3.160 \(\int x (a+b x)^n \left (c+d x^3\right )^3 \, dx\)

Optimal. Leaf size=396 \[ -\frac{21 a d^2 \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+7}}{b^{11} (n+7)}+\frac{3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{n+8}}{b^{11} (n+8)}-\frac{a \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{11} (n+1)}+\frac{\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{11} (n+2)}+\frac{45 a^2 d^3 (a+b x)^{n+9}}{b^{11} (n+9)}-\frac{3 a d \left (40 a^6 d^2-35 a^3 b^3 c d+4 b^6 c^2\right ) (a+b x)^{n+4}}{b^{11} (n+4)}+\frac{3 d \left (70 a^6 d^2-35 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{11} (n+5)}+\frac{63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^{n+6}}{b^{11} (n+6)}+\frac{9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{11} (n+3)}-\frac{10 a d^3 (a+b x)^{n+10}}{b^{11} (n+10)}+\frac{d^3 (a+b x)^{n+11}}{b^{11} (n+11)} \]

[Out]

-((a*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^11*(1 + n))) + ((b^3*c - 10*a^3*d)*
(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^11*(2 + n)) + (9*a^2*d*(2*b^3*c - 5*a^3*
d)*(b^3*c - a^3*d)*(a + b*x)^(3 + n))/(b^11*(3 + n)) - (3*a*d*(4*b^6*c^2 - 35*a^
3*b^3*c*d + 40*a^6*d^2)*(a + b*x)^(4 + n))/(b^11*(4 + n)) + (3*d*(b^6*c^2 - 35*a
^3*b^3*c*d + 70*a^6*d^2)*(a + b*x)^(5 + n))/(b^11*(5 + n)) + (63*a^2*d^2*(b^3*c
- 4*a^3*d)*(a + b*x)^(6 + n))/(b^11*(6 + n)) - (21*a*d^2*(b^3*c - 10*a^3*d)*(a +
b*x)^(7 + n))/(b^11*(7 + n)) + (3*d^2*(b^3*c - 40*a^3*d)*(a + b*x)^(8 + n))/(b^
11*(8 + n)) + (45*a^2*d^3*(a + b*x)^(9 + n))/(b^11*(9 + n)) - (10*a*d^3*(a + b*x
)^(10 + n))/(b^11*(10 + n)) + (d^3*(a + b*x)^(11 + n))/(b^11*(11 + n))

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Rubi [A]  time = 0.587117, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{21 a d^2 \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+7}}{b^{11} (n+7)}+\frac{3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{n+8}}{b^{11} (n+8)}-\frac{a \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{11} (n+1)}+\frac{\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{11} (n+2)}+\frac{45 a^2 d^3 (a+b x)^{n+9}}{b^{11} (n+9)}-\frac{3 a d \left (40 a^6 d^2-35 a^3 b^3 c d+4 b^6 c^2\right ) (a+b x)^{n+4}}{b^{11} (n+4)}+\frac{3 d \left (70 a^6 d^2-35 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{11} (n+5)}+\frac{63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^{n+6}}{b^{11} (n+6)}+\frac{9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{11} (n+3)}-\frac{10 a d^3 (a+b x)^{n+10}}{b^{11} (n+10)}+\frac{d^3 (a+b x)^{n+11}}{b^{11} (n+11)} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^n*(c + d*x^3)^3,x]

[Out]

-((a*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^11*(1 + n))) + ((b^3*c - 10*a^3*d)*
(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^11*(2 + n)) + (9*a^2*d*(2*b^3*c - 5*a^3*
d)*(b^3*c - a^3*d)*(a + b*x)^(3 + n))/(b^11*(3 + n)) - (3*a*d*(4*b^6*c^2 - 35*a^
3*b^3*c*d + 40*a^6*d^2)*(a + b*x)^(4 + n))/(b^11*(4 + n)) + (3*d*(b^6*c^2 - 35*a
^3*b^3*c*d + 70*a^6*d^2)*(a + b*x)^(5 + n))/(b^11*(5 + n)) + (63*a^2*d^2*(b^3*c
- 4*a^3*d)*(a + b*x)^(6 + n))/(b^11*(6 + n)) - (21*a*d^2*(b^3*c - 10*a^3*d)*(a +
b*x)^(7 + n))/(b^11*(7 + n)) + (3*d^2*(b^3*c - 40*a^3*d)*(a + b*x)^(8 + n))/(b^
11*(8 + n)) + (45*a^2*d^3*(a + b*x)^(9 + n))/(b^11*(9 + n)) - (10*a*d^3*(a + b*x
)^(10 + n))/(b^11*(10 + n)) + (d^3*(a + b*x)^(11 + n))/(b^11*(11 + n))

