3.155 \(\int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx\)
Optimal. Leaf size=294 \[ -\frac{2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^9 (n+2)}-\frac{10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^2 (a+b x)^{n+7}}{b^9 (n+7)}+\frac{\left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^9 (n+1)}+\frac{4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}-\frac{8 a d^2 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^2 (a+b x)^{n+9}}{b^9 (n+9)} \]
[Out]
(a^2*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^3*c - 4*a^3*d)
*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^6*c^2 - 20*a^3*b^3*c*d +
28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) + (4*a^2*d*(5*b^3*c - 14*a^3*d)*(a
+ b*x)^(4 + n))/(b^9*(4 + n)) - (10*a*d*(b^3*c - 7*a^3*d)*(a + b*x)^(5 + n))/(b
^9*(5 + n)) + (2*d*(b^3*c - 28*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2
*d^2*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^2*(a + b*x)^(8 + n))/(b^9*(8 + n)
) + (d^2*(a + b*x)^(9 + n))/(b^9*(9 + n))
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Rubi [A] time = 0.418832, antiderivative size = 294, 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
\[ -\frac{2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^9 (n+2)}-\frac{10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^2 (a+b x)^{n+7}}{b^9 (n+7)}+\frac{\left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^9 (n+1)}+\frac{4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}-\frac{8 a d^2 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^2 (a+b x)^{n+9}}{b^9 (n+9)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x)^n*(c + d*x^3)^2,x]
[Out]
(a^2*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^3*c - 4*a^3*d)
*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^6*c^2 - 20*a^3*b^3*c*d +
28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) + (4*a^2*d*(5*b^3*c - 14*a^3*d)*(a
+ b*x)^(4 + n))/(b^9*(4 + n)) - (10*a*d*(b^3*c - 7*a^3*d)*(a + b*x)^(5 + n))/(b
^9*(5 + n)) + (2*d*(b^3*c - 28*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2
*d^2*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^2*(a + b*x)^(8 + n))/(b^9*(8 + n)
) + (d^2*(a + b*x)^(9 + n))/(b^9*(9 + n))
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Rubi in Sympy [A] time = 87.3212, size = 275, normalized size = 0.94 \[ \frac{28 a^{2} d^{2} \left (a + b x\right )^{n + 7}}{b^{9} \left (n + 7\right )} - \frac{4 a^{2} d \left (a + b x\right )^{n + 4} \left (14 a^{3} d - 5 b^{3} c\right )}{b^{9} \left (n + 4\right )} + \frac{a^{2} \left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )^{2}}{b^{9} \left (n + 1\right )} - \frac{8 a d^{2} \left (a + b x\right )^{n + 8}}{b^{9} \left (n + 8\right )} + \frac{10 a d \left (a + b x\right )^{n + 5} \left (7 a^{3} d - b^{3} c\right )}{b^{9} \left (n + 5\right )} - \frac{2 a \left (a + b x\right )^{n + 2} \left (a^{3} d - b^{3} c\right ) \left (4 a^{3} d - b^{3} c\right )}{b^{9} \left (n + 2\right )} + \frac{d^{2} \left (a + b x\right )^{n + 9}}{b^{9} \left (n + 9\right )} - \frac{2 d \left (a + b x\right )^{n + 6} \left (28 a^{3} d - b^{3} c\right )}{b^{9} \left (n + 6\right )} + \frac{\left (a + b x\right )^{n + 3} \left (28 a^{6} d^{2} - 20 a^{3} b^{3} c d + b^{6} c^{2}\right )}{b^{9} \left (n + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**n*(d*x**3+c)**2,x)
[Out]
28*a**2*d**2*(a + b*x)**(n + 7)/(b**9*(n + 7)) - 4*a**2*d*(a + b*x)**(n + 4)*(14
*a**3*d - 5*b**3*c)/(b**9*(n + 4)) + a**2*(a + b*x)**(n + 1)*(a**3*d - b**3*c)**
2/(b**9*(n + 1)) - 8*a*d**2*(a + b*x)**(n + 8)/(b**9*(n + 8)) + 10*a*d*(a + b*x)
**(n + 5)*(7*a**3*d - b**3*c)/(b**9*(n + 5)) - 2*a*(a + b*x)**(n + 2)*(a**3*d -
b**3*c)*(4*a**3*d - b**3*c)/(b**9*(n + 2)) + d**2*(a + b*x)**(n + 9)/(b**9*(n +
9)) - 2*d*(a + b*x)**(n + 6)*(28*a**3*d - b**3*c)/(b**9*(n + 6)) + (a + b*x)**(n
+ 3)*(28*a**6*d**2 - 20*a**3*b**3*c*d + b**6*c**2)/(b**9*(n + 3))
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Mathematica [A] time = 0.571289, size = 534, normalized size = 1.82 \[ \frac{(a+b x)^{n+1} \left (40320 a^8 d^2-40320 a^7 b d^2 (n+1) x+20160 a^6 b^2 d^2 \left (n^2+3 n+2\right ) x^2-240 a^5 b^3 d \left (c \left (n^3+24 n^2+191 n+504\right )+28 d \left (n^3+6 n^2+11 n+6\right ) x^3\right )+240 a^4 b^4 d (n+1) x \left (c \left (n^3+24 n^2+191 n+504\right )+7 d \left (n^3+9 n^2+26 n+24\right ) x^3\right )-24 a^3 b^5 d \left (n^2+3 n+2\right ) x^2 \left (5 c \left (n^3+24 n^2+191 n+504\right )+14 d \left (n^3+12 n^2+47 n+60\right ) x^3\right )+2 a^2 b^6 \left (c^2 \left (n^6+39 n^5+625 n^4+5265 n^3+24574 n^2+60216 n+60480\right )+20 c d \left (n^6+30 n^5+346 n^4+1920 n^3+5269 n^2+6690 n+3024\right ) x^3+28 d^2 \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^6\right )-2 a b^7 \left (n^3+12 n^2+39 n+28\right ) x \left (c^2 \left (n^4+28 n^3+289 n^2+1302 n+2160\right )+5 c d \left (n^4+22 n^3+163 n^2+462 n+432\right ) x^3+4 d^2 \left (n^4+16 n^3+91 n^2+216 n+180\right ) x^6\right )+b^8 \left (n^6+27 n^5+285 n^4+1485 n^3+3954 n^2+4968 n+2240\right ) x^2 \left (c^2 \left (n^2+15 n+54\right )+2 c d \left (n^2+12 n+27\right ) x^3+d^2 \left (n^2+9 n+18\right ) x^6\right )\right )}{b^9 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x)^n*(c + d*x^3)^2,x]
[Out]
((a + b*x)^(1 + n)*(40320*a^8*d^2 - 40320*a^7*b*d^2*(1 + n)*x + 20160*a^6*b^2*d^
2*(2 + 3*n + n^2)*x^2 - 240*a^5*b^3*d*(c*(504 + 191*n + 24*n^2 + n^3) + 28*d*(6
+ 11*n + 6*n^2 + n^3)*x^3) + 240*a^4*b^4*d*(1 + n)*x*(c*(504 + 191*n + 24*n^2 +
n^3) + 7*d*(24 + 26*n + 9*n^2 + n^3)*x^3) - 24*a^3*b^5*d*(2 + 3*n + n^2)*x^2*(5*
