 723680*n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n
^9 + 362880))/(b^2*(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*
n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600)) + (a*n*x^2*(n + 1)*(a +
b*x)^n*(79833600*b^9*c^3 - 19958400*a^9*d^3 + 101378880*b^9*c^3*n + 56231712*b^9*c^3*n^2 + 17893196*b^9*c^3*n^
3 + 3602088*b^9*c^3*n^4 + 476049*b^9*c^3*n^5 + 41328*b^9*c^3*n^6 + 2274*b^9*c^3*n^7 + 72*b^9*c^3*n^8 + b^9*c^3
*n^9 - 119750400*a^3*b^6*c^2*d + 79833600*a^6*b^3*c*d^2 - 78222240*a^3*b^6*c^2*d*n + 21893760*a^6*b^3*c*d^2*n
- 21141720*a^3*b^6*c^2*d*n^2 + 1995840*a^6*b^3*c*d^2*n^2 - 3026700*a^3*b^6*c^2*d*n^3 + 60480*a^6*b^3*c*d^2*n^3
- 242100*a^3*b^6*c^2*d*n^4 - 10260*a^3*b^6*c^2*d*n^5 - 180*a^3*b^6*c^2*d*n^6))/(b^10*(1486442880*n + 19315595
52*n^2 + 1414014888*n^3 + 657206836*n^4 + 206070150*n^5 + 44990231*n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9
+ 2717*n^10 + 78*n^11 + n^12 + 479001600)) + (3*a*d*n*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)*(110
880*a^6*d^2 + 665280*b^6*c^2 + 434568*b^6*c^2*n + 117454*b^6*c^2*n^2 + 16815*b^6*c^2*n^3 + 1345*b^6*c^2*n^4 +
57*b^6*c^2*n^5 + b^6*c^2*n^6 - 443520*a^3*b^3*c...
________________________________________________________________________________________