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Rubi in Sympy [A]  time = 121.254, size = 374, normalized size = 0.94 \[ \frac{45 a^{2} d^{3} \left (a + b x\right )^{n + 9}}{b^{11} \left (n + 9\right )} - \frac{63 a^{2} d^{2} \left (a + b x\right )^{n + 6} \left (4 a^{3} d - b^{3} c\right )}{b^{11} \left (n + 6\right )} + \frac{9 a^{2} d \left (a + b x\right )^{n + 3} \left (a^{3} d - b^{3} c\right ) \left (5 a^{3} d - 2 b^{3} c\right )}{b^{11} \left (n + 3\right )} - \frac{10 a d^{3} \left (a + b x\right )^{n + 10}}{b^{11} \left (n + 10\right )} + \frac{21 a d^{2} \left (a + b x\right )^{n + 7} \left (10 a^{3} d - b^{3} c\right )}{b^{11} \left (n + 7\right )} - \frac{3 a d \left (a + b x\right )^{n + 4} \left (40 a^{6} d^{2} - 35 a^{3} b^{3} c d + 4 b^{6} c^{2}\right )}{b^{11} \left (n + 4\right )} + \frac{a \left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )^{3}}{b^{11} \left (n + 1\right )} + \frac{d^{3} \left (a + b x\right )^{n + 11}}{b^{11} \left (n + 11\right )} - \frac{3 d^{2} \left (a + b x\right )^{n + 8} \left (40 a^{3} d - b^{3} c\right )}{b^{11} \left (n + 8\right )} + \frac{3 d \left (a + b x\right )^{n + 5} \left (70 a^{6} d^{2} - 35 a^{3} b^{3} c d + b^{6} c^{2}\right )}{b^{11} \left (n + 5\right )} - \frac{\left (a + b x\right )^{n + 2} \left (a^{3} d - b^{3} c\right )^{2} \left (10 a^{3} d - b^{3} c\right )}{b^{11} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**n*(d*x**3+c)**3,x)

[Out]

45*a**2*d**3*(a + b*x)**(n + 9)/(b**11*(n + 9)) - 63*a**2*d**2*(a + b*x)**(n + 6
)*(4*a**3*d - b**3*c)/(b**11*(n + 6)) + 9*a**2*d*(a + b*x)**(n + 3)*(a**3*d - b*
*3*c)*(5*a**3*d - 2*b**3*c)/(b**11*(n + 3)) - 10*a*d**3*(a + b*x)**(n + 10)/(b**
11*(n + 10)) + 21*a*d**2*(a + b*x)**(n + 7)*(10*a**3*d - b**3*c)/(b**11*(n + 7))
- 3*a*d*(a + b*x)**(n + 4)*(40*a**6*d**2 - 35*a**3*b**3*c*d + 4*b**6*c**2)/(b**
11*(n + 4)) + a*(a + b*x)**(n + 1)*(a**3*d - b**3*c)**3/(b**11*(n + 1)) + d**3*(
a + b*x)**(n + 11)/(b**11*(n + 11)) - 3*d**2*(a + b*x)**(n + 8)*(40*a**3*d - b**
3*c)/(b**11*(n + 8)) + 3*d*(a + b*x)**(n + 5)*(70*a**6*d**2 - 35*a**3*b**3*c*d +
b**6*c**2)/(b**11*(n + 5)) - (a + b*x)**(n + 2)*(a**3*d - b**3*c)**2*(10*a**3*d
- b**3*c)/(b**11*(n + 2))

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Mathematica [B]  time = 1.34314, size = 903, normalized size = 2.28 \[ \frac{(a+b x)^{n+1} \left (3628800 d^3 a^{10}-3628800 b d^3 (n+1) x a^9+1814400 b^2 d^3 \left (n^2+3 n+2\right ) x^2 a^8-15120 b^3 d^2 \left (40 d \left (n^3+6 n^2+11 n+6\right ) x^3+c \left (n^3+30 n^2+299 n+990\right )\right ) a^7+15120 b^4 d^2 (n+1) x \left (10 d \left (n^3+9 n^2+26 n+24\right ) x^3+c \left (n^3+30 n^2+299 n+990\right )\right ) a^6-7560 b^5 d^2 \left (n^2+3 n+2\right ) x^2 \left (4 d \left (n^3+12 n^2+47 n+60\right ) x^3+c \left (n^3+30 n^2+299 n+990\right )\right ) a^5+72 b^6 d \left (70 d^2 \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^6+35 c d \left (n^6+36 n^5+490 n^4+3120 n^3+9409 n^2+12684 n+5940\right ) x^3+c^2 \left (n^6+51 n^5+1075 n^4+11985 n^3+74524 n^2+245004 n+332640\right )\right ) a^4-18 b^7 d (n+1) x \left (40 d^2 \left (n^6+27 n^5+295 n^4+1665 n^3+5104 n^2+8028 n+5040\right ) x^6+35 c d \left (n^6+39 n^5+595 n^4+4485 n^3+17404 n^2+32916 n+23760\right ) x^3+4 c^2 \left (n^6+51 n^5+1075 n^4+11985 n^3+74524 n^2+245004 n+332640\right )\right ) a^3+18 b^8 d \left (n^2+3 n+2\right ) x^2 \left (5 d^2 \left (n^6+33 n^5+445 n^4+3135 n^3+12154 n^2+24552 n+20160\right ) x^6+7 c d \left (n^6+42 n^5+706 n^4+6048 n^3+27733 n^2+64470 n+59400\right ) x^3+2 c^2 \left (n^6+51 n^5+1075 n^4+11985 n^3+74524 n^2+245004 n+332640\right )\right ) a^2-b^9 \left (n^3+18 n^2+99 n+162\right ) \left (10 d^3 \left (n^6+27 n^5+285 n^4+1485 n^3+3954 n^2+4968 n+2240\right ) x^9+21 c d^2 \left (n^6+33 n^5+411 n^4+2427 n^3+7068 n^2+9420 n+4400\right ) x^6+12 c^2 d \left (n^6+39 n^5+591 n^4+4341 n^3+15600 n^2+24132 n+12320\right ) x^3+c^3 \left (n^6+45 n^5+825 n^4+7875 n^3+41214 n^2+111960 n+123200\right )\right ) a+b^{10} \left (n^7+40 n^6+654 n^5+5620 n^4+27109 n^3+72180 n^2+95436 n+45360\right ) x \left (d^3 \left (n^3+15 n^2+66 n+80\right ) x^9+3 c d^2 \left (n^3+18 n^2+87 n+110\right ) x^6+3 c^2 d \left (n^3+21 n^2+126 n+176\right ) x^3+c^3 \left (n^3+24 n^2+183 n+440\right )\right )\right )}{b^{11} (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9) (n+10) (n+11)} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^n*(c + d*x^3)^3,x]

