3.1472 \(\int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=171 \[ \frac{e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac{15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac{2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac{5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac{3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac{(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac{e^6 (a+b x)^{13}}{13 b^7} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.733821, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac{15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac{2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac{5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac{3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac{(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac{e^6 (a+b x)^{13}}{13 b^7} \]

Antiderivative was successfully verified.

[In]  Int[(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)^7)/(7*b^7) + (3*e*(b*d - a*e)^5*(a + b*x)^8)/(4*b^7) +
(5*e^2*(b*d - a*e)^4*(a + b*x)^9)/(3*b^7) + (2*e^3*(b*d - a*e)^3*(a + b*x)^10)/b
^7 + (15*e^4*(b*d - a*e)^2*(a + b*x)^11)/(11*b^7) + (e^5*(b*d - a*e)*(a + b*x)^1
2)/(2*b^7) + (e^6*(a + b*x)^13)/(13*b^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 113.865, size = 155, normalized size = 0.91 \[ \frac{b^{6} \left (d + e x\right )^{13}}{13 e^{7}} + \frac{b^{5} \left (d + e x\right )^{12} \left (a e - b d\right )}{2 e^{7}} + \frac{15 b^{4} \left (d + e x\right )^{11} \left (a e - b d\right )^{2}}{11 e^{7}} + \frac{2 b^{3} \left (d + e x\right )^{10} \left (a e - b d\right )^{3}}{e^{7}} + \frac{5 b^{2} \left (d + e x\right )^{9} \left (a e - b d\right )^{4}}{3 e^{7}} + \frac{3 b \left (d + e x\right )^{8} \left (a e - b d\right )^{5}}{4 e^{7}} + \frac{\left (d + e x\right )^{7} \left (a e - b d\right )^{6}}{7 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 0.148605, size = 573, normalized size = 3.35 \[ a^6 d^6 x+3 a^5 d^5 x^2 (a e+b d)+\frac{3}{11} b^4 e^4 x^{11} \left (5 a^2 e^2+12 a b d e+5 b^2 d^2\right )+a^4 d^4 x^3 \left (5 a^2 e^2+12 a b d e+5 b^2 d^2\right )+b^3 e^3 x^{10} \left (2 a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+2 b^3 d^3\right )+\frac{5}{2} a^3 d^3 x^4 \left (2 a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+2 b^3 d^3\right )+\frac{5}{3} b^2 e^2 x^9 \left (a^4 e^4+8 a^3 b d e^3+15 a^2 b^2 d^2 e^2+8 a b^3 d^3 e+b^4 d^4\right )+3 a^2 d^2 x^5 \left (a^4 e^4+8 a^3 b d e^3+15 a^2 b^2 d^2 e^2+8 a b^3 d^3 e+b^4 d^4\right )+\frac{3}{4} b e x^8 \left (a^5 e^5+15 a^4 b d e^4+50 a^3 b^2 d^2 e^3+50 a^2 b^3 d^3 e^2+15 a b^4 d^4 e+b^5 d^5\right )+a d x^6 \left (a^5 e^5+15 a^4 b d e^4+50 a^3 b^2 d^2 e^3+50 a^2 b^3 d^3 e^2+15 a b^4 d^4 e+b^5 d^5\right )+\frac{1}{7} x^7 \left (a^6 e^6+36 a^5 b d e^5+225 a^4 b^2 d^2 e^4+400 a^3 b^3 d^3 e^3+225 a^2 b^4 d^4 e^2+36 a b^5 d^5 e+b^6 d^6\right )+\frac{1}{2} b^5 e^5 x^{12} (a e+b d)+\frac{1}{13} b^6 e^6 x^{13} \]

Antiderivative was successfully verified.

