3.1916 \(\int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=173 \[ \frac{6 e^5 (a+b x)^{13} (b d-a e)}{13 b^7}+\frac{5 e^4 (a+b x)^{12} (b d-a e)^2}{4 b^7}+\frac{20 e^3 (a+b x)^{11} (b d-a e)^3}{11 b^7}+\frac{3 e^2 (a+b x)^{10} (b d-a e)^4}{2 b^7}+\frac{2 e (a+b x)^9 (b d-a e)^5}{3 b^7}+\frac{(a+b x)^8 (b d-a e)^6}{8 b^7}+\frac{e^6 (a+b x)^{14}}{14 b^7} \]

[Out]

((b*d - a*e)^6*(a + b*x)^8)/(8*b^7) + (2*e*(b*d - a*e)^5*(a + b*x)^9)/(3*b^7) +
(3*e^2*(b*d - a*e)^4*(a + b*x)^10)/(2*b^7) + (20*e^3*(b*d - a*e)^3*(a + b*x)^11)
/(11*b^7) + (5*e^4*(b*d - a*e)^2*(a + b*x)^12)/(4*b^7) + (6*e^5*(b*d - a*e)*(a +
b*x)^13)/(13*b^7) + (e^6*(a + b*x)^14)/(14*b^7)

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Rubi [A]  time = 0.90284, antiderivative size = 173, 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 \[ \frac{6 e^5 (a+b x)^{13} (b d-a e)}{13 b^7}+\frac{5 e^4 (a+b x)^{12} (b d-a e)^2}{4 b^7}+\frac{20 e^3 (a+b x)^{11} (b d-a e)^3}{11 b^7}+\frac{3 e^2 (a+b x)^{10} (b d-a e)^4}{2 b^7}+\frac{2 e (a+b x)^9 (b d-a e)^5}{3 b^7}+\frac{(a+b x)^8 (b d-a e)^6}{8 b^7}+\frac{e^6 (a+b x)^{14}}{14 b^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^6*(a + b*x)^8)/(8*b^7) + (2*e*(b*d - a*e)^5*(a + b*x)^9)/(3*b^7) +
(3*e^2*(b*d - a*e)^4*(a + b*x)^10)/(2*b^7) + (20*e^3*(b*d - a*e)^3*(a + b*x)^11)
/(11*b^7) + (5*e^4*(b*d - a*e)^2*(a + b*x)^12)/(4*b^7) + (6*e^5*(b*d - a*e)*(a +
b*x)^13)/(13*b^7) + (e^6*(a + b*x)^14)/(14*b^7)

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Rubi in Sympy [A]  time = 128.401, size = 158, normalized size = 0.91 \[ \frac{e^{6} \left (a + b x\right )^{14}}{14 b^{7}} - \frac{6 e^{5} \left (a + b x\right )^{13} \left (a e - b d\right )}{13 b^{7}} + \frac{5 e^{4} \left (a + b x\right )^{12} \left (a e - b d\right )^{2}}{4 b^{7}} - \frac{20 e^{3} \left (a + b x\right )^{11} \left (a e - b d\right )^{3}}{11 b^{7}} + \frac{3 e^{2} \left (a + b x\right )^{10} \left (a e - b d\right )^{4}}{2 b^{7}} - \frac{2 e \left (a + b x\right )^{9} \left (a e - b d\right )^{5}}{3 b^{7}} + \frac{\left (a + b x\right )^{8} \left (a e - b d\right )^{6}}{8 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

e**6*(a + b*x)**14/(14*b**7) - 6*e**5*(a + b*x)**13*(a*e - b*d)/(13*b**7) + 5*e*
*4*(a + b*x)**12*(a*e - b*d)**2/(4*b**7) - 20*e**3*(a + b*x)**11*(a*e - b*d)**3/
(11*b**7) + 3*e**2*(a + b*x)**10*(a*e - b*d)**4/(2*b**7) - 2*e*(a + b*x)**9*(a*e
- b*d)**5/(3*b**7) + (a + b*x)**8*(a*e - b*d)**6/(8*b**7)