[Out]

((a + b*x)^(1 + n)*(3628800*a^10*d^3 - 3628800*a^9*b*d^3*(1 + n)*x + 1814400*a^8
*b^2*d^3*(2 + 3*n + n^2)*x^2 - 15120*a^7*b^3*d^2*(c*(990 + 299*n + 30*n^2 + n^3)
+ 40*d*(6 + 11*n + 6*n^2 + n^3)*x^3) + 15120*a^6*b^4*d^2*(1 + n)*x*(c*(990 + 29
9*n + 30*n^2 + n^3) + 10*d*(24 + 26*n + 9*n^2 + n^3)*x^3) - 7560*a^5*b^5*d^2*(2
+ 3*n + n^2)*x^2*(c*(990 + 299*n + 30*n^2 + n^3) + 4*d*(60 + 47*n + 12*n^2 + n^3
)*x^3) + 72*a^4*b^6*d*(c^2*(332640 + 245004*n + 74524*n^2 + 11985*n^3 + 1075*n^4
+ 51*n^5 + n^6) + 35*c*d*(5940 + 12684*n + 9409*n^2 + 3120*n^3 + 490*n^4 + 36*n
^5 + n^6)*x^3 + 70*d^2*(720 + 1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n
^6)*x^6) - 18*a^3*b^7*d*(1 + n)*x*(4*c^2*(332640 + 245004*n + 74524*n^2 + 11985*
n^3 + 1075*n^4 + 51*n^5 + n^6) + 35*c*d*(23760 + 32916*n + 17404*n^2 + 4485*n^3
+ 595*n^4 + 39*n^5 + n^6)*x^3 + 40*d^2*(5040 + 8028*n + 5104*n^2 + 1665*n^3 + 29
5*n^4 + 27*n^5 + n^6)*x^6) + 18*a^2*b^8*d*(2 + 3*n + n^2)*x^2*(2*c^2*(332640 + 2
45004*n + 74524*n^2 + 11985*n^3 + 1075*n^4 + 51*n^5 + n^6) + 7*c*d*(59400 + 6447
0*n + 27733*n^2 + 6048*n^3 + 706*n^4 + 42*n^5 + n^6)*x^3 + 5*d^2*(20160 + 24552*
n + 12154*n^2 + 3135*n^3 + 445*n^4 + 33*n^5 + n^6)*x^6) + b^10*(45360 + 95436*n
+ 72180*n^2 + 27109*n^3 + 5620*n^4 + 654*n^5 + 40*n^6 + n^7)*x*(c^3*(440 + 183*n
+ 24*n^2 + n^3) + 3*c^2*d*(176 + 126*n + 21*n^2 + n^3)*x^3 + 3*c*d^2*(110 + 87*
n + 18*n^2 + n^3)*x^6 + d^3*(80 + 66*n + 15*n^2 + n^3)*x^9) - a*b^9*(162 + 99*n
+ 18*n^2 + n^3)*(c^3*(123200 + 111960*n + 41214*n^2 + 7875*n^3 + 825*n^4 + 45*n^
5 + n^6) + 12*c^2*d*(12320 + 24132*n + 15600*n^2 + 4341*n^3 + 591*n^4 + 39*n^5 +
n^6)*x^3 + 21*c*d^2*(4400 + 9420*n + 7068*n^2 + 2427*n^3 + 411*n^4 + 33*n^5 + n
^6)*x^6 + 10*d^3*(2240 + 4968*n + 3954*n^2 + 1485*n^3 + 285*n^4 + 27*n^5 + n^6)*
x^9)))/(b^11*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n)*(9
+ n)*(10 + n)*(11 + n))

