3.1473 \(\int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=143 \[ \frac{5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac{e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac{10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac{5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac{(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac{e^5 (a+b x)^{12}}{12 b^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.689161, antiderivative size = 143, 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 \[ \frac{5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac{e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac{10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac{5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac{(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac{e^5 (a+b x)^{12}}{12 b^6} \]

Antiderivative was successfully verified.

[In]  Int[(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)^7)/(7*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^8)/(8*b^6) +
(10*e^2*(b*d - a*e)^3*(a + b*x)^9)/(9*b^6) + (e^3*(b*d - a*e)^2*(a + b*x)^10)/b^
6 + (5*e^4*(b*d - a*e)*(a + b*x)^11)/(11*b^6) + (e^5*(a + b*x)^12)/(12*b^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 82.7841, size = 129, normalized size = 0.9 \[ \frac{e^{5} \left (a + b x\right )^{12}}{12 b^{6}} - \frac{5 e^{4} \left (a + b x\right )^{11} \left (a e - b d\right )}{11 b^{6}} + \frac{e^{3} \left (a + b x\right )^{10} \left (a e - b d\right )^{2}}{b^{6}} - \frac{10 e^{2} \left (a + b x\right )^{9} \left (a e - b d\right )^{3}}{9 b^{6}} + \frac{5 e \left (a + b x\right )^{8} \left (a e - b d\right )^{4}}{8 b^{6}} - \frac{\left (a + b x\right )^{7} \left (a e - b d\right )^{5}}{7 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 0.131894, size = 501, normalized size = 3.5 \[ a^6 d^5 x+\frac{1}{2} a^5 d^4 x^2 (5 a e+6 b d)+\frac{1}{2} b^4 e^3 x^{10} \left (3 a^2 e^2+6 a b d e+2 b^2 d^2\right )+\frac{5}{3} a^4 d^3 x^3 \left (2 a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{5}{9} b^3 e^2 x^9 \left (4 a^3 e^3+15 a^2 b d e^2+12 a b^2 d^2 e+2 b^3 d^3\right )+\frac{5}{4} a^3 d^2 x^4 \left (2 a^3 e^3+12 a^2 b d e^2+15 a b^2 d^2 e+4 b^3 d^3\right )+\frac{5}{8} b^2 e x^8 \left (3 a^4 e^4+20 a^3 b d e^3+30 a^2 b^2 d^2 e^2+12 a b^3 d^3 e+b^4 d^4\right )+a^2 d x^5 \left (a^4 e^4+12 a^3 b d e^3+30 a^2 b^2 d^2 e^2+20 a b^3 d^3 e+3 b^4 d^4\right )+\frac{1}{7} b x^7 \left (6 a^5 e^5+75 a^4 b d e^4+200 a^3 b^2 d^2 e^3+150 a^2 b^3 d^3 e^2+30 a b^4 d^4 e+b^5 d^5\right )+\frac{1}{6} a x^6 \left (a^5 e^5+30 a^4 b d e^4+150 a^3 b^2 d^2 e^3+200 a^2 b^3 d^3 e^2+75 a b^4 d^4 e+6 b^5 d^5\right )+\frac{1}{11} b^5 e^4 x^{11} (6 a e+5 b d)+\frac{1}{12} b^6 e^5 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

a^6*d^5*x + (a^5*d^4*(6*b*d + 5*a*e)*x^2)/2 + (5*a^4*d^3*(3*b^2*d^2 + 6*a*b*d*e
+ 2*a^2*e^2)*x^3)/3 + (5*a^3*d^2*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2 +
2*a^3*e^3)*x^4)/4 + a^2*d*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*
a^3*b*d*e^3 + a^4*e^4)*x^5 + (a*(6*b^5*d^5 + 75*a*b^4*d^4*e + 200*a^2*b^3*d^3*e^
2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5)*x^6)/6 + (b*(b^5*d^5 + 30*a*
b^4*d^4*e + 150*a^2*b^3*d^3*e^2 + 200*a^3*b^2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e
^5)*x^7)/7 + (5*b^2*e*(b^4*d^4 + 12*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 20*a^3*b*
d*e^3 + 3*a^4*e^4)*x^8)/8 + (5*b^3*e^2*(2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^2*b*d*
e^2 + 4*a^3*e^3)*x^9)/9 + (b^4*e^3*(2*b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^10)/2 +
(b^5*e^4*(5*b*d + 6*a*e)*x^11)/11 + (b^6*e^5*x^12)/12