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

[Out]

a^6*d^6*x + 3*a^5*d^5*(b*d + a*e)*x^2 + a^4*d^4*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*
e^2)*x^3 + (5*a^3*d^3*(2*b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3)*x^
4)/2 + 3*a^2*d^2*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 +
a^4*e^4)*x^5 + a*d*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50*a^3*b^2*
d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5)*x^6 + ((b^6*d^6 + 36*a*b^5*d^5*e + 225*a^2*b
^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^
6)*x^7)/7 + (3*b*e*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50*a^3*b^2*d
^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5)*x^8)/4 + (5*b^2*e^2*(b^4*d^4 + 8*a*b^3*d^3*e
+ 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4)*x^9)/3 + b^3*e^3*(2*b^3*d^3 + 9*
a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3)*x^10 + (3*b^4*e^4*(5*b^2*d^2 + 12*a*b*d
*e + 5*a^2*e^2)*x^11)/11 + (b^5*e^5*(b*d + a*e)*x^12)/2 + (b^6*e^6*x^13)/13

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 615, normalized size = 3.6 \[{\frac{{b}^{6}{e}^{6}{x}^{13}}{13}}+{\frac{ \left ( 6\,{e}^{6}a{b}^{5}+6\,d{e}^{5}{b}^{6} \right ){x}^{12}}{12}}+{\frac{ \left ( 15\,{e}^{6}{a}^{2}{b}^{4}+36\,d{e}^{5}a{b}^{5}+15\,{d}^{2}{e}^{4}{b}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 20\,{e}^{6}{a}^{3}{b}^{3}+90\,d{e}^{5}{a}^{2}{b}^{4}+90\,{d}^{2}{e}^{4}a{b}^{5}+20\,{d}^{3}{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 15\,{e}^{6}{b}^{2}{a}^{4}+120\,d{e}^{5}{a}^{3}{b}^{3}+225\,{d}^{2}{e}^{4}{a}^{2}{b}^{4}+120\,{d}^{3}{e}^{3}a{b}^{5}+15\,{d}^{4}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{6}{a}^{5}b+90\,d{e}^{5}{b}^{2}{a}^{4}+300\,{d}^{2}{e}^{4}{a}^{3}{b}^{3}+300\,{d}^{3}{e}^{3}{a}^{2}{b}^{4}+90\,{d}^{4}{e}^{2}a{b}^{5}+6\,{d}^{5}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{6}{a}^{6}+36\,d{e}^{5}{a}^{5}b+225\,{d}^{2}{e}^{4}{b}^{2}{a}^{4}+400\,{d}^{3}{e}^{3}{a}^{3}{b}^{3}+225\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+36\,{d}^{5}ea{b}^{5}+{d}^{6}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,d{e}^{5}{a}^{6}+90\,{d}^{2}{e}^{4}{a}^{5}b+300\,{d}^{3}{e}^{3}{b}^{2}{a}^{4}+300\,{d}^{4}{e}^{2}{a}^{3}{b}^{3}+90\,{d}^{5}e{a}^{2}{b}^{4}+6\,{d}^{6}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{d}^{2}{e}^{4}{a}^{6}+120\,{d}^{3}{e}^{3}{a}^{5}b+225\,{d}^{4}{e}^{2}{b}^{2}{a}^{4}+120\,{d}^{5}e{a}^{3}{b}^{3}+15\,{d}^{6}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{d}^{3}{e}^{3}{a}^{6}+90\,{d}^{4}{e}^{2}{a}^{5}b+90\,{d}^{5}e{b}^{2}{a}^{4}+20\,{d}^{6}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{d}^{4}{e}^{2}{a}^{6}+36\,{d}^{5}e{a}^{5}b+15\,{d}^{6}{b}^{2}{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{d}^{5}e{a}^{6}+6\,{d}^{6}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{6}{a}^{6}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^6*e^6*x^13+1/12*(6*a*b^5*e^6+6*b^6*d*e^5)*x^12+1/11*(15*a^2*b^4*e^6+36*a*
b^5*d*e^5+15*b^6*d^2*e^4)*x^11+1/10*(20*a^3*b^3*e^6+90*a^2*b^4*d*e^5+90*a*b^5*d^
2*e^4+20*b^6*d^3*e^3)*x^10+1/9*(15*a^4*b^2*e^6+120*a^3*b^3*d*e^5+225*a^2*b^4*d^2
*e^4+120*a*b^5*d^3*e^3+15*b^6*d^4*e^2)*x^9+1/8*(6*a^5*b*e^6+90*a^4*b^2*d*e^5+300
*a^3*b^3*d^2*e^4+300*a^2*b^4*d^3*e^3+90*a*b^5*d^4*e^2+6*b^6*d^5*e)*x^8+1/7*(a^6*
e^6+36*a^5*b*d*e^5+225*a^4*b^2*d^2*e^4+400*a^3*b^3*d^3*e^3+225*a^2*b^4*d^4*e^2+3
6*a*b^5*d^5*e+b^6*d^6)*x^7+1/6*(6*a^6*d*e^5+90*a^5*b*d^2*e^4+300*a^4*b^2*d^3*e^3
+300*a^3*b^3*d^4*e^2+90*a^2*b^4*d^5*e+6*a*b^5*d^6)*x^6+1/5*(15*a^6*d^2*e^4+120*a
^5*b*d^3*e^3+225*a^4*b^2*d^4*e^2+120*a^3*b^3*d^5*e+15*a^2*b^4*d^6)*x^5+1/4*(20*a
^6*d^3*e^3+90*a^5*b*d^4*e^2+90*a^4*b^2*d^5*e+20*a^3*b^3*d^6)*x^4+1/3*(15*a^6*d^4
*e^2+36*a^5*b*d^5*e+15*a^4*b^2*d^6)*x^3+1/2*(6*a^6*d^5*e+6*a^5*b*d^6)*x^2+d^6*a^
6*x