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Mathematica [B]  time = 0.418931, size = 581, normalized size = 3.36 \[ \frac{x \left (3432 a^7 \left (7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right )+3003 a^6 b x \left (28 d^6+112 d^5 e x+210 d^4 e^2 x^2+224 d^3 e^3 x^3+140 d^2 e^4 x^4+48 d e^5 x^5+7 e^6 x^6\right )+2002 a^5 b^2 x^2 \left (84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right )+1001 a^4 b^3 x^3 \left (210 d^6+1008 d^5 e x+2100 d^4 e^2 x^2+2400 d^3 e^3 x^3+1575 d^2 e^4 x^4+560 d e^5 x^5+84 e^6 x^6\right )+364 a^3 b^4 x^4 \left (462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right )+91 a^2 b^5 x^5 \left (924 d^6+4752 d^5 e x+10395 d^4 e^2 x^2+12320 d^3 e^3 x^3+8316 d^2 e^4 x^4+3024 d e^5 x^5+462 e^6 x^6\right )+14 a b^6 x^6 \left (1716 d^6+9009 d^5 e x+20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+6006 d e^5 x^5+924 e^6 x^6\right )+b^7 x^7 \left (3003 d^6+16016 d^5 e x+36036 d^4 e^2 x^2+43680 d^3 e^3 x^3+30030 d^2 e^4 x^4+11088 d e^5 x^5+1716 e^6 x^6\right )\right )}{24024} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x*(3432*a^7*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*
x^4 + 7*d*e^5*x^5 + e^6*x^6) + 3003*a^6*b*x*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*
x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 + 48*d*e^5*x^5 + 7*e^6*x^6) + 2002*a^5*b
^2*x^2*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e^4*x
^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 1001*a^4*b^3*x^3*(210*d^6 + 1008*d^5*e*x + 21
00*d^4*e^2*x^2 + 2400*d^3*e^3*x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^
6) + 364*a^3*b^4*x^4*(462*d^6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x
^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6) + 91*a^2*b^5*x^5*(924*d^6
+ 4752*d^5*e*x + 10395*d^4*e^2*x^2 + 12320*d^3*e^3*x^3 + 8316*d^2*e^4*x^4 + 3024
*d*e^5*x^5 + 462*e^6*x^6) + 14*a*b^6*x^6*(1716*d^6 + 9009*d^5*e*x + 20020*d^4*e^
2*x^2 + 24024*d^3*e^3*x^3 + 16380*d^2*e^4*x^4 + 6006*d*e^5*x^5 + 924*e^6*x^6) +
b^7*x^7*(3003*d^6 + 16016*d^5*e*x + 36036*d^4*e^2*x^2 + 43680*d^3*e^3*x^3 + 3003
0*d^2*e^4*x^4 + 11088*d*e^5*x^5 + 1716*e^6*x^6)))/24024

