3.159 \(\int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx\)

Optimal. Leaf size=459 \[ -\frac{6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac{3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+9}}{b^{12} (n+9)}-\frac{a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac{55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}+\frac{\left (b^3 c-a^3 d\right ) \left (55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^{12} (n+3)}-\frac{15 a d \left (22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac{3 d \left (154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac{42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{n+7}}{b^{12} (n+7)}+\frac{a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{12} (n+1)}+\frac{3 a^2 d \left (55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right ) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac{11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac{d^3 (a+b x)^{n+12}}{b^{12} (n+12)} \]

[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)) - (1
5*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))

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Rubi [A]  time = 0.683146, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac{3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+9}}{b^{12} (n+9)}-\frac{a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac{55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}+\frac{\left (b^3 c-a^3 d\right ) \left (55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^{12} (n+3)}-\frac{15 a d \left (22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac{3 d \left (154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac{42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{n+7}}{b^{12} (n+7)}+\frac{a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{12} (n+1)}+\frac{3 a^2 d \left (55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right ) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac{11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac{d^3 (a+b x)^{n+12}}{b^{12} (n+12)} \]

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)) - (1
5*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))

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Rubi in Sympy [A]  time = 144.694, size = 439, normalized size = 0.96 \[ \frac{55 a^{2} d^{3} \left (a + b x\right )^{n + 10}}{b^{12} \left (n + 10\right )} - \frac{42 a^{2} d^{2} \left (a + b x\right )^{n + 7} \left (11 a^{3} d - 2 b^{3} c\right )}{b^{12} \left (n + 7\right )} + \frac{3 a^{2} d \left (a + b x\right )^{n + 4} \left (55 a^{6} d^{2} - 56 a^{3} b^{3} c d + 10 b^{6} c^{2}\right )}{b^{12} \left (n + 4\right )} - \frac{a^{2} \left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )^{3}}{b^{12} \left (n + 1\right )} - \frac{11 a d^{3} \left (a + b x\right )^{n + 11}}{b^{12} \left (n + 11\right )} + \frac{6 a d^{2} \left (a + b x\right )^{n + 8} \left (55 a^{3} d - 4 b^{3} c\right )}{b^{12} \left (n + 8\right )} - \frac{15 a d \left (a + b x\right )^{n + 5} \left (22 a^{6} d^{2} - 14 a^{3} b^{3} c d + b^{6} c^{2}\right )}{b^{12} \left (n + 5\right )} + \frac{a \left (a + b x\right )^{n + 2} \left (a^{3} d - b^{3} c\right )^{2} \left (11 a^{3} d - 2 b^{3} c\right )}{b^{12} \left (n + 2\right )} + \frac{d^{3} \left (a + b x\right )^{n + 12}}{b^{12} \left (n + 12\right )} - \frac{3 d^{2} \left (a + b x\right )^{n + 9} \left (55 a^{3} d - b^{3} c\right )}{b^{12} \left (n + 9\right )} + \frac{3 d \left (a + b x\right )^{n + 6} \left (154 a^{6} d^{2} - 56 a^{3} b^{3} c d + b^{6} c^{2}\right )}{b^{12} \left (n + 6\right )} - \frac{\left (a + b x\right )^{n + 3} \left (a^{3} d - b^{3} c\right ) \left (55 a^{6} d^{2} - 29 a^{3} b^{3} c d + b^{6} c^{2}\right )}{b^{12} \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**n*(d*x**3+c)**3,x)

[Out]