_______________________________________________________________________________________

Maxima [A]  time = 0.691844, size = 809, normalized size = 4.73 \[ \frac{1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac{1}{2} \,{\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac{3}{11} \,{\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} +{\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac{5}{3} \,{\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac{3}{4} \,{\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} +{\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \,{\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \,{\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^6,x, algorithm="maxima")

[Out]

1/13*b^6*e^6*x^13 + a^6*d^6*x + 1/2*(b^6*d*e^5 + a*b^5*e^6)*x^12 + 3/11*(5*b^6*d
^2*e^4 + 12*a*b^5*d*e^5 + 5*a^2*b^4*e^6)*x^11 + (2*b^6*d^3*e^3 + 9*a*b^5*d^2*e^4
+ 9*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^10 + 5/3*(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 +
15*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^9 + 3/4*(b^6*d^5*e + 15*a*
b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + a^5*b
*e^6)*x^8 + 1/7*(b^6*d^6 + 36*a*b^5*d^5*e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^
3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6)*x^7 + (a*b^5*d^6 + 15*a^
2*b^4*d^5*e + 50*a^3*b^3*d^4*e^2 + 50*a^4*b^2*d^3*e^3 + 15*a^5*b*d^2*e^4 + a^6*d
*e^5)*x^6 + 3*(a^2*b^4*d^6 + 8*a^3*b^3*d^5*e + 15*a^4*b^2*d^4*e^2 + 8*a^5*b*d^3*
e^3 + a^6*d^2*e^4)*x^5 + 5/2*(2*a^3*b^3*d^6 + 9*a^4*b^2*d^5*e + 9*a^5*b*d^4*e^2
+ 2*a^6*d^3*e^3)*x^4 + (5*a^4*b^2*d^6 + 12*a^5*b*d^5*e + 5*a^6*d^4*e^2)*x^3 + 3*
(a^5*b*d^6 + a^6*d^5*e)*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.197006, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} e^{6} b^{6} + \frac{1}{2} x^{12} e^{5} d b^{6} + \frac{1}{2} x^{12} e^{6} b^{5} a + \frac{15}{11} x^{11} e^{4} d^{2} b^{6} + \frac{36}{11} x^{11} e^{5} d b^{5} a + \frac{15}{11} x^{11} e^{6} b^{4} a^{2} + 2 x^{10} e^{3} d^{3} b^{6} + 9 x^{10} e^{4} d^{2} b^{5} a + 9 x^{10} e^{5} d b^{4} a^{2} + 2 x^{10} e^{6} b^{3} a^{3} + \frac{5}{3} x^{9} e^{2} d^{4} b^{6} + \frac{40}{3} x^{9} e^{3} d^{3} b^{5} a + 25 x^{9} e^{4} d^{2} b^{4} a^{2} + \frac{40}{3} x^{9} e^{5} d b^{3} a^{3} + \frac{5}{3} x^{9} e^{6} b^{2} a^{4} + \frac{3}{4} x^{8} e d^{5} b^{6} + \frac{45}{4} x^{8} e^{2} d^{4} b^{5} a + \frac{75}{2} x^{8} e^{3} d^{3} b^{4} a^{2} + \frac{75}{2} x^{8} e^{4} d^{2} b^{3} a^{3} + \frac{45}{4} x^{8} e^{5} d b^{2} a^{4} + \frac{3}{4} x^{8} e^{6} b a^{5} + \frac{1}{7} x^{7} d^{6} b^{6} + \frac{36}{7} x^{7} e d^{5} b^{5} a + \frac{225}{7} x^{7} e^{2} d^{4} b^{4} a^{2} + \frac{400}{7} x^{7} e^{3} d^{3} b^{3} a^{3} + \frac{225}{7} x^{7} e^{4} d^{2} b^{2} a^{4} + \frac{36}{7} x^{7} e^{5} d b a^{5} + \frac{1}{7} x^{7} e^{6} a^{6} + x^{6} d^{6} b^{5} a + 15 x^{6} e d^{5} b^{4} a^{2} + 50 x^{6} e^{2} d^{4} b^{3} a^{3} + 50 x^{6} e^{3} d^{3} b^{2} a^{4} + 15 x^{6} e^{4} d^{2} b a^{5} + x^{6} e^{5} d a^{6} + 3 x^{5} d^{6} b^{4} a^{2} + 24 x^{5} e d^{5} b^{3} a^{3} + 45 x^{5} e^{2} d^{4} b^{2} a^{4} + 24 x^{5} e^{3} d^{3} b a^{5} + 3 x^{5} e^{4} d^{2} a^{6} + 5 x^{4} d^{6} b^{3} a^{3} + \frac{45}{2} x^{4} e d^{5} b^{2} a^{4} + \frac{45}{2} x^{4} e^{2} d^{4} b a^{5} + 5 x^{4} e^{3} d^{3} a^{6} + 5 x^{3} d^{6} b^{2} a^{4} + 12 x^{3} e d^{5} b a^{5} + 5 x^{3} e^{2} d^{4} a^{6} + 3 x^{2} d^{6} b a^{5} + 3 x^{2} e d^{5} a^{6} + x d^{6} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^6,x, algorithm="fricas")

