3.359 \(\int x^2 (a+b x)^n \left (c+d x^2\right )^3 \, dx\)
Optimal. Leaf size=343 \[ -\frac{2 a d^2 \left (28 a^2 d+9 b^2 c\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{d^2 \left (28 a^2 d+3 b^2 c\right ) (a+b x)^{n+7}}{b^9 (n+7)}+\frac{a^2 \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left (a^2 d+b^2 c\right )^2 \left (4 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{\left (a^2 d+b^2 c\right ) \left (28 a^4 d^2+17 a^2 b^2 c d+b^4 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}-\frac{4 a d \left (14 a^4 d^2+15 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{d \left (70 a^4 d^2+45 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+5}}{b^9 (n+5)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)} \]
[Out]
(a^2*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^2*c + a^2*d)^2
*(b^2*c + 4*a^2*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^2*c + a^2*d)*(b^4*c^2
+ 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) - (4*a*d*(3*b^4*
c^2 + 15*a^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^(4 + n))/(b^9*(4 + n)) + (d*(3*b^4*
c^2 + 45*a^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^(5 + n))/(b^9*(5 + n)) - (2*a*d^2*(
9*b^2*c + 28*a^2*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (d^2*(3*b^2*c + 28*a^2*d)
*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 + n)) +
(d^3*(a + b*x)^(9 + n))/(b^9*(9 + n))
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Rubi [A] time = 0.431498, antiderivative size = 343, 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
\[ -\frac{2 a d^2 \left (28 a^2 d+9 b^2 c\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{d^2 \left (28 a^2 d+3 b^2 c\right ) (a+b x)^{n+7}}{b^9 (n+7)}+\frac{a^2 \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left (a^2 d+b^2 c\right )^2 \left (4 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{\left (a^2 d+b^2 c\right ) \left (28 a^4 d^2+17 a^2 b^2 c d+b^4 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}-\frac{4 a d \left (14 a^4 d^2+15 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{d \left (70 a^4 d^2+45 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+5}}{b^9 (n+5)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
(a^2*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^2*c + a^2*d)^2
*(b^2*c + 4*a^2*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^2*c + a^2*d)*(b^4*c^2
