3.2146 \(\int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)
Optimal. Leaf size=239 \[ -\frac{7 b^6 (b d-a e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{m+6}}{e^8 (m+6)}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{m+5}}{e^8 (m+5)}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{m+4}}{e^8 (m+4)}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{m+3}}{e^8 (m+3)}-\frac{(b d-a e)^7 (d+e x)^{m+1}}{e^8 (m+1)}+\frac{7 b (b d-a e)^6 (d+e x)^{m+2}}{e^8 (m+2)}+\frac{b^7 (d+e x)^{m+8}}{e^8 (m+8)} \]
[Out]
-(((b*d - a*e)^7*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (7*b*(b*d - a*e)^6*(d + e*x
)^(2 + m))/(e^8*(2 + m)) - (21*b^2*(b*d - a*e)^5*(d + e*x)^(3 + m))/(e^8*(3 + m)
) + (35*b^3*(b*d - a*e)^4*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (35*b^4*(b*d - a*e)
^3*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (21*b^5*(b*d - a*e)^2*(d + e*x)^(6 + m))/(
e^8*(6 + m)) - (7*b^6*(b*d - a*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (b^7*(d + e
*x)^(8 + m))/(e^8*(8 + m))
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Rubi [A] time = 0.310992, antiderivative size = 239, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065
\[ -\frac{7 b^6 (b d-a e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{m+6}}{e^8 (m+6)}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{m+5}}{e^8 (m+5)}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{m+4}}{e^8 (m+4)}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{m+3}}{e^8 (m+3)}-\frac{(b d-a e)^7 (d+e x)^{m+1}}{e^8 (m+1)}+\frac{7 b (b d-a e)^6 (d+e x)^{m+2}}{e^8 (m+2)}+\frac{b^7 (d+e x)^{m+8}}{e^8 (m+8)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
-(((b*d - a*e)^7*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (7*b*(b*d - a*e)^6*(d + e*x
)^(2 + m))/(e^8*(2 + m)) - (21*b^2*(b*d - a*e)^5*(d + e*x)^(3 + m))/(e^8*(3 + m)
) + (35*b^3*(b*d - a*e)^4*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (35*b^4*(b*d - a*e)
^3*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (21*b^5*(b*d - a*e)^2*(d + e*x)^(6 + m))/(
e^8*(6 + m)) - (7*b^6*(b*d - a*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (b^7*(d + e
*x)^(8 + m))/(e^8*(8 + m))