[Out]

1/13*x^13*e^6*b^6 + 1/2*x^12*e^5*d*b^6 + 1/2*x^12*e^6*b^5*a + 15/11*x^11*e^4*d^2
*b^6 + 36/11*x^11*e^5*d*b^5*a + 15/11*x^11*e^6*b^4*a^2 + 2*x^10*e^3*d^3*b^6 + 9*
x^10*e^4*d^2*b^5*a + 9*x^10*e^5*d*b^4*a^2 + 2*x^10*e^6*b^3*a^3 + 5/3*x^9*e^2*d^4
*b^6 + 40/3*x^9*e^3*d^3*b^5*a + 25*x^9*e^4*d^2*b^4*a^2 + 40/3*x^9*e^5*d*b^3*a^3
+ 5/3*x^9*e^6*b^2*a^4 + 3/4*x^8*e*d^5*b^6 + 45/4*x^8*e^2*d^4*b^5*a + 75/2*x^8*e^
3*d^3*b^4*a^2 + 75/2*x^8*e^4*d^2*b^3*a^3 + 45/4*x^8*e^5*d*b^2*a^4 + 3/4*x^8*e^6*
b*a^5 + 1/7*x^7*d^6*b^6 + 36/7*x^7*e*d^5*b^5*a + 225/7*x^7*e^2*d^4*b^4*a^2 + 400
/7*x^7*e^3*d^3*b^3*a^3 + 225/7*x^7*e^4*d^2*b^2*a^4 + 36/7*x^7*e^5*d*b*a^5 + 1/7*
x^7*e^6*a^6 + x^6*d^6*b^5*a + 15*x^6*e*d^5*b^4*a^2 + 50*x^6*e^2*d^4*b^3*a^3 + 50
*x^6*e^3*d^3*b^2*a^4 + 15*x^6*e^4*d^2*b*a^5 + x^6*e^5*d*a^6 + 3*x^5*d^6*b^4*a^2
+ 24*x^5*e*d^5*b^3*a^3 + 45*x^5*e^2*d^4*b^2*a^4 + 24*x^5*e^3*d^3*b*a^5 + 3*x^5*e
^4*d^2*a^6 + 5*x^4*d^6*b^3*a^3 + 45/2*x^4*e*d^5*b^2*a^4 + 45/2*x^4*e^2*d^4*b*a^5
+ 5*x^4*e^3*d^3*a^6 + 5*x^3*d^6*b^2*a^4 + 12*x^3*e*d^5*b*a^5 + 5*x^3*e^2*d^4*a^
6 + 3*x^2*d^6*b*a^5 + 3*x^2*e*d^5*a^6 + x*d^6*a^6