+ 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) - (4*a*d*(3*b^4*
c^2 + 15*a^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^(4 + n))/(b^9*(4 + n)) + (d*(3*b^4*
c^2 + 45*a^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^(5 + n))/(b^9*(5 + n)) - (2*a*d^2*(
9*b^2*c + 28*a^2*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (d^2*(3*b^2*c + 28*a^2*d)
*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 + n)) +
(d^3*(a + b*x)^(9 + n))/(b^9*(9 + n))
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Rubi in Sympy [A] time = 97.5123, size = 328, normalized size = 0.96 \[ \frac{a^{2} \left (a + b x\right )^{n + 1} \left (a^{2} d + b^{2} c\right )^{3}}{b^{9} \left (n + 1\right )} - \frac{8 a d^{3} \left (a + b x\right )^{n + 8}}{b^{9} \left (n + 8\right )} - \frac{2 a d^{2} \left (a + b x\right )^{n + 6} \left (28 a^{2} d + 9 b^{2} c\right )}{b^{9} \left (n + 6\right )} - \frac{4 a d \left (a + b x\right )^{n + 4} \left (14 a^{4} d^{2} + 15 a^{2} b^{2} c d + 3 b^{4} c^{2}\right )}{b^{9} \left (n + 4\right )} - \frac{2 a \left (a + b x\right )^{n + 2} \left (a^{2} d + b^{2} c\right )^{2} \left (4 a^{2} d + b^{2} c\right )}{b^{9} \left (n + 2\right )} + \frac{d^{3} \left (a + b x\right )^{n + 9}}{b^{9} \left (n + 9\right )} + \frac{d^{2} \left (a + b x\right )^{n + 7} \left (28 a^{2} d + 3 b^{2} c\right )}{b^{9} \left (n + 7\right )} + \frac{d \left (a + b x\right )^{n + 5} \left (70 a^{4} d^{2} + 45 a^{2} b^{2} c d + 3 b^{4} c^{2}\right )}{b^{9} \left (n + 5\right )} + \frac{\left (a + b x\right )^{n + 3} \left (a^{2} d + b^{2} c\right ) \left (28 a^{4} d^{2} + 17 a^{2} b^{2} c d + b^{4} c^{2}\right )}{b^{9} \left (n + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**n*(d*x**2+c)**3,x)
[Out]
a**2*(a + b*x)**(n + 1)*(a**2*d + b**2*c)**3/(b**9*(n + 1)) - 8*a*d**3*(a + b*x)
**(n + 8)/(b**9*(n + 8)) - 2*a*d**2*(a + b*x)**(n + 6)*(28*a**2*d + 9*b**2*c)/(b
**9*(n + 6)) - 4*a*d*(a + b*x)**(n + 4)*(14*a**4*d**2 + 15*a**2*b**2*c*d + 3*b**
4*c**2)/(b**9*(n + 4)) - 2*a*(a + b*x)**(n + 2)*(a**2*d + b**2*c)**2*(4*a**2*d +
b**2*c)/(b**9*(n + 2)) + d**3*(a + b*x)**(n + 9)/(b**9*(n + 9)) + d**2*(a + b*x
)**(n + 7)*(28*a**2*d + 3*b**2*c)/(b**9*(n + 7)) + d*(a + b*x)**(n + 5)*(70*a**4
*d**2 + 45*a**2*b**2*c*d + 3*b**4*c**2)/(b**9*(n + 5)) + (a + b*x)**(n + 3)*(a**
