3.1921 \(\int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^8 (b d-a e)}{8 b^2}+\frac{e (a+b x)^9}{9 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0495206, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(a+b x)^8 (b d-a e)}{8 b^2}+\frac{e (a+b x)^9}{9 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.7804, size = 31, normalized size = 0.82 \[ \frac{e \left (a + b x\right )^{9}}{9 b^{2}} - \frac{\left (a + b x\right )^{8} \left (a e - b d\right )}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0396897, size = 151, normalized size = 3.97 \[ a^7 d x+\frac{1}{2} a^6 x^2 (a e+7 b d)+\frac{7}{3} a^5 b x^3 (a e+3 b d)+\frac{7}{4} a^4 b^2 x^4 (3 a e+5 b d)+7 a^3 b^3 x^5 (a e+b d)+\frac{7}{6} a^2 b^4 x^6 (5 a e+3 b d)+\frac{1}{8} b^6 x^8 (7 a e+b d)+a b^5 x^7 (3 a e+b d)+\frac{1}{9} b^7 e x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 250, normalized size = 6.6 \[{\frac{{b}^{7}e{x}^{9}}{9}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{6}+6\,{b}^{6}ea \right ){x}^{8}}{8}}+{\frac{ \left ( ad{b}^{6}+6\, \left ( ae+bd \right ) a{b}^{5}+15\,{b}^{5}e{a}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{a}^{2}d{b}^{5}+15\, \left ( ae+bd \right ){a}^{2}{b}^{4}+20\,{b}^{4}e{a}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{a}^{3}d{b}^{4}+20\, \left ( ae+bd \right ){a}^{3}{b}^{3}+15\,{b}^{3}e{a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{a}^{4}d{b}^{3}+15\, \left ( ae+bd \right ){b}^{2}{a}^{4}+6\,{b}^{2}e{a}^{5} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{a}^{5}d{b}^{2}+6\, \left ( ae+bd \right ){a}^{5}b+be{a}^{6} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{a}^{6}db+ \left ( ae+bd \right ){a}^{6} \right ){x}^{2}}{2}}+{a}^{7}dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.710905, size = 220, normalized size = 5.79 \[ \frac{1}{9} \, b^{7} e x^{9} + a^{7} d x + \frac{1}{8} \,{\left (b^{7} d + 7 \, a b^{6} e\right )} x^{8} +{\left (a b^{6} d + 3 \, a^{2} b^{5} e\right )} x^{7} + \frac{7}{6} \,{\left (3 \, a^{2} b^{5} d + 5 \, a^{3} b^{4} e\right )} x^{6} + 7 \,{\left (a^{3} b^{4} d + a^{4} b^{3} e\right )} x^{5} + \frac{7}{4} \,{\left (5 \, a^{4} b^{3} d + 3 \, a^{5} b^{2} e\right )} x^{4} + \frac{7}{3} \,{\left (3 \, a^{5} b^{2} d + a^{6} b e\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d + a^{7} e\right )} x^{2} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254402, size = 1, normalized size = 0.03 \[ \frac{1}{9} x^{9} e b^{7} + \frac{1}{8} x^{8} d b^{7} + \frac{7}{8} x^{8} e b^{6} a + x^{7} d b^{6} a + 3 x^{7} e b^{5} a^{2} + \frac{7}{2} x^{6} d b^{5} a^{2} + \frac{35}{6} x^{6} e b^{4} a^{3} + 7 x^{5} d b^{4} a^{3} + 7 x^{5} e b^{3} a^{4} + \frac{35}{4} x^{4} d b^{3} a^{4} + \frac{21}{4} x^{4} e b^{2} a^{5} + 7 x^{3} d b^{2} a^{5} + \frac{7}{3} x^{3} e b a^{6} + \frac{7}{2} x^{2} d b a^{6} + \frac{1}{2} x^{2} e a^{7} + x d a^{7} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.204672, size = 178, normalized size = 4.68 \[ a^{7} d x + \frac{b^{7} e x^{9}}{9} + x^{8} \left (\frac{7 a b^{6} e}{8} + \frac{b^{7} d}{8}\right ) + x^{7} \left (3 a^{2} b^{5} e + a b^{6} d\right ) + x^{6} \left (\frac{35 a^{3} b^{4} e}{6} + \frac{7 a^{2} b^{5} d}{2}\right ) + x^{5} \left (7 a^{4} b^{3} e + 7 a^{3} b^{4} d\right ) + x^{4} \left (\frac{21 a^{5} b^{2} e}{4} + \frac{35 a^{4} b^{3} d}{4}\right ) + x^{3} \left (\frac{7 a^{6} b e}{3} + 7 a^{5} b^{2} d\right ) + x^{2} \left (\frac{a^{7} e}{2} + \frac{7 a^{6} b d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285502, size = 239, normalized size = 6.29 \[ \frac{1}{9} \, b^{7} x^{9} e + \frac{1}{8} \, b^{7} d x^{8} + \frac{7}{8} \, a b^{6} x^{8} e + a b^{6} d x^{7} + 3 \, a^{2} b^{5} x^{7} e + \frac{7}{2} \, a^{2} b^{5} d x^{6} + \frac{35}{6} \, a^{3} b^{4} x^{6} e + 7 \, a^{3} b^{4} d x^{5} + 7 \, a^{4} b^{3} x^{5} e + \frac{35}{4} \, a^{4} b^{3} d x^{4} + \frac{21}{4} \, a^{5} b^{2} x^{4} e + 7 \, a^{5} b^{2} d x^{3} + \frac{7}{3} \, a^{6} b x^{3} e + \frac{7}{2} \, a^{6} b d x^{2} + \frac{1}{2} \, a^{7} x^{2} e + a^{7} d x \]

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),x, algorithm="giac")

[Out]

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