_______________________________________________________________________________________

Sympy [A]  time = 0.412863, size = 677, normalized size = 3.96 \[ a^{6} d^{6} x + \frac{b^{6} e^{6} x^{13}}{13} + x^{12} \left (\frac{a b^{5} e^{6}}{2} + \frac{b^{6} d e^{5}}{2}\right ) + x^{11} \left (\frac{15 a^{2} b^{4} e^{6}}{11} + \frac{36 a b^{5} d e^{5}}{11} + \frac{15 b^{6} d^{2} e^{4}}{11}\right ) + x^{10} \left (2 a^{3} b^{3} e^{6} + 9 a^{2} b^{4} d e^{5} + 9 a b^{5} d^{2} e^{4} + 2 b^{6} d^{3} e^{3}\right ) + x^{9} \left (\frac{5 a^{4} b^{2} e^{6}}{3} + \frac{40 a^{3} b^{3} d e^{5}}{3} + 25 a^{2} b^{4} d^{2} e^{4} + \frac{40 a b^{5} d^{3} e^{3}}{3} + \frac{5 b^{6} d^{4} e^{2}}{3}\right ) + x^{8} \left (\frac{3 a^{5} b e^{6}}{4} + \frac{45 a^{4} b^{2} d e^{5}}{4} + \frac{75 a^{3} b^{3} d^{2} e^{4}}{2} + \frac{75 a^{2} b^{4} d^{3} e^{3}}{2} + \frac{45 a b^{5} d^{4} e^{2}}{4} + \frac{3 b^{6} d^{5} e}{4}\right ) + x^{7} \left (\frac{a^{6} e^{6}}{7} + \frac{36 a^{5} b d e^{5}}{7} + \frac{225 a^{4} b^{2} d^{2} e^{4}}{7} + \frac{400 a^{3} b^{3} d^{3} e^{3}}{7} + \frac{225 a^{2} b^{4} d^{4} e^{2}}{7} + \frac{36 a b^{5} d^{5} e}{7} + \frac{b^{6} d^{6}}{7}\right ) + x^{6} \left (a^{6} d e^{5} + 15 a^{5} b d^{2} e^{4} + 50 a^{4} b^{2} d^{3} e^{3} + 50 a^{3} b^{3} d^{4} e^{2} + 15 a^{2} b^{4} d^{5} e + a b^{5} d^{6}\right ) + x^{5} \left (3 a^{6} d^{2} e^{4} + 24 a^{5} b d^{3} e^{3} + 45 a^{4} b^{2} d^{4} e^{2} + 24 a^{3} b^{3} d^{5} e + 3 a^{2} b^{4} d^{6}\right ) + x^{4} \left (5 a^{6} d^{3} e^{3} + \frac{45 a^{5} b d^{4} e^{2}}{2} + \frac{45 a^{4} b^{2} d^{5} e}{2} + 5 a^{3} b^{3} d^{6}\right ) + x^{3} \left (5 a^{6} d^{4} e^{2} + 12 a^{5} b d^{5} e + 5 a^{4} b^{2} d^{6}\right ) + x^{2} \left (3 a^{6} d^{5} e + 3 a^{5} b d^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**6*d**6*x + b**6*e**6*x**13/13 + x**12*(a*b**5*e**6/2 + b**6*d*e**5/2) + x**11
*(15*a**2*b**4*e**6/11 + 36*a*b**5*d*e**5/11 + 15*b**6*d**2*e**4/11) + x**10*(2*
a**3*b**3*e**6 + 9*a**2*b**4*d*e**5 + 9*a*b**5*d**2*e**4 + 2*b**6*d**3*e**3) + x
**9*(5*a**4*b**2*e**6/3 + 40*a**3*b**3*d*e**5/3 + 25*a**2*b**4*d**2*e**4 + 40*a*
b**5*d**3*e**3/3 + 5*b**6*d**4*e**2/3) + x**8*(3*a**5*b*e**6/4 + 45*a**4*b**2*d*
e**5/4 + 75*a**3*b**3*d**2*e**4/2 + 75*a**2*b**4*d**3*e**3/2 + 45*a*b**5*d**4*e*
*2/4 + 3*b**6*d**5*e/4) + x**7*(a**6*e**6/7 + 36*a**5*b*d*e**5/7 + 225*a**4*b**2
*d**2*e**4/7 + 400*a**3*b**3*d**3*e**3/7 + 225*a**2*b**4*d**4*e**2/7 + 36*a*b**5