2*d + b**2*c)*(28*a**4*d**2 + 17*a**2*b**2*c*d + b**4*c**2)/(b**9*(n + 3))
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Mathematica [B] time = 1.1988, size = 746, normalized size = 2.17 \[ \frac{(a+b x)^{n+1} \left (40320 a^8 d^3-40320 a^7 b d^3 (n+1) x+720 a^6 b^2 d^2 \left (3 c \left (n^2+17 n+72\right )+28 d \left (n^2+3 n+2\right ) x^2\right )-240 a^5 b^3 d^2 (n+1) x \left (9 c \left (n^2+17 n+72\right )+28 d \left (n^2+5 n+6\right ) x^2\right )+24 a^4 b^4 d \left (3 c^2 \left (n^4+30 n^3+335 n^2+1650 n+3024\right )+45 c d \left (n^4+20 n^3+125 n^2+250 n+144\right ) x^2+70 d^2 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )-24 a^3 b^5 d (n+1) x \left (3 c^2 \left (n^4+30 n^3+335 n^2+1650 n+3024\right )+15 c d \left (n^4+22 n^3+163 n^2+462 n+432\right ) x^2+14 d^2 \left (n^4+14 n^3+71 n^2+154 n+120\right ) x^4\right )+2 a^2 b^6 \left (c^3 \left (n^6+39 n^5+625 n^4+5265 n^3+24574 n^2+60216 n+60480\right )+18 c^2 d \left (n^6+33 n^5+427 n^4+2715 n^3+8644 n^2+12372 n+6048\right ) x^2+45 c d^2 \left (n^6+27 n^5+277 n^4+1365 n^3+3394 n^2+4008 n+1728\right ) x^4+28 d^3 \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^6\right )-2 a b^7 (n+1) x \left (c^3 \left (n^6+39 n^5+625 n^4+5265 n^3+24574 n^2+60216 n+60480\right )+6 c^2 d \left (n^6+35 n^5+491 n^4+3505 n^3+13284 n^2+25020 n+18144\right ) x^2+9 c d^2 \left (n^6+31 n^5+381 n^4+2369 n^3+7850 n^2+13128 n+8640\right ) x^4+4 d^3 \left (n^6+27 n^5+295 n^4+1665 n^3+5104 n^2+8028 n+5040\right ) x^6\right )+b^8 \left (n^5+21 n^4+160 n^3+540 n^2+784 n+384\right ) x^2 \left (c^3 \left (n^3+21 n^2+143 n+315\right )+3 c^2 d \left (n^3+19 n^2+111 n+189\right ) x^2+3 c d^2 \left (n^3+17 n^2+87 n+135\right ) x^4+d^3 \left (n^3+15 n^2+71 n+105\right ) x^6\right )\right )}{b^9 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*(40320*a^8*d^3 - 40320*a^7*b*d^3*(1 + n)*x + 720*a^6*b^2*d^2*
(3*c*(72 + 17*n + n^2) + 28*d*(2 + 3*n + n^2)*x^2) - 240*a^5*b^3*d^2*(1 + n)*x*(
9*c*(72 + 17*n + n^2) + 28*d*(6 + 5*n + n^2)*x^2) + 24*a^4*b^4*d*(3*c^2*(3024 +
1650*n + 335*n^2 + 30*n^3 + n^4) + 45*c*d*(144 + 250*n + 125*n^2 + 20*n^3 + n^4)
*x^2 + 70*d^2*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4) - 24*a^3*b^5*d*(1 + n)*x*
(3*c^2*(3024 + 1650*n + 335*n^2 + 30*n^3 + n^4) + 15*c*d*(432 + 462*n + 163*n^2
+ 22*n^3 + n^4)*x^2 + 14*d^2*(120 + 154*n + 71*n^2 + 14*n^3 + n^4)*x^4) + b^8*(3
