3.1917 \(\int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=143 \[ \frac{5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac{10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac{e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac{5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac{(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac{e^5 (a+b x)^{13}}{13 b^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.729117, antiderivative size = 143, 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 \[ \frac{5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac{10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac{e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac{5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac{(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac{e^5 (a+b x)^{13}}{13 b^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 98.5269, size = 129, normalized size = 0.9 \[ \frac{e^{5} \left (a + b x\right )^{13}}{13 b^{6}} - \frac{5 e^{4} \left (a + b x\right )^{12} \left (a e - b d\right )}{12 b^{6}} + \frac{10 e^{3} \left (a + b x\right )^{11} \left (a e - b d\right )^{2}}{11 b^{6}} - \frac{e^{2} \left (a + b x\right )^{10} \left (a e - b d\right )^{3}}{b^{6}} + \frac{5 e \left (a + b x\right )^{9} \left (a e - b d\right )^{4}}{9 b^{6}} - \frac{\left (a + b x\right )^{8} \left (a e - b d\right )^{5}}{8 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 0.31905, size = 493, normalized size = 3.45 \[ \frac{x \left (1716 a^7 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+1716 a^6 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+1287 a^5 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+715 a^4 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+286 a^3 b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+78 a^2 b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )+13 a b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )+b^7 x^7 \left (1287 d^5+5720 d^4 e x+10296 d^3 e^2 x^2+9360 d^2 e^3 x^3+4290 d e^4 x^4+792 e^5 x^5\right )\right )}{10296} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x*(1716*a^7*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4
+ e^5*x^5) + 1716*a^6*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x
^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 1287*a^5*b^2*x^2*(56*d^5 + 210*d^4*e*x + 336*d^
3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 715*a^4*b^3*x^3*(126
*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*
x^5) + 286*a^3*b^4*x^4*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3
*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 78*a^2*b^5*x^5*(462*d^5 + 1980*d^4*e*x + 3
465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5) + 13*a*b^6*x^
6*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4
+ 462*e^5*x^5) + b^7*x^7*(1287*d^5 + 5720*d^4*e*x + 10296*d^3*e^2*x^2 + 9360*d^
2*e^3*x^3 + 4290*d*e^4*x^4 + 792*e^5*x^5)))/10296

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 982, normalized size = 6.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.738151, size = 802, normalized size = 5.61 \[ \frac{1}{13} \, b^{7} e^{5} x^{13} + a^{7} d^{5} x + \frac{1}{12} \,{\left (5 \, b^{7} d e^{4} + 7 \, a b^{6} e^{5}\right )} x^{12} + \frac{1}{11} \,{\left (10 \, b^{7} d^{2} e^{3} + 35 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, b^{7} d^{3} e^{2} + 14 \, a b^{6} d^{2} e^{3} + 21 \, a^{2} b^{5} d e^{4} + 7 \, a^{3} b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (b^{7} d^{4} e + 14 \, a b^{6} d^{3} e^{2} + 42 \, a^{2} b^{5} d^{2} e^{3} + 35 \, a^{3} b^{4} d e^{4} + 7 \, a^{4} b^{3} e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{5} + 35 \, a b^{6} d^{4} e + 210 \, a^{2} b^{5} d^{3} e^{2} + 350 \, a^{3} b^{4} d^{2} e^{3} + 175 \, a^{4} b^{3} d e^{4} + 21 \, a^{5} b^{2} e^{5}\right )} x^{8} +{\left (a b^{6} d^{5} + 15 \, a^{2} b^{5} d^{4} e + 50 \, a^{3} b^{4} d^{3} e^{2} + 50 \, a^{4} b^{3} d^{2} e^{3} + 15 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (21 \, a^{2} b^{5} d^{5} + 175 \, a^{3} b^{4} d^{4} e + 350 \, a^{4} b^{3} d^{3} e^{2} + 210 \, a^{5} b^{2} d^{2} e^{3} + 35 \, a^{6} b d e^{4} + a^{7} e^{5}\right )} x^{6} +{\left (7 \, a^{3} b^{4} d^{5} + 35 \, a^{4} b^{3} d^{4} e + 42 \, a^{5} b^{2} d^{3} e^{2} + 14 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (7 \, a^{4} b^{3} d^{5} + 21 \, a^{5} b^{2} d^{4} e + 14 \, a^{6} b d^{3} e^{2} + 2 \, a^{7} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (21 \, a^{5} b^{2} d^{5} + 35 \, a^{6} b d^{4} e + 10 \, a^{7} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{5} + 5 \, a^{7} d^{4} e\right )} x^{2} \]