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Rubi in Sympy [A] time = 162.46, size = 211, normalized size = 0.88 \[ \frac{b^{7} \left (d + e x\right )^{m + 8}}{e^{8} \left (m + 8\right )} + \frac{7 b^{6} \left (d + e x\right )^{m + 7} \left (a e - b d\right )}{e^{8} \left (m + 7\right )} + \frac{21 b^{5} \left (d + e x\right )^{m + 6} \left (a e - b d\right )^{2}}{e^{8} \left (m + 6\right )} + \frac{35 b^{4} \left (d + e x\right )^{m + 5} \left (a e - b d\right )^{3}}{e^{8} \left (m + 5\right )} + \frac{35 b^{3} \left (d + e x\right )^{m + 4} \left (a e - b d\right )^{4}}{e^{8} \left (m + 4\right )} + \frac{21 b^{2} \left (d + e x\right )^{m + 3} \left (a e - b d\right )^{5}}{e^{8} \left (m + 3\right )} + \frac{7 b \left (d + e x\right )^{m + 2} \left (a e - b d\right )^{6}}{e^{8} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right )^{7}}{e^{8} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
b**7*(d + e*x)**(m + 8)/(e**8*(m + 8)) + 7*b**6*(d + e*x)**(m + 7)*(a*e - b*d)/(
e**8*(m + 7)) + 21*b**5*(d + e*x)**(m + 6)*(a*e - b*d)**2/(e**8*(m + 6)) + 35*b*
*4*(d + e*x)**(m + 5)*(a*e - b*d)**3/(e**8*(m + 5)) + 35*b**3*(d + e*x)**(m + 4)
*(a*e - b*d)**4/(e**8*(m + 4)) + 21*b**2*(d + e*x)**(m + 3)*(a*e - b*d)**5/(e**8
*(m + 3)) + 7*b*(d + e*x)**(m + 2)*(a*e - b*d)**6/(e**8*(m + 2)) + (d + e*x)**(m
+ 1)*(a*e - b*d)**7/(e**8*(m + 1))
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Mathematica [B] time = 1.67073, size = 896, normalized size = 3.75 \[ \frac{(d+e x)^{m+1} \left (-\left (5040 d^7-5040 e (m+1) x d^6+2520 e^2 \left (m^2+3 m+2\right ) x^2 d^5-840 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^4+210 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^3-42 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d^2+7 e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6 d-e^7 \left (m^7+28 m^6+322 m^5+1960 m^4+6769 m^3+13132 m^2+13068 m+5040\right ) x^7\right ) b^7+7 a e (m+8) \left (720 d^6-720 e (m+1) x d^5+360 e^2 \left (m^2+3 m+2\right ) x^2 d^4-120 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^3+30 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^2-6 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right ) b^6+21 a^2 e^2 \left (m^2+15 m+56\right ) \left (-120 d^5+120 e (m+1) x d^4-60 e^2 \left (m^2+3 m+2\right ) x^2 d^3+20 