84 + 784*n + 540*n^2 + 160*n^3 + 21*n^4 + n^5)*x^2*(c^3*(315 + 143*n + 21*n^2 +
n^3) + 3*c^2*d*(189 + 111*n + 19*n^2 + n^3)*x^2 + 3*c*d^2*(135 + 87*n + 17*n^2 +
n^3)*x^4 + d^3*(105 + 71*n + 15*n^2 + n^3)*x^6) + 2*a^2*b^6*(c^3*(60480 + 60216
*n + 24574*n^2 + 5265*n^3 + 625*n^4 + 39*n^5 + n^6) + 18*c^2*d*(6048 + 12372*n +
8644*n^2 + 2715*n^3 + 427*n^4 + 33*n^5 + n^6)*x^2 + 45*c*d^2*(1728 + 4008*n + 3
394*n^2 + 1365*n^3 + 277*n^4 + 27*n^5 + n^6)*x^4 + 28*d^3*(720 + 1764*n + 1624*n
^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6)*x^6) - 2*a*b^7*(1 + n)*x*(c^3*(60480 + 60
216*n + 24574*n^2 + 5265*n^3 + 625*n^4 + 39*n^5 + n^6) + 6*c^2*d*(18144 + 25020*
n + 13284*n^2 + 3505*n^3 + 491*n^4 + 35*n^5 + n^6)*x^2 + 9*c*d^2*(8640 + 13128*n
+ 7850*n^2 + 2369*n^3 + 381*n^4 + 31*n^5 + n^6)*x^4 + 4*d^3*(5040 + 8028*n + 51
04*n^2 + 1665*n^3 + 295*n^4 + 27*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 = 0.024, size = 2232, normalized size = 6.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^n*(d*x^2+c)^3,x)
[Out]
(b*x+a)^(1+n)*(b^8*d^3*n^8*x^8+36*b^8*d^3*n^7*x^8-8*a*b^7*d^3*n^7*x^7+3*b^8*c*d^
2*n^8*x^6+546*b^8*d^3*n^6*x^8-224*a*b^7*d^3*n^6*x^7+114*b^8*c*d^2*n^7*x^6+4536*b
^8*d^3*n^5*x^8+56*a^2*b^6*d^3*n^6*x^6-18*a*b^7*c*d^2*n^7*x^5-2576*a*b^7*d^3*n^5*
x^7+3*b^8*c^2*d*n^8*x^4+1812*b^8*c*d^2*n^6*x^6+22449*b^8*d^3*n^4*x^8+1176*a^2*b^
6*d^3*n^5*x^6-576*a*b^7*c*d^2*n^6*x^5-15680*a*b^7*d^3*n^4*x^7+120*b^8*c^2*d*n^7*
x^4+15666*b^8*c*d^2*n^5*x^6+67284*b^8*d^3*n^3*x^8-336*a^3*b^5*d^3*n^5*x^5+90*a^2
*b^6*c*d^2*n^6*x^4+9800*a^2*b^6*d^3*n^4*x^6-12*a*b^7*c^2*d*n^7*x^3-7416*a*b^7*c*
d^2*n^5*x^5-54152*a*b^7*d^3*n^3*x^7+b^8*c^3*n^8*x^2+2010*b^8*c^2*d*n^6*x^4+80157
*b^8*c*d^2*n^4*x^6+118124*b^8*d^3*n^2*x^8-5040*a^3*b^5*d^3*n^4*x^5+2430*a^2*b^6*
c*d^2*n^5*x^4+41160*a^2*b^6*d^3*n^3*x^6-432*a*b^7*c^2*d*n^6*x^3-49500*a*b^7*c*d^
2*n^4*x^5-105056*a*b^7*d^3*n^2*x^7+42*b^8*c^3*n^7*x^2+18300*b^8*c^2*d*n^5*x^4+24
6876*b^8*c*d^2*n^3*x^6+109584*b^8*d^3*n*x^8+1680*a^4*b^4*d^3*n^4*x^4-360*a^3*b^5
*c*d^2*n^5*x^3-28560*a^3*b^5*d^3*n^3*x^5+36*a^2*b^6*c^2*d*n^6*x^2+24930*a^2*b^6*
c*d^2*n^4*x^4+90944*a^2*b^6*d^3*n^2*x^6-2*a*b^7*c^3*n^7*x-6312*a*b^7*c^2*d*n^5*x
^3-183942*a*b^7*c*d^2*n^3*x^5-104544*a*b^7*d^3*n*x^7+744*b^8*c^3*n^6*x^2+98319*b
^8*c^2*d*n^4*x^4+442908*b^8*c*d^2*n^2*x^6+40320*b^8*d^3*x^8+16800*a^4*b^4*d^3*n^