c*(504 + 191*n + 24*n^2 + n^3) + 14*d*(60 + 47*n + 12*n^2 + n^3)*x^3) + b^8*(224
0 + 4968*n + 3954*n^2 + 1485*n^3 + 285*n^4 + 27*n^5 + n^6)*x^2*(c^2*(54 + 15*n +
n^2) + 2*c*d*(27 + 12*n + n^2)*x^3 + d^2*(18 + 9*n + n^2)*x^6) - 2*a*b^7*(28 +
39*n + 12*n^2 + n^3)*x*(c^2*(2160 + 1302*n + 289*n^2 + 28*n^3 + n^4) + 5*c*d*(43
2 + 462*n + 163*n^2 + 22*n^3 + n^4)*x^3 + 4*d^2*(180 + 216*n + 91*n^2 + 16*n^3 +
n^4)*x^6) + 2*a^2*b^6*(c^2*(60480 + 60216*n + 24574*n^2 + 5265*n^3 + 625*n^4 +
39*n^5 + n^6) + 20*c*d*(3024 + 6690*n + 5269*n^2 + 1920*n^3 + 346*n^4 + 30*n^5 +
n^6)*x^3 + 28*d^2*(720 + 1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6)*
x^6)))/(b^9*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n)*(9 +
n))
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Maple [B] time = 0.022, size = 1565, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^n*(d*x^3+c)^2,x)
[Out]
(b*x+a)^(1+n)*(b^8*d^2*n^8*x^8+36*b^8*d^2*n^7*x^8-8*a*b^7*d^2*n^7*x^7+546*b^8*d^
2*n^6*x^8-224*a*b^7*d^2*n^6*x^7+2*b^8*c*d*n^8*x^5+4536*b^8*d^2*n^5*x^8+56*a^2*b^
6*d^2*n^6*x^6-2576*a*b^7*d^2*n^5*x^7+78*b^8*c*d*n^7*x^5+22449*b^8*d^2*n^4*x^8+11
76*a^2*b^6*d^2*n^5*x^6-10*a*b^7*c*d*n^7*x^4-15680*a*b^7*d^2*n^4*x^7+1272*b^8*c*d
*n^6*x^5+67284*b^8*d^2*n^3*x^8-336*a^3*b^5*d^2*n^5*x^5+9800*a^2*b^6*d^2*n^4*x^6-
340*a*b^7*c*d*n^6*x^4-54152*a*b^7*d^2*n^3*x^7+b^8*c^2*n^8*x^2+11268*b^8*c*d*n^5*
x^5+118124*b^8*d^2*n^2*x^8-5040*a^3*b^5*d^2*n^4*x^5+40*a^2*b^6*c*d*n^6*x^3+41160
*a^2*b^6*d^2*n^3*x^6-4660*a*b^7*c*d*n^5*x^4-105056*a*b^7*d^2*n^2*x^7+42*b^8*c^2*
n^7*x^2+58938*b^8*c*d*n^4*x^5+109584*b^8*d^2*n*x^8+1680*a^4*b^4*d^2*n^4*x^4-2856
0*a^3*b^5*d^2*n^3*x^5+1200*a^2*b^6*c*d*n^5*x^3+90944*a^2*b^6*d^2*n^2*x^6-2*a*b^7
*c^2*n^7*x-33040*a*b^7*c*d*n^4*x^4-104544*a*b^7*d^2*n*x^7+744*b^8*c^2*n^6*x^2+18
5022*b^8*c*d*n^3*x^5+40320*b^8*d^2*x^8+16800*a^4*b^4*d^2*n^3*x^4-120*a^3*b^5*c*d
*n^5*x^2-75600*a^3*b^5*d^2*n^2*x^5+13840*a^2*b^6*c*d*n^4*x^3+98784*a^2*b^6*d^2*n
*x^6-80*a*b^7*c^2*n^6*x-129490*a*b^7*c*d*n^3*x^4-40320*a*b^7*d^2*x^7+7218*b^8*c^
2*n^5*x^2+337228*b^8*c*d*n^2*x^5-6720*a^5*b^3*d^2*n^3*x^3+58800*a^4*b^4*d^2*n^2*
x^4-3240*a^3*b^5*c*d*n^4*x^2-92064*a^3*b^5*d^2*n*x^5+2*a^2*b^6*c^2*n^6+76800*a^2
*b^6*c*d*n^3*x^3+40320*a^2*b^6*d^2*x^6-1328*a*b^7*c^2*n^5*x-277660*a*b^7*c*d*n^2
*x^4+41619*b^8*c^2*n^4*x^2+322032*b^8*c*d*n*x^5-40320*a^5*b^3*d^2*n^2*x^3+240*a^
4*b^4*c*d*n^4*x+84000*a^4*b^4*d^2*n*x^4-31800*a^3*b^5*c*d*n^3*x^2-40320*a^3*b^5*
d^2*x^5+78*a^2*b^6*c^2*n^5+210760*a^2*b^6*c*d*n^2*x^3-11780*a*b^7*c^2*n^4*x-2978
40*a*b^7*c*d*n*x^4+144468*b^8*c^2*n^3*x^2+120960*b^8*c*d*x^5+20160*a^6*b^2*d^2*n
^2*x^2-73920*a^5*b^3*d^2*n*x^3+6000*a^4*b^4*c*d*n^3*x+40320*a^4*b^4*d^2*x^4-1350
00*a^3*b^5*c*d*n^2*x^2+1250*a^2*b^6*c^2*n^4+267600*a^2*b^6*c*d*n*x^3-59678*a*b^7
*c^2*n^3*x-120960*a*b^7*c*d*x^4+290276*b^8*c^2*n^2*x^2+60480*a^6*b^2*d^2*n*x^2-2