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Maple [B]  time = 0.003, size = 1165, normalized size = 6.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/14*b^7*e^6*x^14+1/13*((a*e^6+6*b*d*e^5)*b^6+6*b^6*e^6*a)*x^13+1/12*((6*a*d*e^5
+15*b*d^2*e^4)*b^6+6*(a*e^6+6*b*d*e^5)*a*b^5+15*b^5*e^6*a^2)*x^12+1/11*((15*a*d^
2*e^4+20*b*d^3*e^3)*b^6+6*(6*a*d*e^5+15*b*d^2*e^4)*a*b^5+15*(a*e^6+6*b*d*e^5)*a^
2*b^4+20*b^4*e^6*a^3)*x^11+1/10*((20*a*d^3*e^3+15*b*d^4*e^2)*b^6+6*(15*a*d^2*e^4
+20*b*d^3*e^3)*a*b^5+15*(6*a*d*e^5+15*b*d^2*e^4)*a^2*b^4+20*(a*e^6+6*b*d*e^5)*a^
3*b^3+15*b^3*e^6*a^4)*x^10+1/9*((15*a*d^4*e^2+6*b*d^5*e)*b^6+6*(20*a*d^3*e^3+15*
b*d^4*e^2)*a*b^5+15*(15*a*d^2*e^4+20*b*d^3*e^3)*a^2*b^4+20*(6*a*d*e^5+15*b*d^2*e
^4)*a^3*b^3+15*(a*e^6+6*b*d*e^5)*b^2*a^4+6*b^2*e^6*a^5)*x^9+1/8*((6*a*d^5*e+b*d^
6)*b^6+6*(15*a*d^4*e^2+6*b*d^5*e)*a*b^5+15*(20*a*d^3*e^3+15*b*d^4*e^2)*a^2*b^4+2
0*(15*a*d^2*e^4+20*b*d^3*e^3)*a^3*b^3+15*(6*a*d*e^5+15*b*d^2*e^4)*b^2*a^4+6*(a*e
^6+6*b*d*e^5)*a^5*b+a^6*b*e^6)*x^8+1/7*(a*d^6*b^6+6*(6*a*d^5*e+b*d^6)*a*b^5+15*(
15*a*d^4*e^2+6*b*d^5*e)*a^2*b^4+20*(20*a*d^3*e^3+15*b*d^4*e^2)*a^3*b^3+15*(15*a*
d^2*e^4+20*b*d^3*e^3)*b^2*a^4+6*(6*a*d*e^5+15*b*d^2*e^4)*a^5*b+(a*e^6+6*b*d*e^5)
*a^6)*x^7+1/6*(6*a^2*d^6*b^5+15*(6*a*d^5*e+b*d^6)*a^2*b^4+20*(15*a*d^4*e^2+6*b*d
^5*e)*a^3*b^3+15*(20*a*d^3*e^3+15*b*d^4*e^2)*b^2*a^4+6*(15*a*d^2*e^4+20*b*d^3*e^
3)*a^5*b+(6*a*d*e^5+15*b*d^2*e^4)*a^6)*x^6+1/5*(15*a^3*d^6*b^4+20*(6*a*d^5*e+b*d
^6)*a^3*b^3+15*(15*a*d^4*e^2+6*b*d^5*e)*b^2*a^4+6*(20*a*d^3*e^3+15*b*d^4*e^2)*a^
5*b+(15*a*d^2*e^4+20*b*d^3*e^3)*a^6)*x^5+1/4*(20*a^4*d^6*b^3+15*(6*a*d^5*e+b*d^6
)*b^2*a^4+6*(15*a*d^4*e^2+6*b*d^5*e)*a^5*b+(20*a*d^3*e^3+15*b*d^4*e^2)*a^6)*x^4+
1/3*(15*a^5*d^6*b^2+6*(6*a*d^5*e+b*d^6)*a^5*b+(15*a*d^4*e^2+6*b*d^5*e)*a^6)*x^3+
1/2*(6*a^6*d^6*b+(6*a*d^5*e+b*d^6)*a^6)*x^2+a^7*d^6*x

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Maxima [A]  time = 0.72072, size = 953, normalized size = 5.51 \[ \frac{1}{14} \, b^{7} e^{6} x^{14} + a^{7} d^{6} x + \frac{1}{13} \,{\left (6 \, b^{7} d e^{5} + 7 \, a b^{6} e^{6}\right )} x^{13} + \frac{1}{4} \,{\left (5 \, b^{7} d^{2} e^{4} + 14 \, a b^{6} d e^{5} + 7 \, a^{2} b^{5} e^{6}\right )} x^{12} + \frac{1}{11} \,{\left (20 \, b^{7} d^{3} e^{3} + 105 \, a b^{6} d^{2} e^{4} + 126 \, a^{2} b^{5} d e^{5} + 35 \, a^{3} b^{4} e^{6}\right )} x^{11} + \frac{1}{2} \,{\left (3 \, b^{7} d^{4} e^{2} + 28 \, a b^{6} d^{3} e^{3} + 63 \, a^{2} b^{5} d^{2} e^{4} + 42 \, a^{3} b^{4} d e^{5} + 7 \, a^{4} b^{3} e^{6}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, b^{7} d^{5} e + 35 \, a b^{6} d^{4} e^{2} + 140 \, a^{2} b^{5} d^{3} e^{3} + 175 \, a^{3} b^{4} d^{2} e^{4} + 70 \, a^{4} b^{3} d e^{5} + 7 \, a^{5} b^{2} e^{6}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{6} + 42 \, a b^{6} d^{5} e + 315 \, a^{2} b^{5} d^{4} e^{2} + 700 \, a^{3} b^{4} d^{3} e^{3} + 525 \, a^{4} b^{3} d^{2} e^{4} + 126 \, a^{5} b^{2} d e^{5} + 7 \, a^{6} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (7 \, a b^{6} d^{6} + 126 \, a^{2} b^{5} d^{5} e + 525 \, a^{3} b^{4} d^{4} e^{2} + 700 \, a^{4} b^{3} d^{3} e^{3} + 315 \, a^{5} b^{2} d^{2} e^{4} + 42 \, a^{6} b d e^{5} + a^{7} e^{6}\right )} x^{7} + \frac{1}{2} \,{\left (7 \, a^{2} b^{5} d^{6} + 70 \, a^{3} b^{4} d^{5} e + 175 \, a^{4} b^{3} d^{4} e^{2} + 140 \, a^{5} b^{2} d^{3} e^{3} + 35 \, a^{6} b d^{2} e^{4} + 2 \, a^{7} d e^{5}\right )} x^{6} +{\left (7 \, a^{3} b^{4} d^{6} + 42 \, a^{4} b^{3} d^{5} e + 63 \, a^{5} b^{2} d^{4} e^{2} + 28 \, a^{6} b d^{3} e^{3} + 3 \, a^{7} d^{2} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (35 \, a^{4} b^{3} d^{6} + 126 \, a^{5} b^{2} d^{5} e + 105 \, a^{6} b d^{4} e^{2} + 20 \, a^{7} d^{3} e^{3}\right )} x^{4} +{\left (7 \, a^{5} b^{2} d^{6} + 14 \, a^{6} b d^{5} e + 5 \, a^{7} d^{4} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{6} + 6 \, a^{7} d^{5} e\right )} x^{2} \]