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 521, normalized size = 3.6 \[{\frac{{b}^{6}{e}^{5}{x}^{12}}{12}}+{\frac{ \left ( 6\,{e}^{5}a{b}^{5}+5\,d{e}^{4}{b}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 15\,{e}^{5}{a}^{2}{b}^{4}+30\,d{e}^{4}a{b}^{5}+10\,{d}^{2}{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 20\,{e}^{5}{a}^{3}{b}^{3}+75\,d{e}^{4}{a}^{2}{b}^{4}+60\,{d}^{2}{e}^{3}a{b}^{5}+10\,{d}^{3}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 15\,{e}^{5}{b}^{2}{a}^{4}+100\,d{e}^{4}{a}^{3}{b}^{3}+150\,{d}^{2}{e}^{3}{a}^{2}{b}^{4}+60\,{d}^{3}{e}^{2}a{b}^{5}+5\,{d}^{4}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,{e}^{5}{a}^{5}b+75\,d{e}^{4}{b}^{2}{a}^{4}+200\,{d}^{2}{e}^{3}{a}^{3}{b}^{3}+150\,{d}^{3}{e}^{2}{a}^{2}{b}^{4}+30\,{d}^{4}ea{b}^{5}+{d}^{5}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{5}{a}^{6}+30\,d{e}^{4}{a}^{5}b+150\,{d}^{2}{e}^{3}{b}^{2}{a}^{4}+200\,{d}^{3}{e}^{2}{a}^{3}{b}^{3}+75\,{d}^{4}e{a}^{2}{b}^{4}+6\,{d}^{5}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,d{e}^{4}{a}^{6}+60\,{d}^{2}{e}^{3}{a}^{5}b+150\,{d}^{3}{e}^{2}{b}^{2}{a}^{4}+100\,{d}^{4}e{a}^{3}{b}^{3}+15\,{d}^{5}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}{a}^{6}+60\,{d}^{3}{e}^{2}{a}^{5}b+75\,{d}^{4}e{b}^{2}{a}^{4}+20\,{d}^{5}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}{a}^{6}+30\,{d}^{4}e{a}^{5}b+15\,{d}^{5}{b}^{2}{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{d}^{4}e{a}^{6}+6\,{d}^{5}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{5}{a}^{6}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*b^6*e^5*x^12+1/11*(6*a*b^5*e^5+5*b^6*d*e^4)*x^11+1/10*(15*a^2*b^4*e^5+30*a*
b^5*d*e^4+10*b^6*d^2*e^3)*x^10+1/9*(20*a^3*b^3*e^5+75*a^2*b^4*d*e^4+60*a*b^5*d^2
*e^3+10*b^6*d^3*e^2)*x^9+1/8*(15*a^4*b^2*e^5+100*a^3*b^3*d*e^4+150*a^2*b^4*d^2*e
^3+60*a*b^5*d^3*e^2+5*b^6*d^4*e)*x^8+1/7*(6*a^5*b*e^5+75*a^4*b^2*d*e^4+200*a^3*b
^3*d^2*e^3+150*a^2*b^4*d^3*e^2+30*a*b^5*d^4*e+b^6*d^5)*x^7+1/6*(a^6*e^5+30*a^5*b
*d*e^4+150*a^4*b^2*d^2*e^3+200*a^3*b^3*d^3*e^2+75*a^2*b^4*d^4*e+6*a*b^5*d^5)*x^6
+1/5*(5*a^6*d*e^4+60*a^5*b*d^2*e^3+150*a^4*b^2*d^3*e^2+100*a^3*b^3*d^4*e+15*a^2*
b^4*d^5)*x^5+1/4*(10*a^6*d^2*e^3+60*a^5*b*d^3*e^2+75*a^4*b^2*d^4*e+20*a^3*b^3*d^
5)*x^4+1/3*(10*a^6*d^3*e^2+30*a^5*b*d^4*e+15*a^4*b^2*d^5)*x^3+1/2*(5*a^6*d^4*e+6
*a^5*b*d^5)*x^2+d^5*a^6*x