40*a^5*b^3*c*d*n^3-40320*a^5*b^3*d^2*x^3+51600*a^4*b^4*c*d*n^2*x-227280*a^3*b^5*
c*d*n*x^2+10530*a^2*b^6*c^2*n^3+120960*a^2*b^6*c*d*x^3-169580*a*b^7*c^2*n^2*x+30
1872*b^8*c^2*n*x^2-40320*a^7*b*d^2*n*x+40320*a^6*b^2*d^2*x^2-5760*a^5*b^3*c*d*n^
2+166800*a^4*b^4*c*d*n*x-120960*a^3*b^5*c*d*x^2+49148*a^2*b^6*c^2*n^2-241392*a*b
^7*c^2*n*x+120960*b^8*c^2*x^2-40320*a^7*b*d^2*x-45840*a^5*b^3*c*d*n+120960*a^4*b
^4*c*d*x+120432*a^2*b^6*c^2*n-120960*a*b^7*c^2*x+40320*a^8*d^2-120960*a^5*b^3*c*
d+120960*a^2*b^6*c^2)/b^9/(n^9+45*n^8+870*n^7+9450*n^6+63273*n^5+269325*n^4+7236
80*n^3+1172700*n^2+1026576*n+362880)
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Maxima [A] time = 0.716648, size = 811, normalized size = 2.76 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{2 \,{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} +{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \,{\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )}{\left (b x + a\right )}^{n} c d}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac{{\left ({\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{9} x^{9} +{\left (n^{8} + 28 \, n^{7} + 322 \, n^{6} + 1960 \, n^{5} + 6769 \, n^{4} + 13132 \, n^{3} + 13068 \, n^{2} + 5040 \, n\right )} a b^{8} x^{8} - 8 \,{\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a^{2} b^{7} x^{7} + 56 \,{\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{3} b^{6} x^{6} - 336 \,{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{4} b^{5} x^{5} + 1680 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{5} b^{4} x^{4} - 6720 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{6} b^{3} x^{3} + 20160 \,{\left (n^{2} + n\right )} a^{7} b^{2} x^{2} - 40320 \, a^{8} b n x + 40320 \, a^{9}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{9} + 45 \, n^{8} + 870 \, n^{7} + 9450 \, n^{6} + 63273 \, n^{5} + 269325 \, n^{4} + 723680 \, n^{3} + 1172700 \, n^{2} + 1026576 \, n + 362880\right )} b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^2*(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^2/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 2*((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*d/((n^6 + 21*n^5 + 1
75*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^6) + ((n^8 + 36*n^7 + 546*n^6 + 45
36*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*d^2/((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 + 36
2880)*b^9)
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Fricas [A] time = 0.303252, size = 2113, normalized size = 7.19 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^2*(b*x + a)^n*x^2,x, algorithm="fricas")
[Out]
(2*a^3*b^6*c^2*n^6 + 78*a^3*b^6*c^2*n^5 + 1250*a^3*b^6*c^2*n^4 + 120960*a^3*b^6*
c^2 - 120960*a^6*b^3*c*d + 40320*a^9*d^2 + (b^9*d^2*n^8 + 36*b^9*d^2*n^7 + 546*b