55*a**2*d**3*(a + b*x)**(n + 10)/(b**12*(n + 10)) - 42*a**2*d**2*(a + b*x)**(n +
7)*(11*a**3*d - 2*b**3*c)/(b**12*(n + 7)) + 3*a**2*d*(a + b*x)**(n + 4)*(55*a**
6*d**2 - 56*a**3*b**3*c*d + 10*b**6*c**2)/(b**12*(n + 4)) - a**2*(a + b*x)**(n +
1)*(a**3*d - b**3*c)**3/(b**12*(n + 1)) - 11*a*d**3*(a + b*x)**(n + 11)/(b**12*
(n + 11)) + 6*a*d**2*(a + b*x)**(n + 8)*(55*a**3*d - 4*b**3*c)/(b**12*(n + 8)) -
15*a*d*(a + b*x)**(n + 5)*(22*a**6*d**2 - 14*a**3*b**3*c*d + b**6*c**2)/(b**12*
(n + 5)) + a*(a + b*x)**(n + 2)*(a**3*d - b**3*c)**2*(11*a**3*d - 2*b**3*c)/(b**
12*(n + 2)) + d**3*(a + b*x)**(n + 12)/(b**12*(n + 12)) - 3*d**2*(a + b*x)**(n +
9)*(55*a**3*d - b**3*c)/(b**12*(n + 9)) + 3*d*(a + b*x)**(n + 6)*(154*a**6*d**2
- 56*a**3*b**3*c*d + b**6*c**2)/(b**12*(n + 6)) - (a + b*x)**(n + 3)*(a**3*d -
b**3*c)*(55*a**6*d**2 - 29*a**3*b**3*c*d + b**6*c**2)/(b**12*(n + 3))