3*x^4-8280*a^3*b^5*c*d^2*n^4*x^3-75600*a^3*b^5*d^3*n^2*x^5+1188*a^2*b^6*c^2*d*n^
5*x^2+122850*a^2*b^6*c*d^2*n^3*x^4+98784*a^2*b^6*d^3*n*x^6-80*a*b^7*c^3*n^6*x-47
952*a*b^7*c^2*d*n^4*x^3-377604*a*b^7*c*d^2*n^2*x^5-40320*a*b^7*d^3*x^7+7218*b^8*
c^3*n^5*x^2+316380*b^8*c^2*d*n^3*x^4+417744*b^8*c*d^2*n*x^6-6720*a^5*b^3*d^3*n^3
*x^3+1080*a^4*b^4*c*d^2*n^4*x^2+58800*a^4*b^4*d^3*n^2*x^4-72*a^3*b^5*c^2*d*n^5*x
-66600*a^3*b^5*c*d^2*n^3*x^3-92064*a^3*b^5*d^3*n*x^5+2*a^2*b^6*c^3*n^6+15372*a^2
*b^6*c^2*d*n^4*x^2+305460*a^2*b^6*c*d^2*n^2*x^4+40320*a^2*b^6*d^3*x^6-1328*a*b^7
*c^3*n^5*x-201468*a*b^7*c^2*d*n^3*x^3-391824*a*b^7*c*d^2*n*x^5+41619*b^8*c^3*n^4
*x^2+589140*b^8*c^2*d*n^2*x^4+155520*b^8*c*d^2*x^6-40320*a^5*b^3*d^3*n^2*x^3+216
00*a^4*b^4*c*d^2*n^3*x^2+84000*a^4*b^4*d^3*n*x^4-2232*a^3*b^5*c^2*d*n^4*x-225000
*a^3*b^5*c*d^2*n^2*x^3-40320*a^3*b^5*d^3*x^5+78*a^2*b^6*c^3*n^5+97740*a^2*b^6*c^
2*d*n^3*x^2+360720*a^2*b^6*c*d^2*n*x^4-11780*a*b^7*c^3*n^4*x-459648*a*b^7*c^2*d*
n^2*x^3-155520*a*b^7*c*d^2*x^5+144468*b^8*c^3*n^3*x^2+572400*b^8*c^2*d*n*x^4+201
60*a^6*b^2*d^3*n^2*x^2-2160*a^5*b^3*c*d^2*n^3*x-73920*a^5*b^3*d^3*n*x^3+72*a^4*b
^4*c^2*d*n^4+135000*a^4*b^4*c*d^2*n^2*x^2+40320*a^4*b^4*d^3*x^4-26280*a^3*b^5*c^
2*d*n^3*x-321840*a^3*b^5*c*d^2*n*x^3+1250*a^2*b^6*c^3*n^4+311184*a^2*b^6*c^2*d*n
^2*x^2+155520*a^2*b^6*c*d^2*x^4-59678*a*b^7*c^3*n^3*x-517968*a*b^7*c^2*d*n*x^3+2
90276*b^8*c^3*n^2*x^2+217728*b^8*c^2*d*x^4+60480*a^6*b^2*d^3*n*x^2-38880*a^5*b^3
*c*d^2*n^2*x-40320*a^5*b^3*d^3*x^3+2160*a^4*b^4*c^2*d*n^3+270000*a^4*b^4*c*d^2*n
*x^2-142920*a^3*b^5*c^2*d*n^2*x-155520*a^3*b^5*c*d^2*x^3+10530*a^2*b^6*c^3*n^3+4
45392*a^2*b^6*c^2*d*n*x^2-169580*a*b^7*c^3*n^2*x-217728*a*b^7*c^2*d*x^3+301872*b
^8*c^3*n*x^2-40320*a^7*b*d^3*n*x+2160*a^6*b^2*c*d^2*n^2+40320*a^6*b^2*d^3*x^2-19
2240*a^5*b^3*c*d^2*n*x+24120*a^4*b^4*c^2*d*n^2+155520*a^4*b^4*c*d^2*x^2-336528*a
^3*b^5*c^2*d*n*x+49148*a^2*b^6*c^3*n^2+217728*a^2*b^6*c^2*d*x^2-241392*a*b^7*c^3
*n*x+120960*b^8*c^3*x^2-40320*a^7*b*d^3*x+36720*a^6*b^2*c*d^2*n-155520*a^5*b^3*c
*d^2*x+118800*a^4*b^4*c^2*d*n-217728*a^3*b^5*c^2*d*x+120432*a^2*b^6*c^3*n-120960
*a*b^7*c^3*x+40320*a^8*d^3+155520*a^6*b^2*c*d^2+217728*a^4*b^4*c^2*d+120960*a^2*
b^6*c^3)/b^9/(n^9+45*n^8+870*n^7+9450*n^6+63273*n^5+269325*n^4+723680*n^3+117270
0*n^2+1026576*n+362880)