_______________________________________________________________________________________

Maxima [A]  time = 0.690657, size = 698, normalized size = 4.88 \[ \frac{1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac{1}{11} \,{\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac{5}{8} \,{\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^5,x, algorithm="maxima")

[Out]

1/12*b^6*e^5*x^12 + a^6*d^5*x + 1/11*(5*b^6*d*e^4 + 6*a*b^5*e^5)*x^11 + 1/2*(2*b
^6*d^2*e^3 + 6*a*b^5*d*e^4 + 3*a^2*b^4*e^5)*x^10 + 5/9*(2*b^6*d^3*e^2 + 12*a*b^5
*d^2*e^3 + 15*a^2*b^4*d*e^4 + 4*a^3*b^3*e^5)*x^9 + 5/8*(b^6*d^4*e + 12*a*b^5*d^3
*e^2 + 30*a^2*b^4*d^2*e^3 + 20*a^3*b^3*d*e^4 + 3*a^4*b^2*e^5)*x^8 + 1/7*(b^6*d^5
+ 30*a*b^5*d^4*e + 150*a^2*b^4*d^3*e^2 + 200*a^3*b^3*d^2*e^3 + 75*a^4*b^2*d*e^4
+ 6*a^5*b*e^5)*x^7 + 1/6*(6*a*b^5*d^5 + 75*a^2*b^4*d^4*e + 200*a^3*b^3*d^3*e^2
+ 150*a^4*b^2*d^2*e^3 + 30*a^5*b*d*e^4 + a^6*e^5)*x^6 + (3*a^2*b^4*d^5 + 20*a^3*
b^3*d^4*e + 30*a^4*b^2*d^3*e^2 + 12*a^5*b*d^2*e^3 + a^6*d*e^4)*x^5 + 5/4*(4*a^3*
b^3*d^5 + 15*a^4*b^2*d^4*e + 12*a^5*b*d^3*e^2 + 2*a^6*d^2*e^3)*x^4 + 5/3*(3*a^4*
b^2*d^5 + 6*a^5*b*d^4*e + 2*a^6*d^3*e^2)*x^3 + 1/2*(6*a^5*b*d^5 + 5*a^6*d^4*e)*x
^2

_______________________________________________________________________________________

Fricas [A]  time = 0.181844, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} e^{5} b^{6} + \frac{5}{11} x^{11} e^{4} d b^{6} + \frac{6}{11} x^{11} e^{5} b^{5} a + x^{10} e^{3} d^{2} b^{6} + 3 x^{10} e^{4} d b^{5} a + \frac{3}{2} x^{10} e^{5} b^{4} a^{2} + \frac{10}{9} x^{9} e^{2} d^{3} b^{6} + \frac{20}{3} x^{9} e^{3} d^{2} b^{5} a + \frac{25}{3} x^{9} e^{4} d b^{4} a^{2} + \frac{20}{9} x^{9} e^{5} b^{3} a^{3} + \frac{5}{8} x^{8} e d^{4} b^{6} + \frac{15}{2} x^{8} e^{2} d^{3} b^{5} a + \frac{75}{4} x^{8} e^{3} d^{2} b^{4} a^{2} + \frac{25}{2} x^{8} e^{4} d b^{3} a^{3} + \frac{15}{8} x^{8} e^{5} b^{2} a^{4} + \frac{1}{7} x^{7} d^{5} b^{6} + \frac{30}{7} x^{7} e d^{4} b^{5} a + \frac{150}{7} x^{7} e^{2} d^{3} b^{4} a^{2} + \frac{200}{7} x^{7} e^{3} d^{2} b^{3} a^{3} + \frac{75}{7} x^{7} e^{4} d b^{2} a^{4} + \frac{6}{7} x^{7} e^{5} b a^{5} + x^{6} d^{5} b^{5} a + \frac{25}{2} x^{6} e d^{4} b^{4} a^{2} + \frac{100}{3} x^{6} e^{2} d^{3} b^{3} a^{3} + 25 x^{6} e^{3} d^{2} b^{2} a^{4} + 5 x^{6} e^{4} d b a^{5} + \frac{1}{6} x^{6} e^{5} a^{6} + 3 x^{5} d^{5} b^{4} a^{2} + 20 x^{5} e d^{4} b^{3} a^{3} + 30 x^{5} e^{2} d^{3} b^{2} a^{4} + 12 x^{5} e^{3} d^{2} b a^{5} + x^{5} e^{4} d a^{6} + 5 x^{4} d^{5} b^{3} a^{3} + \frac{75}{4} x^{4} e d^{4} b^{2} a^{4} + 15 x^{4} e^{2} d^{3} b a^{5} + \frac{5}{2} x^{4} e^{3} d^{2} a^{6} + 5 x^{3} d^{5} b^{2} a^{4} + 10 x^{3} e d^{4} b a^{5} + \frac{10}{3} x^{3} e^{2} d^{3} a^{6} + 3 x^{2} d^{5} b a^{5} + \frac{5}{2} x^{2} e d^{4} a^{6} + x d^{5} a^{6} \]