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^2-5 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right ) b^5+35 a^3 e^3 \left (m^3+21 m^2+146 m+336\right ) \left (24 d^4-24 e (m+1) x d^3+12 e^2 \left (m^2+3 m+2\right ) x^2 d^2-4 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right ) b^4+35 a^4 e^4 \left (m^4+26 m^3+251 m^2+1066 m+1680\right ) \left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right ) b^3+21 a^5 e^5 \left (m^5+30 m^4+355 m^3+2070 m^2+5944 m+6720\right ) \left (2 d^2-2 e (m+1) x d+e^2 \left (m^2+3 m+2\right ) x^2\right ) b^2-7 a^6 e^6 \left (m^6+33 m^5+445 m^4+3135 m^3+12154 m^2+24552 m+20160\right ) (d-e (m+1) x) b+a^7 e^7 \left (m^7+35 m^6+511 m^5+4025 m^4+18424 m^3+48860 m^2+69264 m+40320\right )\right )}{e^8 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
((d + e*x)^(1 + m)*(a^7*e^7*(40320 + 69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4
+ 511*m^5 + 35*m^6 + m^7) - 7*a^6*b*e^6*(20160 + 24552*m + 12154*m^2 + 3135*m^3
+ 445*m^4 + 33*m^5 + m^6)*(d - e*(1 + m)*x) + 21*a^5*b^2*e^5*(6720 + 5944*m + 20
70*m^2 + 355*m^3 + 30*m^4 + m^5)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*
x^2) + 35*a^4*b^3*e^4*(1680 + 1066*m + 251*m^2 + 26*m^3 + m^4)*(-6*d^3 + 6*d^2*e
*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)*x^3) + 3
5*a^3*b^4*e^3*(336 + 146*m + 21*m^2 + m^3)*(24*d^4 - 24*d^3*e*(1 + m)*x + 12*d^2
*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + e^4*(24 + 50*m
+ 35*m^2 + 10*m^3 + m^4)*x^4) + 21*a^2*b^5*e^2*(56 + 15*m + m^2)*(-120*d^5 + 12
0*d^4*e*(1 + m)*x - 60*d^3*e^2*(2 + 3*m + m^2)*x^2 + 20*d^2*e^3*(6 + 11*m + 6*m^
2 + m^3)*x^3 - 5*d*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 + e^5*(120 + 274*
m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5) + 7*a*b^6*e*(8 + m)*(720*d^6 - 720*d^5
*e*(1 + m)*x + 360*d^4*e^2*(2 + 3*m + m^2)*x^2 - 120*d^3*e^3*(6 + 11*m + 6*m^2 +
m^3)*x^3 + 30*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*d*e^5*(120 +
274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2 + 73
5*m^3 + 175*m^4 + 21*m^5 + m^6)*x^6) - b^7*(5040*d^7 - 5040*d^6*e*(1 + m)*x + 25
20*d^5*e^2*(2 + 3*m + m^2)*x^2 - 840*d^4*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + 210*
d^3*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 42*d^2*e^5*(120 + 274*m + 225*