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Maple [B]  time = 0.033, size = 2972, normalized size = 7.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n*(d*x^3+c)^3,x)

[Out]

(b*x+a)^(1+n)*(b^10*d^3*n^10*x^10+55*b^10*d^3*n^9*x^10-10*a*b^9*d^3*n^9*x^9+1320
*b^10*d^3*n^8*x^10-450*a*b^9*d^3*n^8*x^9+3*b^10*c*d^2*n^10*x^7+18150*b^10*d^3*n^
7*x^10+90*a^2*b^8*d^3*n^8*x^8-8700*a*b^9*d^3*n^7*x^9+174*b^10*c*d^2*n^9*x^7+1577
73*b^10*d^3*n^6*x^10+3240*a^2*b^8*d^3*n^7*x^8-21*a*b^9*c*d^2*n^9*x^6-94500*a*b^9
*d^3*n^6*x^9+4383*b^10*c*d^2*n^8*x^7+902055*b^10*d^3*n^5*x^10-720*a^3*b^7*d^3*n^
7*x^7+49140*a^2*b^8*d^3*n^6*x^8-1071*a*b^9*c*d^2*n^8*x^6-632730*a*b^9*d^3*n^5*x^
9+3*b^10*c^2*d*n^10*x^4+62946*b^10*c*d^2*n^7*x^7+3416930*b^10*d^3*n^4*x^10-20160
*a^3*b^7*d^3*n^6*x^7+126*a^2*b^8*c*d^2*n^8*x^5+408240*a^2*b^8*d^3*n^5*x^8-23184*
a*b^9*c*d^2*n^7*x^6-2693250*a*b^9*d^3*n^4*x^9+183*b^10*c^2*d*n^9*x^4+568701*b^10
*c*d^2*n^6*x^7+8409500*b^10*d^3*n^3*x^10+5040*a^4*b^6*d^3*n^6*x^6-231840*a^3*b^7
*d^3*n^5*x^7+5670*a^2*b^8*c*d^2*n^7*x^5+2020410*a^2*b^8*d^3*n^4*x^8-12*a*b^9*c^2
*d*n^9*x^3-278334*a*b^9*c*d^2*n^6*x^6-7236800*a*b^9*d^3*n^3*x^9+4860*b^10*c^2*d*
n^8*x^4+3363066*b^10*c*d^2*n^5*x^7+12753576*b^10*d^3*n^2*x^10+105840*a^4*b^6*d^3
*n^5*x^6-630*a^3*b^7*c*d^2*n^7*x^4-1411200*a^3*b^7*d^3*n^4*x^7+105084*a^2*b^8*c*
d^2*n^6*x^5+6055560*a^2*b^8*d^3*n^3*x^8-684*a*b^9*c^2*d*n^8*x^3-2032569*a*b^9*c*
d^2*n^5*x^6-11727000*a*b^9*d^3*n^2*x^9+b^10*c^3*n^10*x+73710*b^10*c^2*d*n^7*x^4+
13114077*b^10*c*d^2*n^4*x^7+10628640*b^10*d^3*n*x^10-30240*a^5*b^5*d^3*n^5*x^5+8
82000*a^4*b^6*d^3*n^4*x^6-25200*a^3*b^7*c*d^2*n^6*x^4-4873680*a^3*b^7*d^3*n^3*x^
7+36*a^2*b^8*c^2*d*n^8*x^2+1039500*a^2*b^8*c*d^2*n^5*x^5+10631160*a^2*b^8*d^3*n^
2*x^8-16704*a*b^9*c^2*d*n^7*x^3-9313479*a*b^9*c*d^2*n^4*x^6-10265760*a*b^9*d^3*n
*x^9+64*b^10*c^3*n^9*x+703719*b^10*c^2*d*n^6*x^4+33074574*b^10*c*d^2*n^3*x^7+362
8800*b^10*d^3*x^10-453600*a^5*b^5*d^3*n^4*x^5+2520*a^4*b^6*c*d^2*n^6*x^3+3704400
*a^4*b^6*d^3*n^3*x^6-399420*a^3*b^7*c*d^2*n^5*x^4-9455040*a^3*b^7*d^3*n^2*x^7+19
44*a^2*b^8*c^2*d*n^7*x^2+5958414*a^2*b^8*c*d^2*n^4*x^5+9862560*a^2*b^8*d^3*n*x^8
-a*b^9*c^3*n^9-228024*a*b^9*c^2*d*n^6*x^3-26604186*a*b^9*c*d^2*n^3*x^6-3628800*a
*b^9*d^3*x^9+1797*b^10*c^3*n^8*x+4394079*b^10*c^2*d*n^5*x^4+51177636*b^10*c*d^2*
n^2*x^7+151200*a^6*b^4*d^3*n^4*x^4-2570400*a^5*b^5*d^3*n^3*x^5+90720*a^4*b^6*c*d
^2*n^5*x^3+8184960*a^4*b^6*d^3*n^2*x^6-72*a^3*b^7*c^2*d*n^7*x-3200400*a^3*b^7*c*
d^2*n^4*x^4-9408960*a^3*b^7*d^3*n*x^7+44280*a^2*b^8*c^2*d*n^6*x^2+20130390*a^2*b
^8*c*d^2*n^3*x^5+3628800*a^2*b^8*d^3*x^8-63*a*b^9*c^3*n^8-1902780*a*b^9*c^2*d*n^