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)^5,x, algorithm="fricas")

[Out]

1/12*x^12*e^5*b^6 + 5/11*x^11*e^4*d*b^6 + 6/11*x^11*e^5*b^5*a + x^10*e^3*d^2*b^6
+ 3*x^10*e^4*d*b^5*a + 3/2*x^10*e^5*b^4*a^2 + 10/9*x^9*e^2*d^3*b^6 + 20/3*x^9*e
^3*d^2*b^5*a + 25/3*x^9*e^4*d*b^4*a^2 + 20/9*x^9*e^5*b^3*a^3 + 5/8*x^8*e*d^4*b^6
+ 15/2*x^8*e^2*d^3*b^5*a + 75/4*x^8*e^3*d^2*b^4*a^2 + 25/2*x^8*e^4*d*b^3*a^3 +
15/8*x^8*e^5*b^2*a^4 + 1/7*x^7*d^5*b^6 + 30/7*x^7*e*d^4*b^5*a + 150/7*x^7*e^2*d^
3*b^4*a^2 + 200/7*x^7*e^3*d^2*b^3*a^3 + 75/7*x^7*e^4*d*b^2*a^4 + 6/7*x^7*e^5*b*a
^5 + x^6*d^5*b^5*a + 25/2*x^6*e*d^4*b^4*a^2 + 100/3*x^6*e^2*d^3*b^3*a^3 + 25*x^6
*e^3*d^2*b^2*a^4 + 5*x^6*e^4*d*b*a^5 + 1/6*x^6*e^5*a^6 + 3*x^5*d^5*b^4*a^2 + 20*
x^5*e*d^4*b^3*a^3 + 30*x^5*e^2*d^3*b^2*a^4 + 12*x^5*e^3*d^2*b*a^5 + x^5*e^4*d*a^
6 + 5*x^4*d^5*b^3*a^3 + 75/4*x^4*e*d^4*b^2*a^4 + 15*x^4*e^2*d^3*b*a^5 + 5/2*x^4*
e^3*d^2*a^6 + 5*x^3*d^5*b^2*a^4 + 10*x^3*e*d^4*b*a^5 + 10/3*x^3*e^2*d^3*a^6 + 3*
x^2*d^5*b*a^5 + 5/2*x^2*e*d^4*a^6 + x*d^5*a^6

