3.1920 \(\int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=65 \[ \frac{2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac{(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac{e^2 (a+b x)^{10}}{10 b^3} \]

[Out]

((b*d - a*e)^2*(a + b*x)^8)/(8*b^3) + (2*e*(b*d - a*e)*(a + b*x)^9)/(9*b^3) + (e
^2*(a + b*x)^10)/(10*b^3)

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Rubi [A]  time = 0.311789, antiderivative size = 65, 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 \[ \frac{2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac{(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac{e^2 (a+b x)^{10}}{10 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)^2*(a + b*x)^8)/(8*b^3) + (2*e*(b*d - a*e)*(a + b*x)^9)/(9*b^3) + (e
^2*(a + b*x)^10)/(10*b^3)

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Rubi in Sympy [A]  time = 47.6874, size = 56, normalized size = 0.86 \[ \frac{e^{2} \left (a + b x\right )^{10}}{10 b^{3}} - \frac{2 e \left (a + b x\right )^{9} \left (a e - b d\right )}{9 b^{3}} + \frac{\left (a + b x\right )^{8} \left (a e - b d\right )^{2}}{8 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*(a + b*x)**10/(10*b**3) - 2*e*(a + b*x)**9*(a*e - b*d)/(9*b**3) + (a + b*x)
**8*(a*e - b*d)**2/(8*b**3)

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Mathematica [B]  time = 0.16203, size = 229, normalized size = 3.52 \[ \frac{1}{360} x \left (120 a^7 \left (3 d^2+3 d e x+e^2 x^2\right )+210 a^6 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+252 a^5 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+210 a^4 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+120 a^3 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+45 a^2 b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+10 a b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )+b^7 x^7 \left (45 d^2+80 d e x+36 e^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(120*a^7*(3*d^2 + 3*d*e*x + e^2*x^2) + 210*a^6*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x
^2) + 252*a^5*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 210*a^4*b^3*x^3*(15*d^2
+ 24*d*e*x + 10*e^2*x^2) + 120*a^3*b^4*x^4*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 45
*a^2*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 10*a*b^6*x^6*(36*d^2 + 63*d*e*x
+ 28*e^2*x^2) + b^7*x^7*(45*d^2 + 80*d*e*x + 36*e^2*x^2)))/360

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Maple [B]  time = 0.001, size = 433, normalized size = 6.7 \[{\frac{{b}^{7}{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( a{e}^{2}+2\,bde \right ){b}^{6}+6\,{b}^{6}{e}^{2}a \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,aed+b{d}^{2} \right ){b}^{6}+6\, \left ( a{e}^{2}+2\,bde \right ) a{b}^{5}+15\,{b}^{5}{e}^{2}{a}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( a{d}^{2}{b}^{6}+6\, \left ( 2\,aed+b{d}^{2} \right ) a{b}^{5}+15\, \left ( a{e}^{2}+2\,bde \right ){a}^{2}{b}^{4}+20\,{b}^{4}{e}^{2}{a}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{a}^{2}{d}^{2}{b}^{5}+15\, \left ( 2\,aed+b{d}^{2} \right ){a}^{2}{b}^{4}+20\, \left ( a{e}^{2}+2\,bde \right ){a}^{3}{b}^{3}+15\,{b}^{3}{e}^{2}{a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{a}^{3}{d}^{2}{b}^{4}+20\, \left ( 2\,aed+b{d}^{2} \right ){a}^{3}{b}^{3}+15\, \left ( a{e}^{2}+2\,bde \right ){b}^{2}{a}^{4}+6\,{b}^{2}{e}^{2}{a}^{5} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{a}^{4}{d}^{2}{b}^{3}+15\, \left ( 2\,aed+b{d}^{2} \right ){b}^{2}{a}^{4}+6\, \left ( a{e}^{2}+2\,bde \right ){a}^{5}b+b{e}^{2}{a}^{6} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{a}^{5}{d}^{2}{b}^{2}+6\, \left ( 2\,aed+b{d}^{2} \right ){a}^{5}b+ \left ( a{e}^{2}+2\,bde \right ){a}^{6} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{a}^{6}{d}^{2}b+ \left ( 2\,aed+b{d}^{2} \right ){a}^{6} \right ){x}^{2}}{2}}+{a}^{7}{d}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.713488, size = 369, normalized size = 5.68 \[ \frac{1}{10} \, b^{7} e^{2} x^{10} + a^{7} d^{2} x + \frac{1}{9} \,{\left (2 \, b^{7} d e + 7 \, a b^{6} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{2} + 14 \, a b^{6} d e + 21 \, a^{2} b^{5} e^{2}\right )} x^{8} +{\left (a b^{6} d^{2} + 6 \, a^{2} b^{5} d e + 5 \, a^{3} b^{4} e^{2}\right )} x^{7} + \frac{7}{6} \,{\left (3 \, a^{2} b^{5} d^{2} + 10 \, a^{3} b^{4} d e + 5 \, a^{4} b^{3} e^{2}\right )} x^{6} + \frac{7}{5} \,{\left (5 \, a^{3} b^{4} d^{2} + 10 \, a^{4} b^{3} d e + 3 \, a^{5} b^{2} e^{2}\right )} x^{5} + \frac{7}{4} \,{\left (5 \, a^{4} b^{3} d^{2} + 6 \, a^{5} b^{2} d e + a^{6} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (21 \, a^{5} b^{2} d^{2} + 14 \, a^{6} b d e + a^{7} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{2} + 2 \, a^{7} d e\right )} x^{2} \]