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)^6,x, algorithm="maxima")

[Out]

1/14*b^7*e^6*x^14 + a^7*d^6*x + 1/13*(6*b^7*d*e^5 + 7*a*b^6*e^6)*x^13 + 1/4*(5*b
^7*d^2*e^4 + 14*a*b^6*d*e^5 + 7*a^2*b^5*e^6)*x^12 + 1/11*(20*b^7*d^3*e^3 + 105*a
*b^6*d^2*e^4 + 126*a^2*b^5*d*e^5 + 35*a^3*b^4*e^6)*x^11 + 1/2*(3*b^7*d^4*e^2 + 2
8*a*b^6*d^3*e^3 + 63*a^2*b^5*d^2*e^4 + 42*a^3*b^4*d*e^5 + 7*a^4*b^3*e^6)*x^10 +
1/3*(2*b^7*d^5*e + 35*a*b^6*d^4*e^2 + 140*a^2*b^5*d^3*e^3 + 175*a^3*b^4*d^2*e^4
+ 70*a^4*b^3*d*e^5 + 7*a^5*b^2*e^6)*x^9 + 1/8*(b^7*d^6 + 42*a*b^6*d^5*e + 315*a^
2*b^5*d^4*e^2 + 700*a^3*b^4*d^3*e^3 + 525*a^4*b^3*d^2*e^4 + 126*a^5*b^2*d*e^5 +
7*a^6*b*e^6)*x^8 + 1/7*(7*a*b^6*d^6 + 126*a^2*b^5*d^5*e + 525*a^3*b^4*d^4*e^2 +
700*a^4*b^3*d^3*e^3 + 315*a^5*b^2*d^2*e^4 + 42*a^6*b*d*e^5 + a^7*e^6)*x^7 + 1/2*
(7*a^2*b^5*d^6 + 70*a^3*b^4*d^5*e + 175*a^4*b^3*d^4*e^2 + 140*a^5*b^2*d^3*e^3 +
35*a^6*b*d^2*e^4 + 2*a^7*d*e^5)*x^6 + (7*a^3*b^4*d^6 + 42*a^4*b^3*d^5*e + 63*a^5
*b^2*d^4*e^2 + 28*a^6*b*d^3*e^3 + 3*a^7*d^2*e^4)*x^5 + 1/4*(35*a^4*b^3*d^6 + 126
*a^5*b^2*d^5*e + 105*a^6*b*d^4*e^2 + 20*a^7*d^3*e^3)*x^4 + (7*a^5*b^2*d^6 + 14*a
^6*b*d^5*e + 5*a^7*d^4*e^2)*x^3 + 1/2*(7*a^6*b*d^6 + 6*a^7*d^5*e)*x^2