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Mathematica [B]  time = 2.0107, size = 1134, normalized size = 2.47 \[ \frac{(a+b x)^{n+1} \left (-39916800 d^3 a^{11}[phone] b d^3 (n+1) x a^{10}-19958400 b^2 d^3 \left (n^2+3 n+2\right ) x^2 a^9+120960 b^3 d^2 \left (55 d \left (n^3+6 n^2+11 n+6\right ) x^3+c \left (n^3+33 n^2+362 n+1320\right )\right ) a^8-30240 b^4 d^2 (n+1) x \left (55 d \left (n^3+9 n^2+26 n+24\right ) x^3+4 c \left (n^3+33 n^2+362 n+1320\right )\right ) a^7+30240 b^5 d^2 \left (n^2+3 n+2\right ) x^2 \left (11 d \left (n^3+12 n^2+47 n+60\right ) x^3+2 c \left (n^3+33 n^2+362 n+1320\right )\right ) a^6-360 b^6 d \left (154 d^2 \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^6+56 c d \left (n^6+39 n^5+571 n^4+3861 n^3+12100 n^2+16692 n+7920\right ) x^3+c^2 \left (n^6+57 n^5+1345 n^4+16815 n^3+117454 n^2+434568 n+665280\right )\right ) a^5+360 b^7 d (n+1) x \left (22 d^2 \left (n^6+27 n^5+295 n^4+1665 n^3+5104 n^2+8028 n+5040\right ) x^6+14 c d \left (n^6+42 n^5+685 n^4+5460 n^3+22084 n^2+43008 n+31680\right ) x^3+c^2 \left (n^6+57 n^5+1345 n^4+16815 n^3+117454 n^2+434568 n+665280\right )\right ) a^4-18 b^8 d \left (n^2+3 n+2\right ) x^2 \left (55 d^2 \left (n^6+33 n^5+445 n^4+3135 n^3+12154 n^2+24552 n+20160\right ) x^6+56 c d \left (n^6+45 n^5+805 n^4+7275 n^3+34834 n^2+83760 n+79200\right ) x^3+10 c^2 \left (n^6+57 n^5+1345 n^4+16815 n^3+117454 n^2+434568 n+665280\right )\right ) a^3+2 b^9 \left (55 d^3 \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 ) x^9+84 c d^2 \left (n^9+54 n^8+1230 n^7+15432 n^6+116949 n^5+552426 n^4+1617020 n^3+2806008 n^2+2589120 n+950400\right ) x^6+30 c^2 d \left (n^9+63 n^8+1698 n^7+25518 n^6+233481 n^5+1332327 n^4+4665572 n^3+9476652 n^2+9925488 n+3991680\right ) x^3+c^3 \left (n^9+72 n^8+2274 n^7+41328 n^6+476049 n^5+3602088 n^4+17893196 n^3+56231712 n^2+101378880 n+79833600\right )\right ) a^2-b^{10} \left (n^4+22 n^3+159 n^2+418 n+280\right ) x \left (11 d^3 \left (n^6+33 n^5+435 n^4+2915 n^3+10404 n^2+18612 n+12960\right ) x^9+24 c d^2 \left (n^6+39 n^5+591 n^4+4421 n^3+17160 n^2+32652 n+23760\right ) x^6+15 c^2 d \left (n^6+45 n^5+801 n^4+7115 n^3+32574 n^2+70920 n+57024\right ) x^3+2 c^3 \left (n^6+51 n^5+1065 n^4+11645 n^3+70254 n^2+221544 n+285120\right )\right ) a+b^{11} \left (n^8+48 n^7+962 n^6+10440 n^5+66489 n^4+251352 n^3+541508 n^2+593520 n+246400\right ) x^2 \left (d^3 \left (n^3+18 n^2+99 n+162\right ) x^9+3 c d^2 \left (n^3+21 n^2+126 n+216\right ) x^6+3 c^2 d \left (n^3+24 n^2+171 n+324\right ) x^3+c^3 \left (n^3+27 n^2+234 n+648\right )\right )\right )}{b^{12} (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9) (n+10) (n+11) (n+12)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + n)*(-39916800*a^11*d^3 + 39916800*a^10*b*d^3*(1 + n)*x - 1995840