5*x^3-45292716*a*b^9*c*d^2*n^2*x^6+29076*b^10*c^3*n^7*x+18048210*b^10*c^2*d*n^4*
x^4+43332840*b^10*c*d^2*n*x^7+1512000*a^6*b^4*d^3*n^3*x^4-7560*a^5*b^5*c*d^2*n^5
*x^2-6804000*a^5*b^5*d^3*n^2*x^5+1234800*a^4*b^6*c*d^2*n^4*x^3+8890560*a^4*b^6*d
^3*n*x^6-3744*a^3*b^7*c^2*d*n^6*x-13790070*a^3*b^7*c*d^2*n^3*x^4-3628800*a^3*b^7
*d^3*x^7+551232*a^2*b^8*c^2*d*n^5*x^2+38842776*a^2*b^8*c*d^2*n^2*x^5-1734*a*b^9*
c^3*n^7-9965196*a*b^9*c^2*d*n^4*x^3-41194440*a*b^9*c*d^2*n*x^6+299271*b^10*c^3*n
^6*x+47746140*b^10*c^2*d*n^3*x^4+14968800*b^10*c*d^2*x^7-604800*a^7*b^3*d^3*n^3*
x^3+5292000*a^6*b^4*d^3*n^2*x^4-249480*a^5*b^5*c*d^2*n^4*x^2-8285760*a^5*b^5*d^3
*n*x^5+72*a^4*b^6*c^2*d*n^6+7862400*a^4*b^6*c*d^2*n^3*x^3+3628800*a^4*b^6*d^3*x^
6-81072*a^3*b^7*c^2*d*n^5*x-31701600*a^3*b^7*c*d^2*n^2*x^4+4054644*a^2*b^8*c^2*d
*n^4*x^2+38699640*a^2*b^8*c*d^2*n*x^5-27342*a*b^9*c^3*n^6-32332056*a*b^9*c^2*d*n
^3*x^3-14968800*a*b^9*c*d^2*x^6+2039016*b^10*c^3*n^5*x+77043528*b^10*c^2*d*n^2*x
^4-3628800*a^7*b^3*d^3*n^2*x^3+15120*a^6*b^4*c*d^2*n^4*x+7560000*a^6*b^4*d^3*n*x
^4-2955960*a^5*b^5*c*d^2*n^3*x^2-3628800*a^5*b^5*d^3*x^5+3672*a^4*b^6*c^2*d*n^5+
23710680*a^4*b^6*c*d^2*n^2*x^3-940320*a^3*b^7*c^2*d*n^4*x-35705880*a^3*b^7*c*d^2
*n*x^4+17731656*a^2*b^8*c^2*d*n^3*x^2+14968800*a^2*b^8*c*d^2*x^5-271929*a*b^9*c^
3*n^5-61656336*a*b^9*c^2*d*n^2*x^3+9261503*b^10*c^3*n^4*x+67536288*b^10*c^2*d*n*
x^4+1814400*a^8*b^2*d^3*n^2*x^2-6652800*a^7*b^3*d^3*n*x^3+468720*a^6*b^4*c*d^2*n
^3*x+3628800*a^6*b^4*d^3*x^4-14719320*a^5*b^5*c*d^2*n^2*x^2+77400*a^4*b^6*c^2*d*
n^4+31963680*a^4*b^6*c*d^2*n*x^3-6228648*a^3*b^7*c^2*d*n^3*x-14968800*a^3*b^7*c*
d^2*x^4+43801200*a^2*b^8*c^2*d*n^2*x^2-1767087*a*b^9*c^3*n^4-61548768*a*b^9*c^2*
d*n*x^3+27472724*b^10*c^3*n^3*x+23950080*b^10*c^2*d*x^4+5443200*a^8*b^2*d^3*n*x^
2-15120*a^7*b^3*c*d^2*n^3-3628800*a^7*b^3*d^3*x^3+4974480*a^6*b^4*c*d^2*n^2*x-26
974080*a^5*b^5*c*d^2*n*x^2+862920*a^4*b^6*c^2*d*n^3+14968800*a^4*b^6*c*d^2*x^3-2
3006016*a^3*b^7*c^2*d*n^2*x+53565408*a^2*b^8*c^2*d*n*x^2-7494416*a*b^9*c^3*n^3-2
3950080*a*b^9*c^2*d*x^3+50312628*b^10*c^3*n^2*x-3628800*a^9*b*d^3*n*x+3628800*a^
8*b^2*d^3*x^2-453600*a^7*b^3*c*d^2*n^2+19489680*a^6*b^4*c*d^2*n*x-14968800*a^5*b
^5*c*d^2*x^2+5365728*a^4*b^6*c^2*d*n^2-41590368*a^3*b^7*c^2*d*n*x+23950080*a^2*b
^8*c^2*d*x^2-19978308*a*b^9*c^3*n^2+50292720*b^10*c^3*n*x-3628800*a^9*b*d^3*x-45
20880*a^7*b^3*c*d^2*n+14968800*a^6*b^4*c*d^2*x+17640288*a^4*b^6*c^2*d*n-23950080
*a^3*b^7*c^2*d*x-30334320*a*b^9*c^3*n+19958400*b^10*c^3*x+3628800*a^10*d^3-14968
800*a^7*b^3*c*d^2+23950080*a^4*b^6*c^2*d-19958400*a*b^9*c^3)/b^11/(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)