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Fricas [A]  time = 0.275255, size = 1, normalized size = 0.01 \[ \frac{1}{14} x^{14} e^{6} b^{7} + \frac{6}{13} x^{13} e^{5} d b^{7} + \frac{7}{13} x^{13} e^{6} b^{6} a + \frac{5}{4} x^{12} e^{4} d^{2} b^{7} + \frac{7}{2} x^{12} e^{5} d b^{6} a + \frac{7}{4} x^{12} e^{6} b^{5} a^{2} + \frac{20}{11} x^{11} e^{3} d^{3} b^{7} + \frac{105}{11} x^{11} e^{4} d^{2} b^{6} a + \frac{126}{11} x^{11} e^{5} d b^{5} a^{2} + \frac{35}{11} x^{11} e^{6} b^{4} a^{3} + \frac{3}{2} x^{10} e^{2} d^{4} b^{7} + 14 x^{10} e^{3} d^{3} b^{6} a + \frac{63}{2} x^{10} e^{4} d^{2} b^{5} a^{2} + 21 x^{10} e^{5} d b^{4} a^{3} + \frac{7}{2} x^{10} e^{6} b^{3} a^{4} + \frac{2}{3} x^{9} e d^{5} b^{7} + \frac{35}{3} x^{9} e^{2} d^{4} b^{6} a + \frac{140}{3} x^{9} e^{3} d^{3} b^{5} a^{2} + \frac{175}{3} x^{9} e^{4} d^{2} b^{4} a^{3} + \frac{70}{3} x^{9} e^{5} d b^{3} a^{4} + \frac{7}{3} x^{9} e^{6} b^{2} a^{5} + \frac{1}{8} x^{8} d^{6} b^{7} + \frac{21}{4} x^{8} e d^{5} b^{6} a + \frac{315}{8} x^{8} e^{2} d^{4} b^{5} a^{2} + \frac{175}{2} x^{8} e^{3} d^{3} b^{4} a^{3} + \frac{525}{8} x^{8} e^{4} d^{2} b^{3} a^{4} + \frac{63}{4} x^{8} e^{5} d b^{2} a^{5} + \frac{7}{8} x^{8} e^{6} b a^{6} + x^{7} d^{6} b^{6} a + 18 x^{7} e d^{5} b^{5} a^{2} + 75 x^{7} e^{2} d^{4} b^{4} a^{3} + 100 x^{7} e^{3} d^{3} b^{3} a^{4} + 45 x^{7} e^{4} d^{2} b^{2} a^{5} + 6 x^{7} e^{5} d b a^{6} + \frac{1}{7} x^{7} e^{6} a^{7} + \frac{7}{2} x^{6} d^{6} b^{5} a^{2} + 35 x^{6} e d^{5} b^{4} a^{3} + \frac{175}{2} x^{6} e^{2} d^{4} b^{3} a^{4} + 70 x^{6} e^{3} d^{3} b^{2} a^{5} + \frac{35}{2} x^{6} e^{4} d^{2} b a^{6} + x^{6} e^{5} d a^{7} + 7 x^{5} d^{6} b^{4} a^{3} + 42 x^{5} e d^{5} b^{3} a^{4} + 63 x^{5} e^{2} d^{4} b^{2} a^{5} + 28 x^{5} e^{3} d^{3} b a^{6} + 3 x^{5} e^{4} d^{2} a^{7} + \frac{35}{4} x^{4} d^{6} b^{3} a^{4} + \frac{63}{2} x^{4} e d^{5} b^{2} a^{5} + \frac{105}{4} x^{4} e^{2} d^{4} b a^{6} + 5 x^{4} e^{3} d^{3} a^{7} + 7 x^{3} d^{6} b^{2} a^{5} + 14 x^{3} e d^{5} b a^{6} + 5 x^{3} e^{2} d^{4} a^{7} + \frac{7}{2} x^{2} d^{6} b a^{6} + 3 x^{2} e d^{5} a^{7} + x d^{6} a^{7} \]

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)^6,x, algorithm="fricas")

[Out]