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)^5,x, algorithm="maxima")

[Out]

1/13*b^7*e^5*x^13 + a^7*d^5*x + 1/12*(5*b^7*d*e^4 + 7*a*b^6*e^5)*x^12 + 1/11*(10
*b^7*d^2*e^3 + 35*a*b^6*d*e^4 + 21*a^2*b^5*e^5)*x^11 + 1/2*(2*b^7*d^3*e^2 + 14*a
*b^6*d^2*e^3 + 21*a^2*b^5*d*e^4 + 7*a^3*b^4*e^5)*x^10 + 5/9*(b^7*d^4*e + 14*a*b^
6*d^3*e^2 + 42*a^2*b^5*d^2*e^3 + 35*a^3*b^4*d*e^4 + 7*a^4*b^3*e^5)*x^9 + 1/8*(b^
7*d^5 + 35*a*b^6*d^4*e + 210*a^2*b^5*d^3*e^2 + 350*a^3*b^4*d^2*e^3 + 175*a^4*b^3
*d*e^4 + 21*a^5*b^2*e^5)*x^8 + (a*b^6*d^5 + 15*a^2*b^5*d^4*e + 50*a^3*b^4*d^3*e^
2 + 50*a^4*b^3*d^2*e^3 + 15*a^5*b^2*d*e^4 + a^6*b*e^5)*x^7 + 1/6*(21*a^2*b^5*d^5
+ 175*a^3*b^4*d^4*e + 350*a^4*b^3*d^3*e^2 + 210*a^5*b^2*d^2*e^3 + 35*a^6*b*d*e^
4 + a^7*e^5)*x^6 + (7*a^3*b^4*d^5 + 35*a^4*b^3*d^4*e + 42*a^5*b^2*d^3*e^2 + 14*a
^6*b*d^2*e^3 + a^7*d*e^4)*x^5 + 5/4*(7*a^4*b^3*d^5 + 21*a^5*b^2*d^4*e + 14*a^6*b
*d^3*e^2 + 2*a^7*d^2*e^3)*x^4 + 1/3*(21*a^5*b^2*d^5 + 35*a^6*b*d^4*e + 10*a^7*d^
3*e^2)*x^3 + 1/2*(7*a^6*b*d^5 + 5*a^7*d^4*e)*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.261008, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} e^{5} b^{7} + \frac{5}{12} x^{12} e^{4} d b^{7} + \frac{7}{12} x^{12} e^{5} b^{6} a + \frac{10}{11} x^{11} e^{3} d^{2} b^{7} + \frac{35}{11} x^{11} e^{4} d b^{6} a + \frac{21}{11} x^{11} e^{5} b^{5} a^{2} + x^{10} e^{2} d^{3} b^{7} + 7 x^{10} e^{3} d^{2} b^{6} a + \frac{21}{2} x^{10} e^{4} d b^{5} a^{2} + \frac{7}{2} x^{10} e^{5} b^{4} a^{3} + \frac{5}{9} x^{9} e d^{4} b^{7} + \frac{70}{9} x^{9} e^{2} d^{3} b^{6} a + \frac{70}{3} x^{9} e^{3} d^{2} b^{5} a^{2} + \frac{175}{9} x^{9} e^{4} d b^{4} a^{3} + \frac{35}{9} x^{9} e^{5} b^{3} a^{4} + \frac{1}{8} x^{8} d^{5} b^{7} + \frac{35}{8} x^{8} e d^{4} b^{6} a + \frac{105}{4} x^{8} e^{2} d^{3} b^{5} a^{2} + \frac{175}{4} x^{8} e^{3} d^{2} b^{4} a^{3} + \frac{175}{8} x^{8} e^{4} d b^{3} a^{4} + \frac{21}{8} x^{8} e^{5} b^{2} a^{5} + x^{7} d^{5} b^{6} a + 15 x^{7} e d^{4} b^{5} a^{2} + 50 x^{7} e^{2} d^{3} b^{4} a^{3} + 50 x^{7} e^{3} d^{2} b^{3} a^{4} + 15 x^{7} e^{4} d b^{2} a^{5} + x^{7} e^{5} b a^{6} + \frac{7}{2} x^{6} d^{5} b^{5} a^{2} + \frac{175}{6} x^{6} e d^{4} b^{4} a^{3} + \frac{175}{3} x^{6} e^{2} d^{3} b^{3} a^{4} + 35 x^{6} e^{3} d^{2} b^{2} a^{5} + \frac{35}{6} x^{6} e^{4} d b a^{6} + \frac{1}{6} x^{6} e^{5} a^{7} + 7 x^{5} d^{5} b^{4} a^{3} + 35 x^{5} e d^{4} b^{3} a^{4} + 42 x^{5} e^{2} d^{3} b^{2} a^{5} + 14 x^{5} e^{3} d^{2} b a^{6} + x^{5} e^{4} d a^{7} + \frac{35}{4} x^{4} d^{5} b^{3} a^{4} + \frac{105}{4} x^{4} e d^{4} b^{2} a^{5} + \frac{35}{2} x^{4} e^{2} d^{3} b a^{6} + \frac{5}{2} x^{4} e^{3} d^{2} a^{7} + 7 x^{3} d^{5} b^{2} a^{5} + \frac{35}{3} x^{3} e d^{4} b a^{6} + \frac{10}{3} x^{3} e^{2} d^{3} a^{7} + \frac{7}{2} x^{2} d^{5} b a^{6} + \frac{5}{2} x^{2} e d^{4} a^{7} + x d^{5} a^{7} \]