_______________________________________________________________________________________

Maxima [A]  time = 0.740581, size = 1287, normalized size = 3.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^3*(b*x + a)^n*x,x, algorithm="maxima")

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^3/((n^2 + 3*n + 2)*b^2) + 3*((n^
4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^
4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x
+ 24*a^5)*(b*x + a)^n*c^2*d/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5
) + 3*((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 504
0)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*
b^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^2*b^6*x^6 + 42
*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^3*b^5*x^5 - 210*(n^4 + 6*n^3 + 11*n^2
+ 6*n)*a^4*b^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b
^2*x^2 + 5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*c*d^2/((n^8 + 36*n^7 + 546*n^6 +
4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^8) + ((n^10
+ 55*n^9 + 1320*n^8 + 18150*n^7 + 157773*n^6 + 902055*n^5 + 3416930*n^4 + 84095
00*n^3 + 12753576*n^2 + 10628640*n + 3628800)*b^11*x^11 + (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)*a*b^10*x^10 - 10*(n^9 + 36*n^8 + 546*n^7 + 4536*n^6 + 22449*n^5 + 67
284*n^4 + 118124*n^3 + 109584*n^2 + 40320*n)*a^2*b^9*x^9 + 90*(n^8 + 28*n^7 + 32
2*n^6 + 1960*n^5 + 6769*n^4 + 13132*n^3 + 13068*n^2 + 5040*n)*a^3*b^8*x^8 - 720*
(n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a^4*b^7*x^7 + 5
040*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^5*b^6*x^6 - 30240*(n^5
+ 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^6*b^5*x^5 + 151200*(n^4 + 6*n^3 + 11*n^2 +
6*n)*a^7*b^4*x^4 - 604800*(n^3 + 3*n^2 + 2*n)*a^8*b^3*x^3 + 1814400*(n^2 + n)*a
^9*b^2*x^2 - 3628800*a^10*b*n*x + 3628800*a^11)*(b*x + a)^n*d^3/((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)*b^11)

_______________________________________________________________________________________

Fricas [A]  time = 0.323419, size = 3941, normalized size = 9.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^3*(b*x + a)^n*x,x, algorithm="fricas")

[Out]