m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + 7*d*e^6*(720 + 1764*m + 1624*m^2 + 735*m^3 +
175*m^4 + 21*m^5 + m^6)*x^6 - e^7*(5040 + 13068*m + 13132*m^2 + 6769*m^3 + 1960*
m^4 + 322*m^5 + 28*m^6 + m^7)*x^7)))/(e^8*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m
)*(6 + m)*(7 + m)*(8 + m))
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Maple [B] time = 0.026, size = 3244, normalized size = 13.6 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
(e*x+d)^(1+m)*(b^7*e^7*m^7*x^7+7*a*b^6*e^7*m^7*x^6+28*b^7*e^7*m^6*x^7+21*a^2*b^5
*e^7*m^7*x^5+203*a*b^6*e^7*m^6*x^6-7*b^7*d*e^6*m^6*x^6+322*b^7*e^7*m^5*x^7+35*a^
3*b^4*e^7*m^7*x^4+630*a^2*b^5*e^7*m^6*x^5-42*a*b^6*d*e^6*m^6*x^5+2401*a*b^6*e^7*
m^5*x^6-147*b^7*d*e^6*m^5*x^6+1960*b^7*e^7*m^4*x^7+35*a^4*b^3*e^7*m^7*x^3+1085*a
^3*b^4*e^7*m^6*x^4-105*a^2*b^5*d*e^6*m^6*x^4+7686*a^2*b^5*e^7*m^5*x^5-966*a*b^6*
d*e^6*m^5*x^5+14945*a*b^6*e^7*m^4*x^6+42*b^7*d^2*e^5*m^5*x^5-1225*b^7*d*e^6*m^4*
x^6+6769*b^7*e^7*m^3*x^7+21*a^5*b^2*e^7*m^7*x^2+1120*a^4*b^3*e^7*m^6*x^3-140*a^3
*b^4*d*e^6*m^6*x^3+13685*a^3*b^4*e^7*m^5*x^4-2625*a^2*b^5*d*e^6*m^5*x^4+49140*a^
2*b^5*e^7*m^4*x^5+210*a*b^6*d^2*e^5*m^5*x^4-8610*a*b^6*d*e^6*m^4*x^5+52528*a*b^6
*e^7*m^3*x^6+630*b^7*d^2*e^5*m^4*x^5-5145*b^7*d*e^6*m^3*x^6+13132*b^7*e^7*m^2*x^
7+7*a^6*b*e^7*m^7*x+693*a^5*b^2*e^7*m^6*x^2-105*a^4*b^3*d*e^6*m^6*x^2+14630*a^4*
b^3*e^7*m^5*x^3-3780*a^3*b^4*d*e^6*m^5*x^3+90335*a^3*b^4*e^7*m^4*x^4+420*a^2*b^5
*d^2*e^5*m^5*x^3-25305*a^2*b^5*d*e^6*m^4*x^4+176589*a^2*b^5*e^7*m^3*x^5+3780*a*b
^6*d^2*e^5*m^4*x^4-38010*a*b^6*d*e^6*m^3*x^5+103292*a*b^6*e^7*m^2*x^6-210*b^7*d^
3*e^4*m^4*x^4+3570*b^7*d^2*e^5*m^3*x^5-11368*b^7*d*e^6*m^2*x^6+13068*b^7*e^7*m*x
^7+a^7*e^7*m^7+238*a^6*b*e^7*m^6*x-42*a^5*b^2*d*e^6*m^6*x+9387*a^5*b^2*e^7*m^5*x
^2-3045*a^4*b^3*d*e^6*m^5*x^2+100240*a^4*b^3*e^7*m^4*x^3+420*a^3*b^4*d^2*e^5*m^5
*x^2-39620*a^3*b^4*d*e^6*m^4*x^3+334040*a^3*b^4*e^7*m^3*x^4+8820*a^2*b^5*d^2*e^5
*m^4*x^3-119175*a^2*b^5*d*e^6*m^3*x^4+353430*a^2*b^5*e^7*m^2*x^5-840*a*b^6*d^3*e
^4*m^4*x^3+24150*a*b^6*d^2*e^5*m^3*x^4-87108*a*b^6*d*e^6*m^2*x^5+103824*a*b^6*e^
7*m*x^6-2100*b^7*d^3*e^4*m^3*x^4+9450*b^7*d^2*e^5*m^2*x^5-12348*b^7*d*e^6*m*x^6+
5040*b^7*e^7*x^7+35*a^7*e^7*m^6-7*a^6*b*d*e^6*m^6+3346*a^6*b*e^7*m^5*x-1302*a^5*