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)^5,x, algorithm="fricas")

[Out]

1/13*x^13*e^5*b^7 + 5/12*x^12*e^4*d*b^7 + 7/12*x^12*e^5*b^6*a + 10/11*x^11*e^3*d
^2*b^7 + 35/11*x^11*e^4*d*b^6*a + 21/11*x^11*e^5*b^5*a^2 + x^10*e^2*d^3*b^7 + 7*
x^10*e^3*d^2*b^6*a + 21/2*x^10*e^4*d*b^5*a^2 + 7/2*x^10*e^5*b^4*a^3 + 5/9*x^9*e*
d^4*b^7 + 70/9*x^9*e^2*d^3*b^6*a + 70/3*x^9*e^3*d^2*b^5*a^2 + 175/9*x^9*e^4*d*b^
4*a^3 + 35/9*x^9*e^5*b^3*a^4 + 1/8*x^8*d^5*b^7 + 35/8*x^8*e*d^4*b^6*a + 105/4*x^
8*e^2*d^3*b^5*a^2 + 175/4*x^8*e^3*d^2*b^4*a^3 + 175/8*x^8*e^4*d*b^3*a^4 + 21/8*x
^8*e^5*b^2*a^5 + x^7*d^5*b^6*a + 15*x^7*e*d^4*b^5*a^2 + 50*x^7*e^2*d^3*b^4*a^3 +
50*x^7*e^3*d^2*b^3*a^4 + 15*x^7*e^4*d*b^2*a^5 + x^7*e^5*b*a^6 + 7/2*x^6*d^5*b^5
*a^2 + 175/6*x^6*e*d^4*b^4*a^3 + 175/3*x^6*e^2*d^3*b^3*a^4 + 35*x^6*e^3*d^2*b^2*
a^5 + 35/6*x^6*e^4*d*b*a^6 + 1/6*x^6*e^5*a^7 + 7*x^5*d^5*b^4*a^3 + 35*x^5*e*d^4*
b^3*a^4 + 42*x^5*e^2*d^3*b^2*a^5 + 14*x^5*e^3*d^2*b*a^6 + x^5*e^4*d*a^7 + 35/4*x
^4*d^5*b^3*a^4 + 105/4*x^4*e*d^4*b^2*a^5 + 35/2*x^4*e^2*d^3*b*a^6 + 5/2*x^4*e^3*
d^2*a^7 + 7*x^3*d^5*b^2*a^5 + 35/3*x^3*e*d^4*b*a^6 + 10/3*x^3*e^2*d^3*a^7 + 7/2*
x^2*d^5*b*a^6 + 5/2*x^2*e*d^4*a^7 + x*d^5*a^7