-(a^2*b^9*c^3*n^9 + 63*a^2*b^9*c^3*n^8 + 1734*a^2*b^9*c^3*n^7 + 19958400*a^2*b^9
*c^3 - 23950080*a^5*b^6*c^2*d + 14968800*a^8*b^3*c*d^2 - 3628800*a^11*d^3 - (b^1
1*d^3*n^10 + 55*b^11*d^3*n^9 + 1320*b^11*d^3*n^8 + 18150*b^11*d^3*n^7 + 157773*b
^11*d^3*n^6 + 902055*b^11*d^3*n^5 + 3416930*b^11*d^3*n^4 + 8409500*b^11*d^3*n^3
+ 12753576*b^11*d^3*n^2 + 10628640*b^11*d^3*n + 3628800*b^11*d^3)*x^11 - (a*b^10
*d^3*n^10 + 45*a*b^10*d^3*n^9 + 870*a*b^10*d^3*n^8 + 9450*a*b^10*d^3*n^7 + 63273
*a*b^10*d^3*n^6 + 269325*a*b^10*d^3*n^5 + 723680*a*b^10*d^3*n^4 + 1172700*a*b^10
*d^3*n^3 + 1026576*a*b^10*d^3*n^2 + 362880*a*b^10*d^3*n)*x^10 + 10*(a^2*b^9*d^3*
n^9 + 36*a^2*b^9*d^3*n^8 + 546*a^2*b^9*d^3*n^7 + 4536*a^2*b^9*d^3*n^6 + 22449*a^
2*b^9*d^3*n^5 + 67284*a^2*b^9*d^3*n^4 + 118124*a^2*b^9*d^3*n^3 + 109584*a^2*b^9*
d^3*n^2 + 40320*a^2*b^9*d^3*n)*x^9 - 3*(b^11*c*d^2*n^10 + 58*b^11*c*d^2*n^9 + 49
89600*b^11*c*d^2 + 3*(487*b^11*c*d^2 + 10*a^3*b^8*d^3)*n^8 + 6*(3497*b^11*c*d^2
+ 140*a^3*b^8*d^3)*n^7 + 21*(9027*b^11*c*d^2 + 460*a^3*b^8*d^3)*n^6 + 294*(3813*
b^11*c*d^2 + 200*a^3*b^8*d^3)*n^5 + (4371359*b^11*c*d^2 + 203070*a^3*b^8*d^3)*n^
4 + 2*(5512429*b^11*c*d^2 + 196980*a^3*b^8*d^3)*n^3 + 36*(473867*b^11*c*d^2 + 10
890*a^3*b^8*d^3)*n^2 + 360*(40123*b^11*c*d^2 + 420*a^3*b^8*d^3)*n)*x^8 - 3*(a*b^
10*c*d^2*n^10 + 51*a*b^10*c*d^2*n^9 + 1104*a*b^10*c*d^2*n^8 + 6*(2209*a*b^10*c*d
^2 - 40*a^4*b^7*d^3)*n^7 + 21*(4609*a*b^10*c*d^2 - 240*a^4*b^7*d^3)*n^6 + 21*(21
119*a*b^10*c*d^2 - 2000*a^4*b^7*d^3)*n^5 + 2*(633433*a*b^10*c*d^2 - 88200*a^4*b^
7*d^3)*n^4 + 12*(179733*a*b^10*c*d^2 - 32480*a^4*b^7*d^3)*n^3 + 360*(5449*a*b^10
*c*d^2 - 1176*a^4*b^7*d^3)*n^2 + 21600*(33*a*b^10*c*d^2 - 8*a^4*b^7*d^3)*n)*x^7
+ 18*(1519*a^2*b^9*c^3 - 4*a^5*b^6*c^2*d)*n^6 + 21*(a^2*b^9*c*d^2*n^9 + 45*a^2*b
^9*c*d^2*n^8 + 834*a^2*b^9*c*d^2*n^7 + 30*(275*a^2*b^9*c*d^2 - 8*a^5*b^6*d^3)*n^
6 + 3*(15763*a^2*b^9*c*d^2 - 1200*a^5*b^6*d^3)*n^5 + 15*(10651*a^2*b^9*c*d^2 - 1
360*a^5*b^6*d^3)*n^4 + 4*(77069*a^2*b^9*c*d^2 - 13500*a^5*b^6*d^3)*n^3 + 60*(511
9*a^2*b^9*c*d^2 - 1096*a^5*b^6*d^3)*n^2 + 3600*(33*a^2*b^9*c*d^2 - 8*a^5*b^6*d^3
)*n)*x^6 + 3*(90643*a^2*b^9*c^3 - 1224*a^5*b^6*c^2*d)*n^5 - 3*(b^11*c^2*d*n^10 +
61*b^11*c^2*d*n^9 + 7983360*b^11*c^2*d + 6*(270*b^11*c^2*d + 7*a^3*b^8*c*d^2)*n
^8 + 210*(117*b^11*c^2*d + 8*a^3*b^8*c*d^2)*n^7 + 3*(78191*b^11*c^2*d + 8876*a^3
*b^8*c*d^2)*n^6 + 3*(488231*b^11*c^2*d + 71120*a^3*b^8*c*d^2 - 3360*a^6*b^5*d^3)
*n^5 + 2*(3008035*b^11*c^2*d + 459669*a^3*b^8*c*d^2 - 50400*a^6*b^5*d^3)*n^4 + 2
0*(795769*b^11*c^2*d + 105672*a^3*b^8*c*d^2 - 17640*a^6*b^5*d^3)*n^3 + 72*(35668
3*b^11*c^2*d + 33061*a^3*b^8*c*d^2 - 7000*a^6*b^5*d^3)*n^2 + 288*(78167*b^11*c^2
*d + 3465*a^3*b^8*c*d^2 - 840*a^6*b^5*d^3)*n)*x^5 + 9*(196343*a^2*b^9*c^3 - 8600
