3.1477 \(\int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^7 (b d-a e)}{7 b^2}+\frac{e (a+b x)^8}{8 b^2} \]

[Out]

((b*d - a*e)*(a + b*x)^7)/(7*b^2) + (e*(a + b*x)^8)/(8*b^2)

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Rubi [A]  time = 0.0586906, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^7 (b d-a e)}{7 b^2}+\frac{e (a+b x)^8}{8 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)*(a + b*x)^7)/(7*b^2) + (e*(a + b*x)^8)/(8*b^2)

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Rubi in Sympy [A]  time = 28.0317, size = 31, normalized size = 0.82 \[ \frac{e \left (a + b x\right )^{8}}{8 b^{2}} - \frac{\left (a + b x\right )^{7} \left (a e - b d\right )}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(a + b*x)**8/(8*b**2) - (a + b*x)**7*(a*e - b*d)/(7*b**2)

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Mathematica [B]  time = 0.0662624, size = 122, normalized size = 3.21 \[ \frac{1}{56} x \left (28 a^6 (2 d+e x)+56 a^5 b x (3 d+2 e x)+70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(28*a^6*(2*d + e*x) + 56*a^5*b*x*(3*d + 2*e*x) + 70*a^4*b^2*x^2*(4*d + 3*e*x)
+ 56*a^3*b^3*x^3*(5*d + 4*e*x) + 28*a^2*b^4*x^4*(6*d + 5*e*x) + 8*a*b^5*x^5*(7*
d + 6*e*x) + b^6*x^6*(8*d + 7*e*x)))/56

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Maple [B]  time = 0.002, size = 145, normalized size = 3.8 \[{\frac{e{b}^{6}{x}^{8}}{8}}+{\frac{ \left ( 6\,ea{b}^{5}+d{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 15\,e{a}^{2}{b}^{4}+6\,da{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 20\,e{a}^{3}{b}^{3}+15\,d{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 15\,e{b}^{2}{a}^{4}+20\,d{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,e{a}^{5}b+15\,d{b}^{2}{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{6}+6\,d{a}^{5}b \right ){x}^{2}}{2}}+d{a}^{6}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.678862, size = 192, normalized size = 5.05 \[ \frac{1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac{1}{7} \,{\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} +{\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} +{\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d + a^{6} e\right )} x^{2} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.17862, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} e b^{6} + \frac{1}{7} x^{7} d b^{6} + \frac{6}{7} x^{7} e b^{5} a + x^{6} d b^{5} a + \frac{5}{2} x^{6} e b^{4} a^{2} + 3 x^{5} d b^{4} a^{2} + 4 x^{5} e b^{3} a^{3} + 5 x^{4} d b^{3} a^{3} + \frac{15}{4} x^{4} e b^{2} a^{4} + 5 x^{3} d b^{2} a^{4} + 2 x^{3} e b a^{5} + 3 x^{2} d b a^{5} + \frac{1}{2} x^{2} e a^{6} + x d a^{6} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.18937, size = 148, normalized size = 3.89 \[ a^{6} d x + \frac{b^{6} e x^{8}}{8} + x^{7} \left (\frac{6 a b^{5} e}{7} + \frac{b^{6} d}{7}\right ) + x^{6} \left (\frac{5 a^{2} b^{4} e}{2} + a b^{5} d\right ) + x^{5} \left (4 a^{3} b^{3} e + 3 a^{2} b^{4} d\right ) + x^{4} \left (\frac{15 a^{4} b^{2} e}{4} + 5 a^{3} b^{3} d\right ) + x^{3} \left (2 a^{5} b e + 5 a^{4} b^{2} d\right ) + x^{2} \left (\frac{a^{6} e}{2} + 3 a^{5} b d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209452, size = 205, normalized size = 5.39 \[ \frac{1}{8} \, b^{6} x^{8} e + \frac{1}{7} \, b^{6} d x^{7} + \frac{6}{7} \, a b^{5} x^{7} e + a b^{5} d x^{6} + \frac{5}{2} \, a^{2} b^{4} x^{6} e + 3 \, a^{2} b^{4} d x^{5} + 4 \, a^{3} b^{3} x^{5} e + 5 \, a^{3} b^{3} d x^{4} + \frac{15}{4} \, a^{4} b^{2} x^{4} e + 5 \, a^{4} b^{2} d x^{3} + 2 \, a^{5} b x^{3} e + 3 \, a^{5} b d x^{2} + \frac{1}{2} \, a^{6} x^{2} e + a^{6} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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