_______________________________________________________________________________________

Sympy [A]  time = 0.362983, size = 580, normalized size = 4.06 \[ a^{6} d^{5} x + \frac{b^{6} e^{5} x^{12}}{12} + x^{11} \left (\frac{6 a b^{5} e^{5}}{11} + \frac{5 b^{6} d e^{4}}{11}\right ) + x^{10} \left (\frac{3 a^{2} b^{4} e^{5}}{2} + 3 a b^{5} d e^{4} + b^{6} d^{2} e^{3}\right ) + x^{9} \left (\frac{20 a^{3} b^{3} e^{5}}{9} + \frac{25 a^{2} b^{4} d e^{4}}{3} + \frac{20 a b^{5} d^{2} e^{3}}{3} + \frac{10 b^{6} d^{3} e^{2}}{9}\right ) + x^{8} \left (\frac{15 a^{4} b^{2} e^{5}}{8} + \frac{25 a^{3} b^{3} d e^{4}}{2} + \frac{75 a^{2} b^{4} d^{2} e^{3}}{4} + \frac{15 a b^{5} d^{3} e^{2}}{2} + \frac{5 b^{6} d^{4} e}{8}\right ) + x^{7} \left (\frac{6 a^{5} b e^{5}}{7} + \frac{75 a^{4} b^{2} d e^{4}}{7} + \frac{200 a^{3} b^{3} d^{2} e^{3}}{7} + \frac{150 a^{2} b^{4} d^{3} e^{2}}{7} + \frac{30 a b^{5} d^{4} e}{7} + \frac{b^{6} d^{5}}{7}\right ) + x^{6} \left (\frac{a^{6} e^{5}}{6} + 5 a^{5} b d e^{4} + 25 a^{4} b^{2} d^{2} e^{3} + \frac{100 a^{3} b^{3} d^{3} e^{2}}{3} + \frac{25 a^{2} b^{4} d^{4} e}{2} + a b^{5} d^{5}\right ) + x^{5} \left (a^{6} d e^{4} + 12 a^{5} b d^{2} e^{3} + 30 a^{4} b^{2} d^{3} e^{2} + 20 a^{3} b^{3} d^{4} e + 3 a^{2} b^{4} d^{5}\right ) + x^{4} \left (\frac{5 a^{6} d^{2} e^{3}}{2} + 15 a^{5} b d^{3} e^{2} + \frac{75 a^{4} b^{2} d^{4} e}{4} + 5 a^{3} b^{3} d^{5}\right ) + x^{3} \left (\frac{10 a^{6} d^{3} e^{2}}{3} + 10 a^{5} b d^{4} e + 5 a^{4} b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{6} d^{4} e}{2} + 3 a^{5} b d^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**6*d**5*x + b**6*e**5*x**12/12 + x**11*(6*a*b**5*e**5/11 + 5*b**6*d*e**4/11) +
x**10*(3*a**2*b**4*e**5/2 + 3*a*b**5*d*e**4 + b**6*d**2*e**3) + x**9*(20*a**3*b
**3*e**5/9 + 25*a**2*b**4*d*e**4/3 + 20*a*b**5*d**2*e**3/3 + 10*b**6*d**3*e**2/9
) + x**8*(15*a**4*b**2*e**5/8 + 25*a**3*b**3*d*e**4/2 + 75*a**2*b**4*d**2*e**3/4
+ 15*a*b**5*d**3*e**2/2 + 5*b**6*d**4*e/8) + x**7*(6*a**5*b*e**5/7 + 75*a**4*b*
*2*d*e**4/7 + 200*a**3*b**3*d**2*e**3/7 + 150*a**2*b**4*d**3*e**2/7 + 30*a*b**5*
d**4*e/7 + b**6*d**5/7) + x**6*(a**6*e**5/6 + 5*a**5*b*d*e**4 + 25*a**4*b**2*d**
2*e**3 + 100*a**3*b**3*d**3*e**2/3 + 25*a**2*b**4*d**4*e/2 + a*b**5*d**5) + x**5
*(a**6*d*e**4 + 12*a**5*b*d**2*e**3 + 30*a**4*b**2*d**3*e**2 + 20*a**3*b**3*d**4
*e + 3*a**2*b**4*d**5) + x**4*(5*a**6*d**2*e**3/2 + 15*a**5*b*d**3*e**2 + 75*a**
4*b**2*d**4*e/4 + 5*a**3*b**3*d**5) + x**3*(10*a**6*d**3*e**2/3 + 10*a**5*b*d**4
*e + 5*a**4*b**2*d**5) + x**2*(5*a**6*d**4*e/2 + 3*a**5*b*d**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207418, size = 753, normalized size = 5.27 \[ \frac{1}{12} \, b^{6} x^{12} e^{5} + \frac{5}{11} \, b^{6} d x^{11} e^{4} + b^{6} d^{2} x^{10} e^{3} + \frac{10}{9} \, b^{6} d^{3} x^{9} e^{2} + \frac{5}{8} \, b^{6} d^{4} x^{8} e + \frac{1}{7} \, b^{6} d^{5} x^{7} + \frac{6}{11} \, a b^{5} x^{11} e^{5} + 3 \, a b^{5} d x^{10} e^{4} + \frac{20}{3} \, a b^{5} d^{2} x^{9} e^{3} + \frac{15}{2} \, a b^{5} d^{3} x^{8} e^{2} + \frac{30}{7} \, a b^{5} d^{4} x^{7} e + a b^{5} d^{5} x^{6} + \frac{3}{2} \, a^{2} b^{4} x^{10} e^{5} + \frac{25}{3} \, a^{2} b^{4} d x^{9} e^{4} + \frac{75}{4} \, a^{2} b^{4} d^{2} x^{8} e^{3} + \frac{150}{7} \, a^{2} b^{4} d^{3} x^{7} e^{2} + \frac{25}{2} \, a^{2} b^{4} d^{4} x^{6} e + 3 \, a^{2} b^{4} d^{5} x^{5} + \frac{20}{9} \, a^{3} b^{3} x^{9} e^{5} + \frac{25}{2} \, a^{3} b^{3} d x^{8} e^{4} + \frac{200}{7} \, a^{3} b^{3} d^{2} x^{7} e^{3} + \frac{100}{3} \, a^{3} b^{3} d^{3} x^{6} e^{2} + 20 \, a^{3} b^{3} d^{4} x^{5} e + 5 \, a^{3} b^{3} d^{5} x^{4} + \frac{15}{8} \, a^{4} b^{2} x^{8} e^{5} + \frac{75}{7} \, a^{4} b^{2} d x^{7} e^{4} + 25 \, a^{4} b^{2} d^{2} x^{6} e^{3} + 30 \, a^{4} b^{2} d^{3} x^{5} e^{2} + \frac{75}{4} \, a^{4} b^{2} d^{4} x^{4} e + 5 \, a^{4} b^{2} d^{5} x^{3} + \frac{6}{7} \, a^{5} b x^{7} e^{5} + 5 \, a^{5} b d x^{6} e^{4} + 12 \, a^{5} b d^{2} x^{5} e^{3} + 15 \, a^{5} b d^{3} x^{4} e^{2} + 10 \, a^{5} b d^{4} x^{3} e + 3 \, a^{5} b d^{5} x^{2} + \frac{1}{6} \, a^{6} x^{6} e^{5} + a^{6} d x^{5} e^{4} + \frac{5}{2} \, a^{6} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{6} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{6} d^{4} x^{2} e + a^{6} d^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^5,x, algorithm="giac")