b^2*d*e^6*m^5*x+67095*a^5*b^2*e^7*m^4*x^2+210*a^4*b^3*d^2*e^5*m^5*x-34755*a^4*b^
3*d*e^6*m^4*x^2+384755*a^4*b^3*e^7*m^3*x^3+10080*a^3*b^4*d^2*e^5*m^4*x^2-202860*
a^3*b^4*d*e^6*m^3*x^3+684740*a^3*b^4*e^7*m^2*x^4-1260*a^2*b^5*d^3*e^4*m^4*x^2+65
940*a^2*b^5*d^2*e^5*m^3*x^3-287070*a^2*b^5*d*e^6*m^2*x^4+360024*a^2*b^5*e^7*m*x^
5-11760*a*b^6*d^3*e^4*m^3*x^3+69300*a*b^6*d^2*e^5*m^2*x^4-97104*a*b^6*d*e^6*m*x^
5+40320*a*b^6*e^7*x^6+840*b^7*d^4*e^3*m^3*x^3-7350*b^7*d^3*e^4*m^2*x^4+11508*b^7
*d^2*e^5*m*x^5-5040*b^7*d*e^6*x^6+511*a^7*e^7*m^5-231*a^6*b*d*e^6*m^5+25060*a^6*
b*e^7*m^4*x+42*a^5*b^2*d^2*e^5*m^5-16170*a^5*b^2*d*e^6*m^4*x+270144*a^5*b^2*e^7*
m^3*x^2+5670*a^4*b^3*d^2*e^5*m^4*x-196455*a^4*b^3*d*e^6*m^3*x^2+815920*a^4*b^3*e
^7*m^2*x^3-840*a^3*b^4*d^3*e^4*m^4*x+88620*a^3*b^4*d^2*e^5*m^3*x^2-524720*a^3*b^
4*d*e^6*m^2*x^3+710640*a^3*b^4*e^7*m*x^4-22680*a^2*b^5*d^3*e^4*m^3*x^2+212940*a^
2*b^5*d^2*e^5*m^2*x^3-331800*a^2*b^5*d*e^6*m*x^4+141120*a^2*b^5*e^7*x^5+2520*a*b
^6*d^4*e^3*m^3*x^2-49560*a*b^6*d^3*e^4*m^2*x^3+89040*a*b^6*d^2*e^5*m*x^4-40320*a
*b^6*d*e^6*x^5+5040*b^7*d^4*e^3*m^2*x^3-10500*b^7*d^3*e^4*m*x^4+5040*b^7*d^2*e^5
*x^5+4025*a^7*e^7*m^4-3115*a^6*b*d*e^6*m^4+107023*a^6*b*e^7*m^3*x+1260*a^5*b^2*d
^2*e^5*m^4-101850*a^5*b^2*d*e^6*m^3*x+602532*a^5*b^2*e^7*m^2*x^2-210*a^4*b^3*d^3
*e^4*m^4+58170*a^4*b^3*d^2*e^5*m^3*x-564900*a^4*b^3*d*e^6*m^2*x^2+870660*a^4*b^3
*e^7*m*x^3-18480*a^3*b^4*d^3*e^4*m^3*x+342720*a^3*b^4*d^2*e^5*m^2*x^2-640080*a^3
*b^4*d*e^6*m*x^3+282240*a^3*b^4*e^7*x^4+2520*a^2*b^5*d^4*e^3*m^3*x-129780*a^2*b^
5*d^3*e^4*m^2*x^2+296520*a^2*b^5*d^2*e^5*m*x^3-141120*a^2*b^5*d*e^6*x^4+27720*a*
b^6*d^4*e^3*m^2*x^2-78960*a*b^6*d^3*e^4*m*x^3+40320*a*b^6*d^2*e^5*x^4-2520*b^7*d
^5*e^2*m^2*x^2+9240*b^7*d^4*e^3*m*x^3-5040*b^7*d^3*e^4*x^4+18424*a^7*e^7*m^3-219
45*a^6*b*d*e^6*m^3+256942*a^6*b*e^7*m^2*x+14910*a^5*b^2*d^2*e^5*m^3-336588*a^5*b
^2*d*e^6*m^2*x+673008*a^5*b^2*e^7*m*x^2-5460*a^4*b^3*d^3*e^4*m^3+276570*a^4*b^3*
d^2*e^5*m^2*x-753060*a^4*b^3*d*e^6*m*x^2+352800*a^4*b^3*e^7*x^3+840*a^3*b^4*d^4*
e^3*m^3-140280*a^3*b^4*d^3*e^4*m^2*x+546000*a^3*b^4*d^2*e^5*m*x^2-282240*a^3*b^4
*d*e^6*x^3+40320*a^2*b^5*d^4*e^3*m^2*x-249480*a^2*b^5*d^3*e^4*m*x^2+141120*a^2*b
^5*d^2*e^5*x^3-5040*a*b^6*d^5*e^2*m^2*x+65520*a*b^6*d^4*e^3*m*x^2-40320*a*b^6*d^