_______________________________________________________________________________________

Sympy [A]  time = 0.432192, size = 673, normalized size = 4.71 \[ a^{7} d^{5} x + \frac{b^{7} e^{5} x^{13}}{13} + x^{12} \left (\frac{7 a b^{6} e^{5}}{12} + \frac{5 b^{7} d e^{4}}{12}\right ) + x^{11} \left (\frac{21 a^{2} b^{5} e^{5}}{11} + \frac{35 a b^{6} d e^{4}}{11} + \frac{10 b^{7} d^{2} e^{3}}{11}\right ) + x^{10} \left (\frac{7 a^{3} b^{4} e^{5}}{2} + \frac{21 a^{2} b^{5} d e^{4}}{2} + 7 a b^{6} d^{2} e^{3} + b^{7} d^{3} e^{2}\right ) + x^{9} \left (\frac{35 a^{4} b^{3} e^{5}}{9} + \frac{175 a^{3} b^{4} d e^{4}}{9} + \frac{70 a^{2} b^{5} d^{2} e^{3}}{3} + \frac{70 a b^{6} d^{3} e^{2}}{9} + \frac{5 b^{7} d^{4} e}{9}\right ) + x^{8} \left (\frac{21 a^{5} b^{2} e^{5}}{8} + \frac{175 a^{4} b^{3} d e^{4}}{8} + \frac{175 a^{3} b^{4} d^{2} e^{3}}{4} + \frac{105 a^{2} b^{5} d^{3} e^{2}}{4} + \frac{35 a b^{6} d^{4} e}{8} + \frac{b^{7} d^{5}}{8}\right ) + x^{7} \left (a^{6} b e^{5} + 15 a^{5} b^{2} d e^{4} + 50 a^{4} b^{3} d^{2} e^{3} + 50 a^{3} b^{4} d^{3} e^{2} + 15 a^{2} b^{5} d^{4} e + a b^{6} d^{5}\right ) + x^{6} \left (\frac{a^{7} e^{5}}{6} + \frac{35 a^{6} b d e^{4}}{6} + 35 a^{5} b^{2} d^{2} e^{3} + \frac{175 a^{4} b^{3} d^{3} e^{2}}{3} + \frac{175 a^{3} b^{4} d^{4} e}{6} + \frac{7 a^{2} b^{5} d^{5}}{2}\right ) + x^{5} \left (a^{7} d e^{4} + 14 a^{6} b d^{2} e^{3} + 42 a^{5} b^{2} d^{3} e^{2} + 35 a^{4} b^{3} d^{4} e + 7 a^{3} b^{4} d^{5}\right ) + x^{4} \left (\frac{5 a^{7} d^{2} e^{3}}{2} + \frac{35 a^{6} b d^{3} e^{2}}{2} + \frac{105 a^{5} b^{2} d^{4} e}{4} + \frac{35 a^{4} b^{3} d^{5}}{4}\right ) + x^{3} \left (\frac{10 a^{7} d^{3} e^{2}}{3} + \frac{35 a^{6} b d^{4} e}{3} + 7 a^{5} b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{7} d^{4} e}{2} + \frac{7 a^{6} b d^{5}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**7*d**5*x + b**7*e**5*x**13/13 + x**12*(7*a*b**6*e**5/12 + 5*b**7*d*e**4/12) +
x**11*(21*a**2*b**5*e**5/11 + 35*a*b**6*d*e**4/11 + 10*b**7*d**2*e**3/11) + x**
10*(7*a**3*b**4*e**5/2 + 21*a**2*b**5*d*e**4/2 + 7*a*b**6*d**2*e**3 + b**7*d**3*
e**2) + x**9*(35*a**4*b**3*e**5/9 + 175*a**3*b**4*d*e**4/9 + 70*a**2*b**5*d**2*e
**3/3 + 70*a*b**6*d**3*e**2/9 + 5*b**7*d**4*e/9) + x**8*(21*a**5*b**2*e**5/8 + 1
75*a**4*b**3*d*e**4/8 + 175*a**3*b**4*d**2*e**3/4 + 105*a**2*b**5*d**3*e**2/4 +
35*a*b**6*d**4*e/8 + b**7*d**5/8) + x**7*(a**6*b*e**5 + 15*a**5*b**2*d*e**4 + 50
*a**4*b**3*d**2*e**3 + 50*a**3*b**4*d**3*e**2 + 15*a**2*b**5*d**4*e + a*b**6*d**