*a^5*b^6*c^2*d)*n^4 - 3*(a*b^10*c^2*d*n^10 + 57*a*b^10*c^2*d*n^9 + 1392*a*b^10*c
^2*d*n^8 + 6*(3167*a*b^10*c^2*d - 35*a^4*b^7*c*d^2)*n^7 + 15*(10571*a*b^10*c^2*d
- 504*a^4*b^7*c*d^2)*n^6 + 3*(276811*a*b^10*c^2*d - 34300*a^4*b^7*c*d^2)*n^5 +
2*(1347169*a*b^10*c^2*d - 327600*a^4*b^7*c*d^2 + 25200*a^7*b^4*d^3)*n^4 + 42*(12
2334*a*b^10*c^2*d - 47045*a^4*b^7*c*d^2 + 7200*a^7*b^4*d^3)*n^3 + 72*(71237*a*b^
10*c^2*d - 36995*a^4*b^7*c*d^2 + 7700*a^7*b^4*d^3)*n^2 + 7560*(264*a*b^10*c^2*d
- 165*a^4*b^7*c*d^2 + 40*a^7*b^4*d^3)*n)*x^4 + 8*(936802*a^2*b^9*c^3 - 107865*a^
5*b^6*c^2*d + 1890*a^8*b^3*c*d^2)*n^3 + 12*(a^2*b^9*c^2*d*n^9 + 54*a^2*b^9*c^2*d
*n^8 + 1230*a^2*b^9*c^2*d*n^7 + 6*(2552*a^2*b^9*c^2*d - 35*a^5*b^6*c*d^2)*n^6 +
33*(3413*a^2*b^9*c^2*d - 210*a^5*b^6*c*d^2)*n^5 + 6*(82091*a^2*b^9*c^2*d - 13685
*a^5*b^6*c*d^2)*n^4 + 10*(121670*a^2*b^9*c^2*d - 40887*a^5*b^6*c*d^2 + 5040*a^8*
b^3*d^3)*n^3 + 24*(61997*a^2*b^9*c^2*d - 31220*a^5*b^6*c*d^2 + 6300*a^8*b^3*d^3)
*n^2 + 2520*(264*a^2*b^9*c^2*d - 165*a^5*b^6*c*d^2 + 40*a^8*b^3*d^3)*n)*x^3 + 36
*(554953*a^2*b^9*c^3 - 149048*a^5*b^6*c^2*d + 12600*a^8*b^3*c*d^2)*n^2 - (b^11*c
^3*n^10 + 64*b^11*c^3*n^9 + 19958400*b^11*c^3 + 3*(599*b^11*c^3 + 12*a^3*b^8*c^2
*d)*n^8 + 12*(2423*b^11*c^3 + 156*a^3*b^8*c^2*d)*n^7 + 3*(99757*b^11*c^3 + 13512
*a^3*b^8*c^2*d)*n^6 + 24*(84959*b^11*c^3 + 19590*a^3*b^8*c^2*d - 315*a^6*b^5*c*d
^2)*n^5 + (9261503*b^11*c^3 + 3114324*a^3*b^8*c^2*d - 234360*a^6*b^5*c*d^2)*n^4
+ 4*(6868181*b^11*c^3 + 2875752*a^3*b^8*c^2*d - 621810*a^6*b^5*c*d^2)*n^3 + 36*(
1397573*b^11*c^3 + 577644*a^3*b^8*c^2*d - 270690*a^6*b^5*c*d^2 + 50400*a^9*b^2*d
^3)*n^2 + 720*(69851*b^11*c^3 + 16632*a^3*b^8*c^2*d - 10395*a^6*b^5*c*d^2 + 2520
*a^9*b^2*d^3)*n)*x^2 + 144*(210655*a^2*b^9*c^3 - 122502*a^5*b^6*c^2*d + 31395*a^
8*b^3*c*d^2)*n - (a*b^10*c^3*n^10 + 63*a*b^10*c^3*n^9 + 1734*a*b^10*c^3*n^8 + 18
*(1519*a*b^10*c^3 - 4*a^4*b^7*c^2*d)*n^7 + 3*(90643*a*b^10*c^3 - 1224*a^4*b^7*c^
2*d)*n^6 + 9*(196343*a*b^10*c^3 - 8600*a^4*b^7*c^2*d)*n^5 + 8*(936802*a*b^10*c^3
- 107865*a^4*b^7*c^2*d + 1890*a^7*b^4*c*d^2)*n^4 + 36*(554953*a*b^10*c^3 - 1490
48*a^4*b^7*c^2*d + 12600*a^7*b^4*c*d^2)*n^3 + 144*(210655*a*b^10*c^3 - 122502*a^
4*b^7*c^2*d + 31395*a^7*b^4*c*d^2)*n^2 + 90720*(220*a*b^10*c^3 - 264*a^4*b^7*c^2
*d + 165*a^7*b^4*c*d^2 - 40*a^10*b*d^3)*n)*x)*(b*x + a)^n/(b^11*n^11 + 66*b^11*n
^10 + 1925*b^11*n^9 + 32670*b^11*n^8 + 357423*b^11*n^7 + 2637558*b^11*n^6 + 1333
9535*b^11*n^5 + 45995730*b^11*n^4 + 105258076*b^11*n^3 + 150917976*b^11*n^2 + 12
0543840*b^11*n + 39916800*b^11)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**n*(d*x**3+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.29768, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^3*(b*x + a)^n*x,x, algorithm="giac")

[Out]

Done