3*e^4*x^3-7560*b^7*d^5*e^2*m*x^2+5040*b^7*d^4*e^3*x^3+48860*a^7*e^7*m^2-85078*a^
6*b*d*e^6*m^2+312984*a^6*b*e^7*m*x+86940*a^5*b^2*d^2*e^5*m^2-531888*a^5*b^2*d*e^
6*m*x+282240*a^5*b^2*e^7*x^2-52710*a^4*b^3*d^3*e^4*m^2+576660*a^4*b^3*d^2*e^5*m*
x-352800*a^4*b^3*d*e^6*x^2+17640*a^3*b^4*d^4*e^3*m^2-404880*a^3*b^4*d^3*e^4*m*x+
282240*a^3*b^4*d^2*e^5*x^2-2520*a^2*b^5*d^5*e^2*m^2+178920*a^2*b^5*d^4*e^3*m*x-1
41120*a^2*b^5*d^3*e^4*x^2-45360*a*b^6*d^5*e^2*m*x+40320*a*b^6*d^4*e^3*x^2+5040*b
^7*d^6*e*m*x-5040*b^7*d^5*e^2*x^2+69264*a^7*e^7*m-171864*a^6*b*d*e^6*m+141120*a^
6*b*e^7*x+249648*a^5*b^2*d^2*e^5*m-282240*a^5*b^2*d*e^6*x-223860*a^4*b^3*d^3*e^4
*m+352800*a^4*b^3*d^2*e^5*x+122640*a^3*b^4*d^4*e^3*m-282240*a^3*b^4*d^3*e^4*x-37
800*a^2*b^5*d^5*e^2*m+141120*a^2*b^5*d^4*e^3*x+5040*a*b^6*d^6*e*m-40320*a*b^6*d^
5*e^2*x+5040*b^7*d^6*e*x+40320*a^7*e^7-141120*a^6*b*d*e^6+282240*a^5*b^2*d^2*e^5
-352800*a^4*b^3*d^3*e^4+282240*a^3*b^4*d^4*e^3-141120*a^2*b^5*d^5*e^2+40320*a*b^
6*d^6*e-5040*b^7*d^7)/e^8/(m^8+36*m^7+546*m^6+4536*m^5+22449*m^4+67284*m^3+11812
4*m^2+109584*m+40320)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.325996, size = 4321, normalized size = 18.08 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a^7*d*e^7*m^7 - 5040*b^7*d^8 + 40320*a*b^6*d^7*e - 141120*a^2*b^5*d^6*e^2 + 282
240*a^3*b^4*d^5*e^3 - 352800*a^4*b^3*d^4*e^4 + 282240*a^5*b^2*d^3*e^5 - 141120*a
^6*b*d^2*e^6 + 40320*a^7*d*e^7 + (b^7*e^8*m^7 + 28*b^7*e^8*m^6 + 322*b^7*e^8*m^5
+ 1960*b^7*e^8*m^4 + 6769*b^7*e^8*m^3 + 13132*b^7*e^8*m^2 + 13068*b^7*e^8*m + 5
040*b^7*e^8)*x^8 + (40320*a*b^6*e^8 + (b^7*d*e^7 + 7*a*b^6*e^8)*m^7 + 7*(3*b^7*d
*e^7 + 29*a*b^6*e^8)*m^6 + 7*(25*b^7*d*e^7 + 343*a*b^6*e^8)*m^5 + 245*(3*b^7*d*e
^7 + 61*a*b^6*e^8)*m^4 + 56*(29*b^7*d*e^7 + 938*a*b^6*e^8)*m^3 + 196*(9*b^7*d*e^
7 + 527*a*b^6*e^8)*m^2 + 144*(5*b^7*d*e^7 + 721*a*b^6*e^8)*m)*x^7 - 7*(a^6*b*d^2
*e^6 - 5*a^7*d*e^7)*m^6 + 7*(20160*a^2*b^5*e^8 + (a*b^6*d*e^7 + 3*a^2*b^5*e^8)*m
^7 - (b^7*d^2*e^6 - 23*a*b^6*d*e^7 - 90*a^2*b^5*e^8)*m^6 - (15*b^7*d^2*e^6 - 205
*a*b^6*d*e^7 - 1098*a^2*b^5*e^8)*m^5 - 5*(17*b^7*d^2*e^6 - 181*a*b^6*d*e^7 - 140
4*a^2*b^5*e^8)*m^4 - (225*b^7*d^2*e^6 - 2074*a*b^6*d*e^7 - 25227*a^2*b^5*e^8)*m^
3 - 2*(137*b^7*d^2*e^6 - 1156*a*b^6*d*e^7 - 25245*a^2*b^5*e^8)*m^2 - 24*(5*b^7*d