5) + x**6*(a**7*e**5/6 + 35*a**6*b*d*e**4/6 + 35*a**5*b**2*d**2*e**3 + 175*a**4*
b**3*d**3*e**2/3 + 175*a**3*b**4*d**4*e/6 + 7*a**2*b**5*d**5/2) + x**5*(a**7*d*e
**4 + 14*a**6*b*d**2*e**3 + 42*a**5*b**2*d**3*e**2 + 35*a**4*b**3*d**4*e + 7*a**
3*b**4*d**5) + x**4*(5*a**7*d**2*e**3/2 + 35*a**6*b*d**3*e**2/2 + 105*a**5*b**2*
d**4*e/4 + 35*a**4*b**3*d**5/4) + x**3*(10*a**7*d**3*e**2/3 + 35*a**6*b*d**4*e/3
+ 7*a**5*b**2*d**5) + x**2*(5*a**7*d**4*e/2 + 7*a**6*b*d**5/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27849, size = 872, normalized size = 6.1 \[ \frac{1}{13} \, b^{7} x^{13} e^{5} + \frac{5}{12} \, b^{7} d x^{12} e^{4} + \frac{10}{11} \, b^{7} d^{2} x^{11} e^{3} + b^{7} d^{3} x^{10} e^{2} + \frac{5}{9} \, b^{7} d^{4} x^{9} e + \frac{1}{8} \, b^{7} d^{5} x^{8} + \frac{7}{12} \, a b^{6} x^{12} e^{5} + \frac{35}{11} \, a b^{6} d x^{11} e^{4} + 7 \, a b^{6} d^{2} x^{10} e^{3} + \frac{70}{9} \, a b^{6} d^{3} x^{9} e^{2} + \frac{35}{8} \, a b^{6} d^{4} x^{8} e + a b^{6} d^{5} x^{7} + \frac{21}{11} \, a^{2} b^{5} x^{11} e^{5} + \frac{21}{2} \, a^{2} b^{5} d x^{10} e^{4} + \frac{70}{3} \, a^{2} b^{5} d^{2} x^{9} e^{3} + \frac{105}{4} \, a^{2} b^{5} d^{3} x^{8} e^{2} + 15 \, a^{2} b^{5} d^{4} x^{7} e + \frac{7}{2} \, a^{2} b^{5} d^{5} x^{6} + \frac{7}{2} \, a^{3} b^{4} x^{10} e^{5} + \frac{175}{9} \, a^{3} b^{4} d x^{9} e^{4} + \frac{175}{4} \, a^{3} b^{4} d^{2} x^{8} e^{3} + 50 \, a^{3} b^{4} d^{3} x^{7} e^{2} + \frac{175}{6} \, a^{3} b^{4} d^{4} x^{6} e + 7 \, a^{3} b^{4} d^{5} x^{5} + \frac{35}{9} \, a^{4} b^{3} x^{9} e^{5} + \frac{175}{8} \, a^{4} b^{3} d x^{8} e^{4} + 50 \, a^{4} b^{3} d^{2} x^{7} e^{3} + \frac{175}{3} \, a^{4} b^{3} d^{3} x^{6} e^{2} + 35 \, a^{4} b^{3} d^{4} x^{5} e + \frac{35}{4} \, a^{4} b^{3} d^{5} x^{4} + \frac{21}{8} \, a^{5} b^{2} x^{8} e^{5} + 15 \, a^{5} b^{2} d x^{7} e^{4} + 35 \, a^{5} b^{2} d^{2} x^{6} e^{3} + 42 \, a^{5} b^{2} d^{3} x^{5} e^{2} + \frac{105}{4} \, a^{5} b^{2} d^{4} x^{4} e + 7 \, a^{5} b^{2} d^{5} x^{3} + a^{6} b x^{7} e^{5} + \frac{35}{6} \, a^{6} b d x^{6} e^{4} + 14 \, a^{6} b d^{2} x^{5} e^{3} + \frac{35}{2} \, a^{6} b d^{3} x^{4} e^{2} + \frac{35}{3} \, a^{6} b d^{4} x^{3} e + \frac{7}{2} \, a^{6} b d^{5} x^{2} + \frac{1}{6} \, a^{7} x^{6} e^{5} + a^{7} d x^{5} e^{4} + \frac{5}{2} \, a^{7} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{7} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{7} d^{4} x^{2} e + a^{7} d^{5} x \]