1/14*x^14*e^6*b^7 + 6/13*x^13*e^5*d*b^7 + 7/13*x^13*e^6*b^6*a + 5/4*x^12*e^4*d^2
*b^7 + 7/2*x^12*e^5*d*b^6*a + 7/4*x^12*e^6*b^5*a^2 + 20/11*x^11*e^3*d^3*b^7 + 10
5/11*x^11*e^4*d^2*b^6*a + 126/11*x^11*e^5*d*b^5*a^2 + 35/11*x^11*e^6*b^4*a^3 + 3
/2*x^10*e^2*d^4*b^7 + 14*x^10*e^3*d^3*b^6*a + 63/2*x^10*e^4*d^2*b^5*a^2 + 21*x^1
0*e^5*d*b^4*a^3 + 7/2*x^10*e^6*b^3*a^4 + 2/3*x^9*e*d^5*b^7 + 35/3*x^9*e^2*d^4*b^
6*a + 140/3*x^9*e^3*d^3*b^5*a^2 + 175/3*x^9*e^4*d^2*b^4*a^3 + 70/3*x^9*e^5*d*b^3
*a^4 + 7/3*x^9*e^6*b^2*a^5 + 1/8*x^8*d^6*b^7 + 21/4*x^8*e*d^5*b^6*a + 315/8*x^8*
e^2*d^4*b^5*a^2 + 175/2*x^8*e^3*d^3*b^4*a^3 + 525/8*x^8*e^4*d^2*b^3*a^4 + 63/4*x
^8*e^5*d*b^2*a^5 + 7/8*x^8*e^6*b*a^6 + x^7*d^6*b^6*a + 18*x^7*e*d^5*b^5*a^2 + 75
*x^7*e^2*d^4*b^4*a^3 + 100*x^7*e^3*d^3*b^3*a^4 + 45*x^7*e^4*d^2*b^2*a^5 + 6*x^7*
e^5*d*b*a^6 + 1/7*x^7*e^6*a^7 + 7/2*x^6*d^6*b^5*a^2 + 35*x^6*e*d^5*b^4*a^3 + 175
/2*x^6*e^2*d^4*b^3*a^4 + 70*x^6*e^3*d^3*b^2*a^5 + 35/2*x^6*e^4*d^2*b*a^6 + x^6*e
^5*d*a^7 + 7*x^5*d^6*b^4*a^3 + 42*x^5*e*d^5*b^3*a^4 + 63*x^5*e^2*d^4*b^2*a^5 + 2
8*x^5*e^3*d^3*b*a^6 + 3*x^5*e^4*d^2*a^7 + 35/4*x^4*d^6*b^3*a^4 + 63/2*x^4*e*d^5*
b^2*a^5 + 105/4*x^4*e^2*d^4*b*a^6 + 5*x^4*e^3*d^3*a^7 + 7*x^3*d^6*b^2*a^5 + 14*x
^3*e*d^5*b*a^6 + 5*x^3*e^2*d^4*a^7 + 7/2*x^2*d^6*b*a^6 + 3*x^2*e*d^5*a^7 + x*d^6
*a^7