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)^2,x, algorithm="maxima")

[Out]

1/10*b^7*e^2*x^10 + a^7*d^2*x + 1/9*(2*b^7*d*e + 7*a*b^6*e^2)*x^9 + 1/8*(b^7*d^2
+ 14*a*b^6*d*e + 21*a^2*b^5*e^2)*x^8 + (a*b^6*d^2 + 6*a^2*b^5*d*e + 5*a^3*b^4*e
^2)*x^7 + 7/6*(3*a^2*b^5*d^2 + 10*a^3*b^4*d*e + 5*a^4*b^3*e^2)*x^6 + 7/5*(5*a^3*
b^4*d^2 + 10*a^4*b^3*d*e + 3*a^5*b^2*e^2)*x^5 + 7/4*(5*a^4*b^3*d^2 + 6*a^5*b^2*d
*e + a^6*b*e^2)*x^4 + 1/3*(21*a^5*b^2*d^2 + 14*a^6*b*d*e + a^7*e^2)*x^3 + 1/2*(7
*a^6*b*d^2 + 2*a^7*d*e)*x^2

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Fricas [A]  time = 0.256896, size = 1, normalized size = 0.02 \[ \frac{1}{10} x^{10} e^{2} b^{7} + \frac{2}{9} x^{9} e d b^{7} + \frac{7}{9} x^{9} e^{2} b^{6} a + \frac{1}{8} x^{8} d^{2} b^{7} + \frac{7}{4} x^{8} e d b^{6} a + \frac{21}{8} x^{8} e^{2} b^{5} a^{2} + x^{7} d^{2} b^{6} a + 6 x^{7} e d b^{5} a^{2} + 5 x^{7} e^{2} b^{4} a^{3} + \frac{7}{2} x^{6} d^{2} b^{5} a^{2} + \frac{35}{3} x^{6} e d b^{4} a^{3} + \frac{35}{6} x^{6} e^{2} b^{3} a^{4} + 7 x^{5} d^{2} b^{4} a^{3} + 14 x^{5} e d b^{3} a^{4} + \frac{21}{5} x^{5} e^{2} b^{2} a^{5} + \frac{35}{4} x^{4} d^{2} b^{3} a^{4} + \frac{21}{2} x^{4} e d b^{2} a^{5} + \frac{7}{4} x^{4} e^{2} b a^{6} + 7 x^{3} d^{2} b^{2} a^{5} + \frac{14}{3} x^{3} e d b a^{6} + \frac{1}{3} x^{3} e^{2} a^{7} + \frac{7}{2} x^{2} d^{2} b a^{6} + x^{2} e d a^{7} + x d^{2} a^{7} \]

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)^2,x, algorithm="fricas")

[Out]

1/10*x^10*e^2*b^7 + 2/9*x^9*e*d*b^7 + 7/9*x^9*e^2*b^6*a + 1/8*x^8*d^2*b^7 + 7/4*
x^8*e*d*b^6*a + 21/8*x^8*e^2*b^5*a^2 + x^7*d^2*b^6*a + 6*x^7*e*d*b^5*a^2 + 5*x^7
*e^2*b^4*a^3 + 7/2*x^6*d^2*b^5*a^2 + 35/3*x^6*e*d*b^4*a^3 + 35/6*x^6*e^2*b^3*a^4
+ 7*x^5*d^2*b^4*a^3 + 14*x^5*e*d*b^3*a^4 + 21/5*x^5*e^2*b^2*a^5 + 35/4*x^4*d^2*
b^3*a^4 + 21/2*x^4*e*d*b^2*a^5 + 7/4*x^4*e^2*b*a^6 + 7*x^3*d^2*b^2*a^5 + 14/3*x^
3*e*d*b*a^6 + 1/3*x^3*e^2*a^7 + 7/2*x^2*d^2*b*a^6 + x^2*e*d*a^7 + x*d^2*a^7