0*a^9*b^2*d^3*(2 + 3*n + n^2)*x^2 + 120960*a^8*b^3*d^2*(c*(1320 + 362*n + 33*n^2
+ n^3) + 55*d*(6 + 11*n + 6*n^2 + n^3)*x^3) - 30240*a^7*b^4*d^2*(1 + n)*x*(4*c*
(1320 + 362*n + 33*n^2 + n^3) + 55*d*(24 + 26*n + 9*n^2 + n^3)*x^3) + 30240*a^6*
b^5*d^2*(2 + 3*n + n^2)*x^2*(2*c*(1320 + 362*n + 33*n^2 + n^3) + 11*d*(60 + 47*n
+ 12*n^2 + n^3)*x^3) - 360*a^5*b^6*d*(c^2*(665280 + 434568*n + 117454*n^2 + 168
15*n^3 + 1345*n^4 + 57*n^5 + n^6) + 56*c*d*(7920 + 16692*n + 12100*n^2 + 3861*n^
3 + 571*n^4 + 39*n^5 + n^6)*x^3 + 154*d^2*(720 + 1764*n + 1624*n^2 + 735*n^3 + 1
75*n^4 + 21*n^5 + n^6)*x^6) + 360*a^4*b^7*d*(1 + n)*x*(c^2*(665280 + 434568*n +
117454*n^2 + 16815*n^3 + 1345*n^4 + 57*n^5 + n^6) + 14*c*d*(31680 + 43008*n + 22
084*n^2 + 5460*n^3 + 685*n^4 + 42*n^5 + n^6)*x^3 + 22*d^2*(5040 + 8028*n + 5104*
n^2 + 1665*n^3 + 295*n^4 + 27*n^5 + n^6)*x^6) - 18*a^3*b^8*d*(2 + 3*n + n^2)*x^2
*(10*c^2*(665280 + 434568*n + 117454*n^2 + 16815*n^3 + 1345*n^4 + 57*n^5 + n^6)
+ 56*c*d*(79200 + 83760*n + 34834*n^2 + 7275*n^3 + 805*n^4 + 45*n^5 + n^6)*x^3 +
55*d^2*(20160 + 24552*n + 12154*n^2 + 3135*n^3 + 445*n^4 + 33*n^5 + n^6)*x^6) +
b^11*(246400 + 593520*n + 541508*n^2 + 251352*n^3 + 66489*n^4 + 10440*n^5 + 962
*n^6 + 48*n^7 + n^8)*x^2*(c^3*(648 + 234*n + 27*n^2 + n^3) + 3*c^2*d*(324 + 171*
n + 24*n^2 + n^3)*x^3 + 3*c*d^2*(216 + 126*n + 21*n^2 + n^3)*x^6 + d^3*(162 + 99
*n + 18*n^2 + n^3)*x^9) - a*b^10*(280 + 418*n + 159*n^2 + 22*n^3 + n^4)*x*(2*c^3
*(285120 + 221544*n + 70254*n^2 + 11645*n^3 + 1065*n^4 + 51*n^5 + n^6) + 15*c^2*
d*(57024 + 70920*n + 32574*n^2 + 7115*n^3 + 801*n^4 + 45*n^5 + n^6)*x^3 + 24*c*d
^2*(23760 + 32652*n + 17160*n^2 + 4421*n^3 + 591*n^4 + 39*n^5 + n^6)*x^6 + 11*d^
3*(12960 + 18612*n + 10404*n^2 + 2915*n^3 + 435*n^4 + 33*n^5 + n^6)*x^9) + 2*a^2
*b^9*(c^3*(79833600 + 101378880*n + 56231712*n^2 + 17893196*n^3 + 3602088*n^4 +
476049*n^5 + 41328*n^6 + 2274*n^7 + 72*n^8 + n^9) + 30*c^2*d*(3991680 + 9925488*
n + 9476652*n^2 + 4665572*n^3 + 1332327*n^4 + 233481*n^5 + 25518*n^6 + 1698*n^7
+ 63*n^8 + n^9)*x^3 + 84*c*d^2*(950400 + 2589120*n + 2806008*n^2 + 1617020*n^3 +
552426*n^4 + 116949*n^5 + 15432*n^6 + 1230*n^7 + 54*n^8 + n^9)*x^6 + 55*d^3*(36
2880 + 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)*x^9)))/(b^12*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(
6 + n)*(7 + n)*(8 + n)*(9 + n)*(10 + n)*(11 + n)*(12 + n))