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Sympy [A]  time = 0.491001, size = 796, normalized size = 4.6 \[ a^{7} d^{6} x + \frac{b^{7} e^{6} x^{14}}{14} + x^{13} \left (\frac{7 a b^{6} e^{6}}{13} + \frac{6 b^{7} d e^{5}}{13}\right ) + x^{12} \left (\frac{7 a^{2} b^{5} e^{6}}{4} + \frac{7 a b^{6} d e^{5}}{2} + \frac{5 b^{7} d^{2} e^{4}}{4}\right ) + x^{11} \left (\frac{35 a^{3} b^{4} e^{6}}{11} + \frac{126 a^{2} b^{5} d e^{5}}{11} + \frac{105 a b^{6} d^{2} e^{4}}{11} + \frac{20 b^{7} d^{3} e^{3}}{11}\right ) + x^{10} \left (\frac{7 a^{4} b^{3} e^{6}}{2} + 21 a^{3} b^{4} d e^{5} + \frac{63 a^{2} b^{5} d^{2} e^{4}}{2} + 14 a b^{6} d^{3} e^{3} + \frac{3 b^{7} d^{4} e^{2}}{2}\right ) + x^{9} \left (\frac{7 a^{5} b^{2} e^{6}}{3} + \frac{70 a^{4} b^{3} d e^{5}}{3} + \frac{175 a^{3} b^{4} d^{2} e^{4}}{3} + \frac{140 a^{2} b^{5} d^{3} e^{3}}{3} + \frac{35 a b^{6} d^{4} e^{2}}{3} + \frac{2 b^{7} d^{5} e}{3}\right ) + x^{8} \left (\frac{7 a^{6} b e^{6}}{8} + \frac{63 a^{5} b^{2} d e^{5}}{4} + \frac{525 a^{4} b^{3} d^{2} e^{4}}{8} + \frac{175 a^{3} b^{4} d^{3} e^{3}}{2} + \frac{315 a^{2} b^{5} d^{4} e^{2}}{8} + \frac{21 a b^{6} d^{5} e}{4} + \frac{b^{7} d^{6}}{8}\right ) + x^{7} \left (\frac{a^{7} e^{6}}{7} + 6 a^{6} b d e^{5} + 45 a^{5} b^{2} d^{2} e^{4} + 100 a^{4} b^{3} d^{3} e^{3} + 75 a^{3} b^{4} d^{4} e^{2} + 18 a^{2} b^{5} d^{5} e + a b^{6} d^{6}\right ) + x^{6} \left (a^{7} d e^{5} + \frac{35 a^{6} b d^{2} e^{4}}{2} + 70 a^{5} b^{2} d^{3} e^{3} + \frac{175 a^{4} b^{3} d^{4} e^{2}}{2} + 35 a^{3} b^{4} d^{5} e + \frac{7 a^{2} b^{5} d^{6}}{2}\right ) + x^{5} \left (3 a^{7} d^{2} e^{4} + 28 a^{6} b d^{3} e^{3} + 63 a^{5} b^{2} d^{4} e^{2} + 42 a^{4} b^{3} d^{5} e + 7 a^{3} b^{4} d^{6}\right ) + x^{4} \left (5 a^{7} d^{3} e^{3} + \frac{105 a^{6} b d^{4} e^{2}}{4} + \frac{63 a^{5} b^{2} d^{5} e}{2} + \frac{35 a^{4} b^{3} d^{6}}{4}\right ) + x^{3} \left (5 a^{7} d^{4} e^{2} + 14 a^{6} b d^{5} e + 7 a^{5} b^{2} d^{6}\right ) + x^{2} \left (3 a^{7} d^{5} e + \frac{7 a^{6} b d^{6}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**7*d**6*x + b**7*e**6*x**14/14 + x**13*(7*a*b**6*e**6/13 + 6*b**7*d*e**5/13) +
x**12*(7*a**2*b**5*e**6/4 + 7*a*b**6*d*e**5/2 + 5*b**7*d**2*e**4/4) + x**11*(35
*a**3*b**4*e**6/11 + 126*a**2*b**5*d*e**5/11 + 105*a*b**6*d**2*e**4/11 + 20*b**7
*d**3*e**3/11) + x**10*(7*a**4*b**3*e**6/2 + 21*a**3*b**4*d*e**5 + 63*a**2*b**5*
d**2*e**4/2 + 14*a*b**6*d**3*e**3 + 3*b**7*d**4*e**2/2) + x**9*(7*a**5*b**2*e**6
/3 + 70*a**4*b**3*d*e**5/3 + 175*a**3*b**4*d**2*e**4/3 + 140*a**2*b**5*d**3*e**3
/3 + 35*a*b**6*d**4*e**2/3 + 2*b**7*d**5*e/3) + x**8*(7*a**6*b*e**6/8 + 63*a**5*
b**2*d*e**5/4 + 525*a**4*b**3*d**2*e**4/8 + 175*a**3*b**4*d**3*e**3/2 + 315*a**2
*b**5*d**4*e**2/8 + 21*a*b**6*d**5*e/4 + b**7*d**6/8) + x**7*(a**7*e**6/7 + 6*a*
*6*b*d*e**5 + 45*a**5*b**2*d**2*e**4 + 100*a**4*b**3*d**3*e**3 + 75*a**3*b**4*d*
*4*e**2 + 18*a**2*b**5*d**5*e + a*b**6*d**6) + x**6*(a**7*d*e**5 + 35*a**6*b*d**
2*e**4/2 + 70*a**5*b**2*d**3*e**3 + 175*a**4*b**3*d**4*e**2/2 + 35*a**3*b**4*d**
5*e + 7*a**2*b**5*d**6/2) + x**5*(3*a**7*d**2*e**4 + 28*a**6*b*d**3*e**3 + 63*a*
*5*b**2*d**4*e**2 + 42*a**4*b**3*d**5*e + 7*a**3*b**4*d**6) + x**4*(5*a**7*d**3*
e**3 + 105*a**6*b*d**4*e**2/4 + 63*a**5*b**2*d**5*e/2 + 35*a**4*b**3*d**6/4) + x
**3*(5*a**7*d**4*e**2 + 14*a**6*b*d**5*e + 7*a**5*b**2*d**6) + x**2*(3*a**7*d**5
*e + 7*a**6*b*d**6/2)