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Sympy [A]  time = 0.275768, size = 303, normalized size = 4.66 \[ a^{7} d^{2} x + \frac{b^{7} e^{2} x^{10}}{10} + x^{9} \left (\frac{7 a b^{6} e^{2}}{9} + \frac{2 b^{7} d e}{9}\right ) + x^{8} \left (\frac{21 a^{2} b^{5} e^{2}}{8} + \frac{7 a b^{6} d e}{4} + \frac{b^{7} d^{2}}{8}\right ) + x^{7} \left (5 a^{3} b^{4} e^{2} + 6 a^{2} b^{5} d e + a b^{6} d^{2}\right ) + x^{6} \left (\frac{35 a^{4} b^{3} e^{2}}{6} + \frac{35 a^{3} b^{4} d e}{3} + \frac{7 a^{2} b^{5} d^{2}}{2}\right ) + x^{5} \left (\frac{21 a^{5} b^{2} e^{2}}{5} + 14 a^{4} b^{3} d e + 7 a^{3} b^{4} d^{2}\right ) + x^{4} \left (\frac{7 a^{6} b e^{2}}{4} + \frac{21 a^{5} b^{2} d e}{2} + \frac{35 a^{4} b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{7} e^{2}}{3} + \frac{14 a^{6} b d e}{3} + 7 a^{5} b^{2} d^{2}\right ) + x^{2} \left (a^{7} d e + \frac{7 a^{6} b d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**7*d**2*x + b**7*e**2*x**10/10 + x**9*(7*a*b**6*e**2/9 + 2*b**7*d*e/9) + x**8*
(21*a**2*b**5*e**2/8 + 7*a*b**6*d*e/4 + b**7*d**2/8) + x**7*(5*a**3*b**4*e**2 +
6*a**2*b**5*d*e + a*b**6*d**2) + x**6*(35*a**4*b**3*e**2/6 + 35*a**3*b**4*d*e/3
+ 7*a**2*b**5*d**2/2) + x**5*(21*a**5*b**2*e**2/5 + 14*a**4*b**3*d*e + 7*a**3*b*
*4*d**2) + x**4*(7*a**6*b*e**2/4 + 21*a**5*b**2*d*e/2 + 35*a**4*b**3*d**2/4) + x
**3*(a**7*e**2/3 + 14*a**6*b*d*e/3 + 7*a**5*b**2*d**2) + x**2*(a**7*d*e + 7*a**6
*b*d**2/2)

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GIAC/XCAS [A]  time = 0.28228, size = 397, normalized size = 6.11 \[ \frac{1}{10} \, b^{7} x^{10} e^{2} + \frac{2}{9} \, b^{7} d x^{9} e + \frac{1}{8} \, b^{7} d^{2} x^{8} + \frac{7}{9} \, a b^{6} x^{9} e^{2} + \frac{7}{4} \, a b^{6} d x^{8} e + a b^{6} d^{2} x^{7} + \frac{21}{8} \, a^{2} b^{5} x^{8} e^{2} + 6 \, a^{2} b^{5} d x^{7} e + \frac{7}{2} \, a^{2} b^{5} d^{2} x^{6} + 5 \, a^{3} b^{4} x^{7} e^{2} + \frac{35}{3} \, a^{3} b^{4} d x^{6} e + 7 \, a^{3} b^{4} d^{2} x^{5} + \frac{35}{6} \, a^{4} b^{3} x^{6} e^{2} + 14 \, a^{4} b^{3} d x^{5} e + \frac{35}{4} \, a^{4} b^{3} d^{2} x^{4} + \frac{21}{5} \, a^{5} b^{2} x^{5} e^{2} + \frac{21}{2} \, a^{5} b^{2} d x^{4} e + 7 \, a^{5} b^{2} d^{2} x^{3} + \frac{7}{4} \, a^{6} b x^{4} e^{2} + \frac{14}{3} \, a^{6} b d x^{3} e + \frac{7}{2} \, a^{6} b d^{2} x^{2} + \frac{1}{3} \, a^{7} x^{3} e^{2} + a^{7} d x^{2} e + a^{7} d^{2} x \]

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)^2,x, algorithm="giac")

[Out]

1/10*b^7*x^10*e^2 + 2/9*b^7*d*x^9*e + 1/8*b^7*d^2*x^8 + 7/9*a*b^6*x^9*e^2 + 7/4*
a*b^6*d*x^8*e + a*b^6*d^2*x^7 + 21/8*a^2*b^5*x^8*e^2 + 6*a^2*b^5*d*x^7*e + 7/2*a
^2*b^5*d^2*x^6 + 5*a^3*b^4*x^7*e^2 + 35/3*a^3*b^4*d*x^6*e + 7*a^3*b^4*d^2*x^5 +
35/6*a^4*b^3*x^6*e^2 + 14*a^4*b^3*d*x^5*e + 35/4*a^4*b^3*d^2*x^4 + 21/5*a^5*b^2*
x^5*e^2 + 21/2*a^5*b^2*d*x^4*e + 7*a^5*b^2*d^2*x^3 + 7/4*a^6*b*x^4*e^2 + 14/3*a^
6*b*d*x^3*e + 7/2*a^6*b*d^2*x^2 + 1/3*a^7*x^3*e^2 + a^7*d*x^2*e + a^7*d^2*x