^9*d^2*n^6 + 4536*b^9*d^2*n^5 + 22449*b^9*d^2*n^4 + 67284*b^9*d^2*n^3 + 118124*b
^9*d^2*n^2 + 109584*b^9*d^2*n + 40320*b^9*d^2)*x^9 + (a*b^8*d^2*n^8 + 28*a*b^8*d
^2*n^7 + 322*a*b^8*d^2*n^6 + 1960*a*b^8*d^2*n^5 + 6769*a*b^8*d^2*n^4 + 13132*a*b
^8*d^2*n^3 + 13068*a*b^8*d^2*n^2 + 5040*a*b^8*d^2*n)*x^8 - 8*(a^2*b^7*d^2*n^7 +
21*a^2*b^7*d^2*n^6 + 175*a^2*b^7*d^2*n^5 + 735*a^2*b^7*d^2*n^4 + 1624*a^2*b^7*d^
2*n^3 + 1764*a^2*b^7*d^2*n^2 + 720*a^2*b^7*d^2*n)*x^7 + 2*(b^9*c*d*n^8 + 39*b^9*
c*d*n^7 + 60480*b^9*c*d + 4*(159*b^9*c*d + 7*a^3*b^6*d^2)*n^6 + 6*(939*b^9*c*d +
70*a^3*b^6*d^2)*n^5 + (29469*b^9*c*d + 2380*a^3*b^6*d^2)*n^4 + 9*(10279*b^9*c*d
+ 700*a^3*b^6*d^2)*n^3 + 2*(84307*b^9*c*d + 3836*a^3*b^6*d^2)*n^2 + 24*(6709*b^
9*c*d + 140*a^3*b^6*d^2)*n)*x^6 + 2*(a*b^8*c*d*n^8 + 34*a*b^8*c*d*n^7 + 466*a*b^
8*c*d*n^6 + 56*(59*a*b^8*c*d - 3*a^4*b^5*d^2)*n^5 + (12949*a*b^8*c*d - 1680*a^4*
b^5*d^2)*n^4 + 2*(13883*a*b^8*c*d - 2940*a^4*b^5*d^2)*n^3 + 24*(1241*a*b^8*c*d -
350*a^4*b^5*d^2)*n^2 + 4032*(3*a*b^8*c*d - a^4*b^5*d^2)*n)*x^5 - 10*(a^2*b^7*c*
d*n^7 + 30*a^2*b^7*c*d*n^6 + 346*a^2*b^7*c*d*n^5 + 24*(80*a^2*b^7*c*d - 7*a^5*b^
4*d^2)*n^4 + (5269*a^2*b^7*c*d - 1008*a^5*b^4*d^2)*n^3 + 6*(1115*a^2*b^7*c*d - 3
08*a^5*b^4*d^2)*n^2 + 1008*(3*a^2*b^7*c*d - a^5*b^4*d^2)*n)*x^4 + 30*(351*a^3*b^
6*c^2 - 8*a^6*b^3*c*d)*n^3 + (b^9*c^2*n^8 + 42*b^9*c^2*n^7 + 120960*b^9*c^2 + 8*
(93*b^9*c^2 + 5*a^3*b^6*c*d)*n^6 + 18*(401*b^9*c^2 + 60*a^3*b^6*c*d)*n^5 + (4161
9*b^9*c^2 + 10600*a^3*b^6*c*d)*n^4 + 12*(12039*b^9*c^2 + 3750*a^3*b^6*c*d - 560*
a^6*b^3*d^2)*n^3 + 4*(72569*b^9*c^2 + 18940*a^3*b^6*c*d - 5040*a^6*b^3*d^2)*n^2
+ 48*(6289*b^9*c^2 + 840*a^3*b^6*c*d - 280*a^6*b^3*d^2)*n)*x^3 + 4*(12287*a^3*b^
6*c^2 - 1440*a^6*b^3*c*d)*n^2 + (a*b^8*c^2*n^8 + 40*a*b^8*c^2*n^7 + 664*a*b^8*c^
2*n^6 + 10*(589*a*b^8*c^2 - 12*a^4*b^5*c*d)*n^5 + (29839*a*b^8*c^2 - 3000*a^4*b^
5*c*d)*n^4 + 10*(8479*a*b^8*c^2 - 2580*a^4*b^5*c*d)*n^3 + 24*(5029*a*b^8*c^2 - 3
475*a^4*b^5*c*d + 840*a^7*b^2*d^2)*n^2 + 20160*(3*a*b^8*c^2 - 3*a^4*b^5*c*d + a^
7*b^2*d^2)*n)*x^2 + 48*(2509*a^3*b^6*c^2 - 955*a^6*b^3*c*d)*n - 2*(a^2*b^7*c^2*n
^7 + 39*a^2*b^7*c^2*n^6 + 625*a^2*b^7*c^2*n^5 + 15*(351*a^2*b^7*c^2 - 8*a^5*b^4*
c*d)*n^4 + 2*(12287*a^2*b^7*c^2 - 1440*a^5*b^4*c*d)*n^3 + 24*(2509*a^2*b^7*c^2 -
955*a^5*b^4*c*d)*n^2 + 20160*(3*a^2*b^7*c^2 - 3*a^5*b^4*c*d + a^8*b*d^2)*n)*x)*
(b*x + a)^n/(b^9*n^9 + 45*b^9*n^8 + 870*b^9*n^7 + 9450*b^9*n^6 + 63273*b^9*n^5 +
269325*b^9*n^4 + 723680*b^9*n^3 + 1172700*b^9*n^2 + 1026576*b^9*n + 362880*b^9)
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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)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.273453, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^2*(b*x + a)^n*x^2,x, algorithm="giac")
[Out]
Done