^2*e^6 - 40*a*b^6*d*e^7 - 2143*a^2*b^5*e^8)*m)*x^6 + 7*(6*a^5*b^2*d^3*e^5 - 33*a
^6*b*d^2*e^6 + 73*a^7*d*e^7)*m^5 + 7*(40320*a^3*b^4*e^8 + (3*a^2*b^5*d*e^7 + 5*a
^3*b^4*e^8)*m^7 - (6*a*b^6*d^2*e^6 - 75*a^2*b^5*d*e^7 - 155*a^3*b^4*e^8)*m^6 + (
6*b^7*d^3*e^5 - 108*a*b^6*d^2*e^6 + 723*a^2*b^5*d*e^7 + 1955*a^3*b^4*e^8)*m^5 +
5*(12*b^7*d^3*e^5 - 138*a*b^6*d^2*e^6 + 681*a^2*b^5*d*e^7 + 2581*a^3*b^4*e^8)*m^
4 + 2*(105*b^7*d^3*e^5 - 990*a*b^6*d^2*e^6 + 4101*a^2*b^5*d*e^7 + 23860*a^3*b^4*
e^8)*m^3 + 4*(75*b^7*d^3*e^5 - 636*a*b^6*d^2*e^6 + 2370*a^2*b^5*d*e^7 + 24455*a^
3*b^4*e^8)*m^2 + 144*(b^7*d^3*e^5 - 8*a*b^6*d^2*e^6 + 28*a^2*b^5*d*e^7 + 705*a^3
*b^4*e^8)*m)*x^5 - 35*(6*a^4*b^3*d^4*e^4 - 36*a^5*b^2*d^3*e^5 + 89*a^6*b*d^2*e^6
- 115*a^7*d*e^7)*m^4 + 35*(10080*a^4*b^3*e^8 + (a^3*b^4*d*e^7 + a^4*b^3*e^8)*m^
7 - (3*a^2*b^5*d^2*e^6 - 27*a^3*b^4*d*e^7 - 32*a^4*b^3*e^8)*m^6 + (6*a*b^6*d^3*e
^5 - 63*a^2*b^5*d^2*e^6 + 283*a^3*b^4*d*e^7 + 418*a^4*b^3*e^8)*m^5 - (6*b^7*d^4*
e^4 - 84*a*b^6*d^3*e^5 + 471*a^2*b^5*d^2*e^6 - 1449*a^3*b^4*d*e^7 - 2864*a^4*b^3
*e^8)*m^4 - (36*b^7*d^4*e^4 - 354*a*b^6*d^3*e^5 + 1521*a^2*b^5*d^2*e^6 - 3748*a^
3*b^4*d*e^7 - 10993*a^4*b^3*e^8)*m^3 - 2*(33*b^7*d^4*e^4 - 282*a*b^6*d^3*e^5 + 1
059*a^2*b^5*d^2*e^6 - 2286*a^3*b^4*d*e^7 - 11656*a^4*b^3*e^8)*m^2 - 36*(b^7*d^4*
e^4 - 8*a*b^6*d^3*e^5 + 28*a^2*b^5*d^2*e^6 - 56*a^3*b^4*d*e^7 - 691*a^4*b^3*e^8)
*m)*x^4 + 7*(120*a^3*b^4*d^5*e^3 - 780*a^4*b^3*d^4*e^4 + 2130*a^5*b^2*d^3*e^5 -
3135*a^6*b*d^2*e^6 + 2632*a^7*d*e^7)*m^3 + 7*(40320*a^5*b^2*e^8 + (5*a^4*b^3*d*e
^7 + 3*a^5*b^2*e^8)*m^7 - (20*a^3*b^4*d^2*e^6 - 145*a^4*b^3*d*e^7 - 99*a^5*b^2*e
^8)*m^6 + (60*a^2*b^5*d^3*e^5 - 480*a^3*b^4*d^2*e^6 + 1655*a^4*b^3*d*e^7 + 1341*
a^5*b^2*e^8)*m^5 - 5*(24*a*b^6*d^4*e^4 - 216*a^2*b^5*d^3*e^5 + 844*a^3*b^4*d^2*e
^6 - 1871*a^4*b^3*d*e^7 - 1917*a^5*b^2*e^8)*m^4 + 4*(30*b^7*d^5*e^3 - 330*a*b^6*
d^4*e^4 + 1545*a^2*b^5*d^3*e^5 - 4080*a^3*b^4*d^2*e^6 + 6725*a^4*b^3*d*e^7 + 964
8*a^5*b^2*e^8)*m^3 + 4*(90*b^7*d^5*e^3 - 780*a*b^6*d^4*e^4 + 2970*a^2*b^5*d^3*e^
5 - 6500*a^3*b^4*d^2*e^6 + 8965*a^4*b^3*d*e^7 + 21519*a^5*b^2*e^8)*m^2 + 48*(5*b
^7*d^5*e^3 - 40*a*b^6*d^4*e^4 + 140*a^2*b^5*d^3*e^5 - 280*a^3*b^4*d^2*e^6 + 350*
a^4*b^3*d*e^7 + 2003*a^5*b^2*e^8)*m)*x^3 - 14*(180*a^2*b^5*d^6*e^2 - 1260*a^3*b^