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)^5,x, algorithm="giac")

[Out]

1/13*b^7*x^13*e^5 + 5/12*b^7*d*x^12*e^4 + 10/11*b^7*d^2*x^11*e^3 + b^7*d^3*x^10*
e^2 + 5/9*b^7*d^4*x^9*e + 1/8*b^7*d^5*x^8 + 7/12*a*b^6*x^12*e^5 + 35/11*a*b^6*d*
x^11*e^4 + 7*a*b^6*d^2*x^10*e^3 + 70/9*a*b^6*d^3*x^9*e^2 + 35/8*a*b^6*d^4*x^8*e
+ a*b^6*d^5*x^7 + 21/11*a^2*b^5*x^11*e^5 + 21/2*a^2*b^5*d*x^10*e^4 + 70/3*a^2*b^
5*d^2*x^9*e^3 + 105/4*a^2*b^5*d^3*x^8*e^2 + 15*a^2*b^5*d^4*x^7*e + 7/2*a^2*b^5*d
^5*x^6 + 7/2*a^3*b^4*x^10*e^5 + 175/9*a^3*b^4*d*x^9*e^4 + 175/4*a^3*b^4*d^2*x^8*
e^3 + 50*a^3*b^4*d^3*x^7*e^2 + 175/6*a^3*b^4*d^4*x^6*e + 7*a^3*b^4*d^5*x^5 + 35/
9*a^4*b^3*x^9*e^5 + 175/8*a^4*b^3*d*x^8*e^4 + 50*a^4*b^3*d^2*x^7*e^3 + 175/3*a^4
*b^3*d^3*x^6*e^2 + 35*a^4*b^3*d^4*x^5*e + 35/4*a^4*b^3*d^5*x^4 + 21/8*a^5*b^2*x^
8*e^5 + 15*a^5*b^2*d*x^7*e^4 + 35*a^5*b^2*d^2*x^6*e^3 + 42*a^5*b^2*d^3*x^5*e^2 +
105/4*a^5*b^2*d^4*x^4*e + 7*a^5*b^2*d^5*x^3 + a^6*b*x^7*e^5 + 35/6*a^6*b*d*x^6*
e^4 + 14*a^6*b*d^2*x^5*e^3 + 35/2*a^6*b*d^3*x^4*e^2 + 35/3*a^6*b*d^4*x^3*e + 7/2
*a^6*b*d^5*x^2 + 1/6*a^7*x^6*e^5 + a^7*d*x^5*e^4 + 5/2*a^7*d^2*x^4*e^3 + 10/3*a^
7*d^3*x^3*e^2 + 5/2*a^7*d^4*x^2*e + a^7*d^5*x