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Maxima [A] time = 0.725939, size = 1073, normalized size = 3.13 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + 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^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^6 + 21*n^5 + 175*n^4 + 73
5*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 27
4*n^2 + 120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5
+ 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3
*x^3 + 360*(n^2 + n)*a^5*b^2*x^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a)^n*c*d^2/((
n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7)
+ ((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109
584*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^3/((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)*b^9)
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Fricas [A] time = 0.294397, size = 2923, normalized size = 8.52 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n*x^2,x, algorithm="fricas")
[Out]
(2*a^3*b^6*c^3*n^6 + 78*a^3*b^6*c^3*n^5 + 120960*a^3*b^6*c^3 + 217728*a^5*b^4*c^
2*d + 155520*a^7*b^2*c*d^2 + 40320*a^9*d^3 + (b^9*d^3*n^8 + 36*b^9*d^3*n^7 + 546
*b^9*d^3*n^6 + 4536*b^9*d^3*n^5 + 22449*b^9*d^3*n^4 + 67284*b^9*d^3*n^3 + 118124
*b^9*d^3*n^2 + 109584*b^9*d^3*n + 40320*b^9*d^3)*x^9 + (a*b^8*d^3*n^8 + 28*a*b^8
*d^3*n^7 + 322*a*b^8*d^3*n^6 + 1960*a*b^8*d^3*n^5 + 6769*a*b^8*d^3*n^4 + 13132*a
*b^8*d^3*n^3 + 13068*a*b^8*d^3*n^2 + 5040*a*b^8*d^3*n)*x^8 + (3*b^9*c*d^2*n^8 +
155520*b^9*c*d^2 + 2*(57*b^9*c*d^2 - 4*a^2*b^7*d^3)*n^7 + 12*(151*b^9*c*d^2 - 14
*a^2*b^7*d^3)*n^6 + 14*(1119*b^9*c*d^2 - 100*a^2*b^7*d^3)*n^5 + 21*(3817*b^9*c*d
^2 - 280*a^2*b^7*d^3)*n^4 + 28*(8817*b^9*c*d^2 - 464*a^2*b^7*d^3)*n^3 + 36*(1230
3*b^9*c*d^2 - 392*a^2*b^7*d^3)*n^2 + 144*(2901*b^9*c*d^2 - 40*a^2*b^7*d^3)*n)*x^
7 + (3*a*b^8*c*d^2*n^8 + 96*a*b^8*c*d^2*n^7 + 4*(309*a*b^8*c*d^2 + 14*a^3*b^6*d^
3)*n^6 + 30*(275*a*b^8*c*d^2 + 28*a^3*b^6*d^3)*n^5 + (30657*a*b^8*c*d^2 + 4760*a
^3*b^6*d^3)*n^4 + 6*(10489*a*b^8*c*d^2 + 2100*a^3*b^6*d^3)*n^3 + 8*(8163*a*b^8*c
*d^2 + 1918*a^3*b^6*d^3)*n^2 + 960*(27*a*b^8*c*d^2 + 7*a^3*b^6*d^3)*n)*x^6 + 3*(
b^9*c^2*d*n^8 + 72576*b^9*c^2*d + 2*(20*b^9*c^2*d - 3*a^2*b^7*c*d^2)*n^7 + 2*(33
5*b^9*c^2*d - 81*a^2*b^7*c*d^2)*n^6 + 2*(3050*b^9*c^2*d - 831*a^2*b^7*c*d^2 - 56
*a^4*b^5*d^3)*n^5 + (32773*b^9*c^2*d - 8190*a^2*b^7*c*d^2 - 1120*a^4*b^5*d^3)*n^
4 + 4*(26365*b^9*c^2*d - 5091*a^2*b^7*c*d^2 - 980*a^4*b^5*d^3)*n^3 + 4*(49095*b^