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Maple [B]  time = 0.043, size = 3780, normalized size = 8.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^n*(d*x^3+c)^3,x)

[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^1
0-1925*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^1
1*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+173
5503*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^8-1333953
5*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^8-45995730*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+3758623
0*a*b^10*d^3*n^4*x^10-6855*b^11*c^2*d*n^9*x^5-9512559*b^11*c*d^2*n^6*x^8-1052580
76*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+44906
40*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^8-150
917976*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+4
8384*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+1
16915040*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^5-405
697080*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+9702
000*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^10*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^4-104
005440*a^4*b^7*d^3*n^2*x^7+10800*a^3*b^8*c^2*d*n^7*x^2+58735152*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^
3-271659360*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
[phone]*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^4-10349856
0*a^4*b^7*d^3*n*x^7+273240*a^3*b^8*c^2*d*n^6*x^2+204434496*a^3*b^8*c*d^2*n^3*x^5
[phone]*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^4-3991680
0*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+408360960*a^3*b^8*c*d^2*n*x^5-82656*a^2*b^9*c^3*n^6-279934
320*a^2*b^9*c^2*d*n^3*x^3-159667200*a^2*b^9*c*d^2*x^6+8156274*a*b^10*c^3*n^5*x+7
17481440*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^5-39916800*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^2-39916800*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-376427520*a^4
*b^7*c*d^2*n*x^4+147700800*a^3*b^8*c^2*d*n^3*x^2+159667200*a^3*b^8*c*d^2*x^5-952
098*a^2*b^9*c^3*n^5-568599120*a^2*b^9*c^2*d*n^2*x^3+42990568*a*b^10*c^3*n^4*x+65
5404480*a*b^10*c^2*d*n*x^4-305860408*b^11*c^3*n^3*x^2-239500800*b^11*c^2*d*x^5+1
9958400*a^9*b^2*d^3*n^2*x^2-73180800*a^8*b^3*d^3*n*x^3+4112640*a^7*b^4*c*d^2*n^3
*x+39916800*a^7*b^4*d^3*x^4-149506560*a^6*b^5*c*d^2*n^2*x^2+484200*a^5*b^6*c^2*d
*n^4+336510720*a^5*b^6*c*d^2*n*x^3-48336840*a^4*b^7*c^2*d*n^3*x-159667200*a^4*b^
7*c*d^2*x^4+396700560*a^3*b^8*c^2*d*n^2*x^2-7204176*a^2*b^9*c^3*n^4-595529280*a^
2*b^9*c^2*d*n*x^3+148249816*a*b^10*c^3*n^3*x+239500800*a*b^10*c^2*d*x^4-49643366
4*b^11*c^3*n^2*x^2+59875200*a^9*b^2*d^3*n*x^2-120960*a^8*b^3*c*d^2*n^3-39916800*
a^8*b^3*d^3*x^3+47779200*a^7*b^4*c*d^2*n^2*x-283288320*a^6*b^5*c*d^2*n*x^2+60534
00*a^5*b^6*c^2*d*n^3+159667200*a^5*b^6*c*d^2*x^3-198727920*a^4*b^7*c^2*d*n^2*x+5
15695680*a^3*b^8*c^2*d*n*x^2-35786392*a^2*b^9*c^3*n^3-239500800*a^2*b^9*c^2*d*x^
3+315221184*a*b^10*c^3*n^2*x-442258560*b^11*c^3*n*x^2-39916800*a^10*b*d^3*n*x+39
916800*a^9*b^2*d^3*x^2-3991680*a^8*b^3*c*d^2*n^2+203454720*a^7*b^4*c*d^2*n*x-159
667200*a^6*b^5*c*d^2*x^2+42283440*a^5*b^6*c^2*d*n^2-395945280*a^4*b^7*c^2*d*n*x+
239500800*a^3*b^8*c^2*d*x^2-112463424*a^2*b^9*c^3*n^2+362424960*a*b^10*c^3*n*x-1
59667200*b^11*c^3*x^2-39916800*a^10*b*d^3*x-43787520*a^8*b^3*c*d^2*n+159667200*a
^7*b^4*c*d^2*x+156444480*a^5*b^6*c^2*d*n-239500800*a^4*b^7*c^2*d*x-202757760*a^2
*b^9*c^3*n+159667200*a*b^10*c^3*x+39916800*a^11*d^3-159667200*a^8*b^3*c*d^2+2395
00800*a^5*b^6*c^2*d-159667200*a^2*b^9*c^3)/b^12/(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
[phone]*n^2+1486442880*n+479001600)

_______________________________________________________________________________________

Maxima [A]  time = 0.73345, size = 1557, normalized size = 3.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^3*(b*x + a)^n*x^2,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 + 21*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 + 274*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 + 8
70*n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 1172700*n^2 + 1026576*
n + 362880)*b^9) + ((n^11 + 66*n^10 + 1925*n^9 + 32670*n^8 + 357423*n^7 + 263755
8*n^6 + 13339535*n^5 + 45995730*n^4 + 105258076*n^3 + 150917976*n^2 + 120543840*
n + 39916800)*b^12*x^12 + (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
)*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 + 1172700*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 + 14140
14888*n^3 + 1931559552*n^2 + 1486442880*n + 479001600)*b^12)