*d**5*e/7 + b**6*d**6/7) + x**6*(a**6*d*e**5 + 15*a**5*b*d**2*e**4 + 50*a**4*b**
2*d**3*e**3 + 50*a**3*b**3*d**4*e**2 + 15*a**2*b**4*d**5*e + a*b**5*d**6) + x**5
*(3*a**6*d**2*e**4 + 24*a**5*b*d**3*e**3 + 45*a**4*b**2*d**4*e**2 + 24*a**3*b**3
*d**5*e + 3*a**2*b**4*d**6) + x**4*(5*a**6*d**3*e**3 + 45*a**5*b*d**4*e**2/2 + 4
5*a**4*b**2*d**5*e/2 + 5*a**3*b**3*d**6) + x**3*(5*a**6*d**4*e**2 + 12*a**5*b*d*
*5*e + 5*a**4*b**2*d**6) + x**2*(3*a**6*d**5*e + 3*a**5*b*d**6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211489, size = 892, normalized size = 5.22 \[ \frac{1}{13} \, b^{6} x^{13} e^{6} + \frac{1}{2} \, b^{6} d x^{12} e^{5} + \frac{15}{11} \, b^{6} d^{2} x^{11} e^{4} + 2 \, b^{6} d^{3} x^{10} e^{3} + \frac{5}{3} \, b^{6} d^{4} x^{9} e^{2} + \frac{3}{4} \, b^{6} d^{5} x^{8} e + \frac{1}{7} \, b^{6} d^{6} x^{7} + \frac{1}{2} \, a b^{5} x^{12} e^{6} + \frac{36}{11} \, a b^{5} d x^{11} e^{5} + 9 \, a b^{5} d^{2} x^{10} e^{4} + \frac{40}{3} \, a b^{5} d^{3} x^{9} e^{3} + \frac{45}{4} \, a b^{5} d^{4} x^{8} e^{2} + \frac{36}{7} \, a b^{5} d^{5} x^{7} e + a b^{5} d^{6} x^{6} + \frac{15}{11} \, a^{2} b^{4} x^{11} e^{6} + 9 \, a^{2} b^{4} d x^{10} e^{5} + 25 \, a^{2} b^{4} d^{2} x^{9} e^{4} + \frac{75}{2} \, a^{2} b^{4} d^{3} x^{8} e^{3} + \frac{225}{7} \, a^{2} b^{4} d^{4} x^{7} e^{2} + 15 \, a^{2} b^{4} d^{5} x^{6} e + 3 \, a^{2} b^{4} d^{6} x^{5} + 2 \, a^{3} b^{3} x^{10} e^{6} + \frac{40}{3} \, a^{3} b^{3} d x^{9} e^{5} + \frac{75}{2} \, a^{3} b^{3} d^{2} x^{8} e^{4} + \frac{400}{7} \, a^{3} b^{3} d^{3} x^{7} e^{3} + 50 \, a^{3} b^{3} d^{4} x^{6} e^{2} + 24 \, a^{3} b^{3} d^{5} x^{5} e + 5 \, a^{3} b^{3} d^{6} x^{4} + \frac{5}{3} \, a^{4} b^{2} x^{9} e^{6} + \frac{45}{4} \, a^{4} b^{2} d x^{8} e^{5} + \frac{225}{7} \, a^{4} b^{2} d^{2} x^{7} e^{4} + 50 \, a^{4} b^{2} d^{3} x^{6} e^{3} + 45 \, a^{4} b^{2} d^{4} x^{5} e^{2} + \frac{45}{2} \, a^{4} b^{2} d^{5} x^{4} e + 5 \, a^{4} b^{2} d^{6} x^{3} + \frac{3}{4} \, a^{5} b x^{8} e^{6} + \frac{36}{7} \, a^{5} b d x^{7} e^{5} + 15 \, a^{5} b d^{2} x^{6} e^{4} + 24 \, a^{5} b d^{3} x^{5} e^{3} + \frac{45}{2} \, a^{5} b d^{4} x^{4} e^{2} + 12 \, a^{5} b d^{5} x^{3} e + 3 \, a^{5} b d^{6} x^{2} + \frac{1}{7} \, a^{6} x^{7} e^{6} + a^{6} d x^{6} e^{5} + 3 \, a^{6} d^{2} x^{5} e^{4} + 5 \, a^{6} d^{3} x^{4} e^{3} + 5 \, a^{6} d^{4} x^{3} e^{2} + 3 \, a^{6} d^{5} x^{2} e + a^{6} d^{6} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^6,x, algorithm="giac")