9*c^2*d - 6012*a^2*b^7*c*d^2 - 1400*a^4*b^5*d^3)*n^2 + 48*(3975*b^9*c^2*d - 216*
a^2*b^7*c*d^2 - 56*a^4*b^5*d^3)*n)*x^5 + 2*(625*a^3*b^6*c^3 + 36*a^5*b^4*c^2*d)*
n^4 + 3*(a*b^8*c^2*d*n^8 + 36*a*b^8*c^2*d*n^7 + 2*(263*a*b^8*c^2*d + 15*a^3*b^6*
c*d^2)*n^6 + 6*(666*a*b^8*c^2*d + 115*a^3*b^6*c*d^2)*n^5 + (16789*a*b^8*c^2*d +
5550*a^3*b^6*c*d^2 + 560*a^5*b^4*d^3)*n^4 + 6*(6384*a*b^8*c^2*d + 3125*a^3*b^6*c
*d^2 + 560*a^5*b^4*d^3)*n^3 + 4*(10791*a*b^8*c^2*d + 6705*a^3*b^6*c*d^2 + 1540*a
^5*b^4*d^3)*n^2 + 96*(189*a*b^8*c^2*d + 135*a^3*b^6*c*d^2 + 35*a^5*b^4*d^3)*n)*x
^4 + 270*(39*a^3*b^6*c^3 + 8*a^5*b^4*c^2*d)*n^3 + (b^9*c^3*n^8 + 120960*b^9*c^3
+ 6*(7*b^9*c^3 - 2*a^2*b^7*c^2*d)*n^7 + 12*(62*b^9*c^3 - 33*a^2*b^7*c^2*d)*n^6 +
6*(1203*b^9*c^3 - 854*a^2*b^7*c^2*d - 60*a^4*b^5*c*d^2)*n^5 + 3*(13873*b^9*c^3
- 10860*a^2*b^7*c^2*d - 2400*a^4*b^5*c*d^2)*n^4 + 12*(12039*b^9*c^3 - 8644*a^2*b
^7*c^2*d - 3750*a^4*b^5*c*d^2 - 560*a^6*b^3*d^3)*n^3 + 4*(72569*b^9*c^3 - 37116*
a^2*b^7*c^2*d - 22500*a^4*b^5*c*d^2 - 5040*a^6*b^3*d^3)*n^2 + 48*(6289*b^9*c^3 -
1512*a^2*b^7*c^2*d - 1080*a^4*b^5*c*d^2 - 280*a^6*b^3*d^3)*n)*x^3 + 4*(12287*a^
3*b^6*c^3 + 6030*a^5*b^4*c^2*d + 540*a^7*b^2*c*d^2)*n^2 + (a*b^8*c^3*n^8 + 40*a*
b^8*c^3*n^7 + 4*(166*a*b^8*c^3 + 9*a^3*b^6*c^2*d)*n^6 + 62*(95*a*b^8*c^3 + 18*a^
3*b^6*c^2*d)*n^5 + (29839*a*b^8*c^3 + 13140*a^3*b^6*c^2*d + 1080*a^5*b^4*c*d^2)*
n^4 + 10*(8479*a*b^8*c^3 + 7146*a^3*b^6*c^2*d + 1944*a^5*b^4*c*d^2)*n^3 + 24*(50
29*a*b^8*c^3 + 7011*a^3*b^6*c^2*d + 4005*a^5*b^4*c*d^2 + 840*a^7*b^2*d^3)*n^2 +
576*(105*a*b^8*c^3 + 189*a^3*b^6*c^2*d + 135*a^5*b^4*c*d^2 + 35*a^7*b^2*d^3)*n)*
x^2 + 48*(2509*a^3*b^6*c^3 + 2475*a^5*b^4*c^2*d + 765*a^7*b^2*c*d^2)*n - 2*(a^2*
b^7*c^3*n^7 + 39*a^2*b^7*c^3*n^6 + (625*a^2*b^7*c^3 + 36*a^4*b^5*c^2*d)*n^5 + 13
5*(39*a^2*b^7*c^3 + 8*a^4*b^5*c^2*d)*n^4 + 2*(12287*a^2*b^7*c^3 + 6030*a^4*b^5*c
^2*d + 540*a^6*b^3*c*d^2)*n^3 + 24*(2509*a^2*b^7*c^3 + 2475*a^4*b^5*c^2*d + 765*
a^6*b^3*c*d^2)*n^2 + 576*(105*a^2*b^7*c^3 + 189*a^4*b^5*c^2*d + 135*a^6*b^3*c*d^
2 + 35*a^8*b*d^3)*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 + 102
6576*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**2+c)**3,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.308134, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n*x^2,x, algorithm="giac")
[Out]
Done