[Out]

1/12*b^6*x^12*e^5 + 5/11*b^6*d*x^11*e^4 + b^6*d^2*x^10*e^3 + 10/9*b^6*d^3*x^9*e^
2 + 5/8*b^6*d^4*x^8*e + 1/7*b^6*d^5*x^7 + 6/11*a*b^5*x^11*e^5 + 3*a*b^5*d*x^10*e
^4 + 20/3*a*b^5*d^2*x^9*e^3 + 15/2*a*b^5*d^3*x^8*e^2 + 30/7*a*b^5*d^4*x^7*e + a*
b^5*d^5*x^6 + 3/2*a^2*b^4*x^10*e^5 + 25/3*a^2*b^4*d*x^9*e^4 + 75/4*a^2*b^4*d^2*x
^8*e^3 + 150/7*a^2*b^4*d^3*x^7*e^2 + 25/2*a^2*b^4*d^4*x^6*e + 3*a^2*b^4*d^5*x^5
+ 20/9*a^3*b^3*x^9*e^5 + 25/2*a^3*b^3*d*x^8*e^4 + 200/7*a^3*b^3*d^2*x^7*e^3 + 10
0/3*a^3*b^3*d^3*x^6*e^2 + 20*a^3*b^3*d^4*x^5*e + 5*a^3*b^3*d^5*x^4 + 15/8*a^4*b^
2*x^8*e^5 + 75/7*a^4*b^2*d*x^7*e^4 + 25*a^4*b^2*d^2*x^6*e^3 + 30*a^4*b^2*d^3*x^5
*e^2 + 75/4*a^4*b^2*d^4*x^4*e + 5*a^4*b^2*d^5*x^3 + 6/7*a^5*b*x^7*e^5 + 5*a^5*b*
d*x^6*e^4 + 12*a^5*b*d^2*x^5*e^3 + 15*a^5*b*d^3*x^4*e^2 + 10*a^5*b*d^4*x^3*e + 3
*a^5*b*d^5*x^2 + 1/6*a^6*x^6*e^5 + a^6*d*x^5*e^4 + 5/2*a^6*d^2*x^4*e^3 + 10/3*a^
6*d^3*x^3*e^2 + 5/2*a^6*d^4*x^2*e + a^6*d^5*x