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GIAC/XCAS [A]  time = 0.273438, size = 1034, normalized size = 5.98 \[ \text{result too large to display} \]

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)^6,x, algorithm="giac")

[Out]

1/14*b^7*x^14*e^6 + 6/13*b^7*d*x^13*e^5 + 5/4*b^7*d^2*x^12*e^4 + 20/11*b^7*d^3*x
^11*e^3 + 3/2*b^7*d^4*x^10*e^2 + 2/3*b^7*d^5*x^9*e + 1/8*b^7*d^6*x^8 + 7/13*a*b^
6*x^13*e^6 + 7/2*a*b^6*d*x^12*e^5 + 105/11*a*b^6*d^2*x^11*e^4 + 14*a*b^6*d^3*x^1
0*e^3 + 35/3*a*b^6*d^4*x^9*e^2 + 21/4*a*b^6*d^5*x^8*e + a*b^6*d^6*x^7 + 7/4*a^2*
b^5*x^12*e^6 + 126/11*a^2*b^5*d*x^11*e^5 + 63/2*a^2*b^5*d^2*x^10*e^4 + 140/3*a^2
*b^5*d^3*x^9*e^3 + 315/8*a^2*b^5*d^4*x^8*e^2 + 18*a^2*b^5*d^5*x^7*e + 7/2*a^2*b^
5*d^6*x^6 + 35/11*a^3*b^4*x^11*e^6 + 21*a^3*b^4*d*x^10*e^5 + 175/3*a^3*b^4*d^2*x
^9*e^4 + 175/2*a^3*b^4*d^3*x^8*e^3 + 75*a^3*b^4*d^4*x^7*e^2 + 35*a^3*b^4*d^5*x^6
*e + 7*a^3*b^4*d^6*x^5 + 7/2*a^4*b^3*x^10*e^6 + 70/3*a^4*b^3*d*x^9*e^5 + 525/8*a
^4*b^3*d^2*x^8*e^4 + 100*a^4*b^3*d^3*x^7*e^3 + 175/2*a^4*b^3*d^4*x^6*e^2 + 42*a^
4*b^3*d^5*x^5*e + 35/4*a^4*b^3*d^6*x^4 + 7/3*a^5*b^2*x^9*e^6 + 63/4*a^5*b^2*d*x^
8*e^5 + 45*a^5*b^2*d^2*x^7*e^4 + 70*a^5*b^2*d^3*x^6*e^3 + 63*a^5*b^2*d^4*x^5*e^2
+ 63/2*a^5*b^2*d^5*x^4*e + 7*a^5*b^2*d^6*x^3 + 7/8*a^6*b*x^8*e^6 + 6*a^6*b*d*x^
7*e^5 + 35/2*a^6*b*d^2*x^6*e^4 + 28*a^6*b*d^3*x^5*e^3 + 105/4*a^6*b*d^4*x^4*e^2
+ 14*a^6*b*d^5*x^3*e + 7/2*a^6*b*d^6*x^2 + 1/7*a^7*x^7*e^6 + a^7*d*x^6*e^5 + 3*a
^7*d^2*x^5*e^4 + 5*a^7*d^3*x^4*e^3 + 5*a^7*d^4*x^3*e^2 + 3*a^7*d^5*x^2*e + a^7*d
^6*x