[Out]

1/13*b^6*x^13*e^6 + 1/2*b^6*d*x^12*e^5 + 15/11*b^6*d^2*x^11*e^4 + 2*b^6*d^3*x^10
*e^3 + 5/3*b^6*d^4*x^9*e^2 + 3/4*b^6*d^5*x^8*e + 1/7*b^6*d^6*x^7 + 1/2*a*b^5*x^1
2*e^6 + 36/11*a*b^5*d*x^11*e^5 + 9*a*b^5*d^2*x^10*e^4 + 40/3*a*b^5*d^3*x^9*e^3 +
45/4*a*b^5*d^4*x^8*e^2 + 36/7*a*b^5*d^5*x^7*e + a*b^5*d^6*x^6 + 15/11*a^2*b^4*x
^11*e^6 + 9*a^2*b^4*d*x^10*e^5 + 25*a^2*b^4*d^2*x^9*e^4 + 75/2*a^2*b^4*d^3*x^8*e
^3 + 225/7*a^2*b^4*d^4*x^7*e^2 + 15*a^2*b^4*d^5*x^6*e + 3*a^2*b^4*d^6*x^5 + 2*a^
3*b^3*x^10*e^6 + 40/3*a^3*b^3*d*x^9*e^5 + 75/2*a^3*b^3*d^2*x^8*e^4 + 400/7*a^3*b
^3*d^3*x^7*e^3 + 50*a^3*b^3*d^4*x^6*e^2 + 24*a^3*b^3*d^5*x^5*e + 5*a^3*b^3*d^6*x
^4 + 5/3*a^4*b^2*x^9*e^6 + 45/4*a^4*b^2*d*x^8*e^5 + 225/7*a^4*b^2*d^2*x^7*e^4 +
50*a^4*b^2*d^3*x^6*e^3 + 45*a^4*b^2*d^4*x^5*e^2 + 45/2*a^4*b^2*d^5*x^4*e + 5*a^4
*b^2*d^6*x^3 + 3/4*a^5*b*x^8*e^6 + 36/7*a^5*b*d*x^7*e^5 + 15*a^5*b*d^2*x^6*e^4 +
24*a^5*b*d^3*x^5*e^3 + 45/2*a^5*b*d^4*x^4*e^2 + 12*a^5*b*d^5*x^3*e + 3*a^5*b*d^
6*x^2 + 1/7*a^6*x^7*e^6 + a^6*d*x^6*e^5 + 3*a^6*d^2*x^5*e^4 + 5*a^6*d^3*x^4*e^3
+ 5*a^6*d^4*x^3*e^2 + 3*a^6*d^5*x^2*e + a^6*d^6*x