4*d^5*e^3 + 3765*a^4*b^3*d^4*e^4 - 6210*a^5*b^2*d^3*e^5 + 6077*a^6*b*d^2*e^6 - 3
490*a^7*d*e^7)*m^2 + 7*(20160*a^6*b*e^8 + (3*a^5*b^2*d*e^7 + a^6*b*e^8)*m^7 - (1
5*a^4*b^3*d^2*e^6 - 93*a^5*b^2*d*e^7 - 34*a^6*b*e^8)*m^6 + (60*a^3*b^4*d^3*e^5 -
405*a^4*b^3*d^2*e^6 + 1155*a^5*b^2*d*e^7 + 478*a^6*b*e^8)*m^5 - 5*(36*a^2*b^5*d
^4*e^4 - 264*a^3*b^4*d^3*e^5 + 831*a^4*b^3*d^2*e^6 - 1455*a^5*b^2*d*e^7 - 716*a^
6*b*e^8)*m^4 + (360*a*b^6*d^5*e^3 - 2880*a^2*b^5*d^4*e^4 + 10020*a^3*b^4*d^3*e^5
- 19755*a^4*b^3*d^2*e^6 + 24042*a^5*b^2*d*e^7 + 15289*a^6*b*e^8)*m^3 - 2*(180*b
^7*d^6*e^2 - 1620*a*b^6*d^5*e^3 + 6390*a^2*b^5*d^4*e^4 - 14460*a^3*b^4*d^3*e^5 +
20595*a^4*b^3*d^2*e^6 - 18996*a^5*b^2*d*e^7 - 18353*a^6*b*e^8)*m^2 - 72*(5*b^7*
d^6*e^2 - 40*a*b^6*d^5*e^3 + 140*a^2*b^5*d^4*e^4 - 280*a^3*b^4*d^3*e^5 + 350*a^4
*b^3*d^2*e^6 - 280*a^5*b^2*d*e^7 - 621*a^6*b*e^8)*m)*x^2 + 12*(420*a*b^6*d^7*e -
3150*a^2*b^5*d^6*e^2 + 10220*a^3*b^4*d^5*e^3 - 18655*a^4*b^3*d^4*e^4 + 20804*a^
5*b^2*d^3*e^5 - 14322*a^6*b*d^2*e^6 + 5772*a^7*d*e^7)*m + (40320*a^7*e^8 + (7*a^
6*b*d*e^7 + a^7*e^8)*m^7 - 7*(6*a^5*b^2*d^2*e^6 - 33*a^6*b*d*e^7 - 5*a^7*e^8)*m^
6 + 7*(30*a^4*b^3*d^3*e^5 - 180*a^5*b^2*d^2*e^6 + 445*a^6*b*d*e^7 + 73*a^7*e^8)*
m^5 - 35*(24*a^3*b^4*d^4*e^4 - 156*a^4*b^3*d^3*e^5 + 426*a^5*b^2*d^2*e^6 - 627*a
^6*b*d*e^7 - 115*a^7*e^8)*m^4 + 14*(180*a^2*b^5*d^5*e^3 - 1260*a^3*b^4*d^4*e^4 +
3765*a^4*b^3*d^3*e^5 - 6210*a^5*b^2*d^2*e^6 + 6077*a^6*b*d*e^7 + 1316*a^7*e^8)*
m^3 - 28*(180*a*b^6*d^6*e^2 - 1350*a^2*b^5*d^5*e^3 + 4380*a^3*b^4*d^4*e^4 - 7995
*a^4*b^3*d^3*e^5 + 8916*a^5*b^2*d^2*e^6 - 6138*a^6*b*d*e^7 - 1745*a^7*e^8)*m^2 +
144*(35*b^7*d^7*e - 280*a*b^6*d^6*e^2 + 980*a^2*b^5*d^5*e^3 - 1960*a^3*b^4*d^4*
e^4 + 2450*a^4*b^3*d^3*e^5 - 1960*a^5*b^2*d^2*e^6 + 980*a^6*b*d*e^7 + 481*a^7*e^
8)*m)*x)*(e*x + d)^m/(e^8*m^8 + 36*e^8*m^7 + 546*e^8*m^6 + 4536*e^8*m^5 + 22449*
e^8*m^4 + 67284*e^8*m^3 + 118124*e^8*m^2 + 109584*e^8*m + 40320*e^8)
_______________________________________________________________________________________
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((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.307599, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*(e*x + d)^m,x, algorithm="giac")
[Out]
Done