_______________________________________________________________________________________

Fricas [A]  time = 0.330684, size = 4811, normalized size = 10.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^3*(b*x + a)^n*x^2,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 + 32670*b^12*d^3*n^8 + 3
57423*b^12*d^3*n^7 + 2637558*b^12*d^3*n^6 + 13339535*b^12*d^3*n^5 + 45995730*b^1
2*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 + 4
5*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*(2051
288*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*(2
572*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*(6509445*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*(81401*b^12*c^2*d + 3080*a^3*b^9*c*d^2 - 770*a^6*b^6*d^3)*n)*x^6 + 6*(15
8683*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 - 81
424*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 + 266
1120*(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 - 63
952*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 + 2772
0*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 + 15120*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 + 30
360*a^3*b^9*c^2*d)*n^7 + 9*(568099*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*(38232551*b^12*c^3 + 16529190*a^3*b^9*c^2*d - 6229440*a^6*b^6*c*d^2 + 831
600*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^3)*n)*x^3 + 144*(780996*a^3*b^9*c^3 - 2936
35*a^6*b^6*c^2*d + 27720*a^9*b^3*c*d^2)*n^2 + (a*b^11*c^3*n^11 + 73*a*b^11*c^3*n
^10 + 2346*a*b^11*c^3*n^9 + 6*(7267*a*b^11*c^3 - 30*a^4*b^8*c^2*d)*n^8 + 3*(1724
59*a*b^11*c^3 - 3480*a^4*b^8*c^2*d)*n^7 + 3*(1359379*a*b^11*c^3 - 84120*a^4*b^8*
c^2*d)*n^6 + 4*(5373821*a*b^11*c^3 - 817200*a^4*b^8*c^2*d + 15120*a^7*b^5*c*d^2)
*n^5 + 4*(18531227*a*b^11*c^3 - 6042105*a^4*b^8*c^2*d + 514080*a^7*b^5*c*d^2)*n^
4 + 72*(2189036*a*b^11*c^3 - 1380055*a^4*b^8*c^2*d + 331800*a^7*b^5*c*d^2)*n^3 +
1440*(125842*a*b^11*c^3 - 137481*a^4*b^8*c^2*d + 70644*a^7*b^5*c*d^2 - 13860*a^
10*b^2*d^3)*n^2 + 19958400*(4*a*b^11*c^3 - 6*a^4*b^8*c^2*d + 4*a^7*b^5*c*d^2 - a
^10*b^2*d^3)*n)*x^2 + 2880*(70402*a^3*b^9*c^3 - 54321*a^6*b^6*c^2*d + 15204*a^9*
b^3*c*d^2)*n - 2*(a^2*b^10*c^3*n^10 + 72*a^2*b^10*c^3*n^9 + 2274*a^2*b^10*c^3*n^
8 + 36*(1148*a^2*b^10*c^3 - 5*a^5*b^7*c^2*d)*n^7 + 3*(158683*a^2*b^10*c^3 - 3420
*a^5*b^7*c^2*d)*n^6 + 36*(100058*a^2*b^10*c^3 - 6725*a^5*b^7*c^2*d)*n^5 + 4*(447
3299*a^2*b^10*c^3 - 756675*a^5*b^7*c^2*d + 15120*a^8*b^4*c*d^2)*n^4 + 72*(780996
*a^2*b^10*c^3 - 293635*a^5*b^7*c^2*d + 27720*a^8*b^4*c*d^2)*n^3 + 1440*(70402*a^
2*b^10*c^3 - 54321*a^5*b^7*c^2*d + 15204*a^8*b^4*c*d^2)*n^2 + 19958400*(4*a^2*b^
10*c^3 - 6*a^5*b^7*c^2*d + 4*a^8*b^4*c*d^2 - a^11*b*d^3)*n)*x)*(b*x + a)^n/(b^12
*n^12 + 78*b^12*n^11 + 2717*b^12*n^10 + 55770*b^12*n^9 + 749463*b^12*n^8 + 69266
34*b^12*n^7 + 44990231*b^12*n^6 + 206070150*b^12*n^5 + 657206836*b^12*n^4 + 1414
014888*b^12*n^3 + 1931559552*b^12*n^2 + 1486442880*b^12*n + 479001600*b^12)

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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**2*(b*x+a)**n*(d*x**3+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^3*(b*x + a)^n*x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError