3.1478 \(\int \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=14 \[ \frac{(a+b x)^7}{7 b} \]

[Out]

(a + b*x)^7/(7*b)

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Rubi [A]  time = 0.010427, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^7/(7*b)

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Rubi in Sympy [A]  time = 1.90319, size = 29, normalized size = 2.07 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{14 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**3/(14*b)

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Mathematica [A]  time = 0.00265618, size = 14, normalized size = 1. \[ \frac{(a+b x)^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^7/(7*b)

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Maple [B]  time = 0.003, size = 65, normalized size = 4.6 \[{\frac{{b}^{6}{x}^{7}}{7}}+{x}^{6}a{b}^{5}+3\,{x}^{5}{a}^{2}{b}^{4}+5\,{x}^{4}{a}^{3}{b}^{3}+5\,{x}^{3}{a}^{4}{b}^{2}+3\,{x}^{2}{a}^{5}b+{a}^{6}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/7*b^6*x^7+x^6*a*b^5+3*x^5*a^2*b^4+5*x^4*a^3*b^3+5*x^3*a^4*b^2+3*x^2*a^5*b+a^6*
x

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Maxima [A]  time = 0.686692, size = 131, normalized size = 9.36 \[ \frac{1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + \frac{12}{5} \, a^{2} b^{4} x^{5} + 2 \, a^{3} b^{3} x^{4} + a^{6} x +{\left (b^{2} x^{3} + 3 \, a b x^{2}\right )} a^{4} + \frac{1}{5} \,{\left (3 \, b^{4} x^{5} + 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 12/5*a^2*b^4*x^5 + 2*a^3*b^3*x^4 + a^6*x + (b^2*x^3 +
3*a*b*x^2)*a^4 + 1/5*(3*b^4*x^5 + 15*a*b^3*x^4 + 20*a^2*b^2*x^3)*a^2

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Fricas [A]  time = 0.178272, size = 1, normalized size = 0.07 \[ \frac{1}{7} x^{7} b^{6} + x^{6} b^{5} a + 3 x^{5} b^{4} a^{2} + 5 x^{4} b^{3} a^{3} + 5 x^{3} b^{2} a^{4} + 3 x^{2} b a^{5} + x a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*b^6 + x^6*b^5*a + 3*x^5*b^4*a^2 + 5*x^4*b^3*a^3 + 5*x^3*b^2*a^4 + 3*x^2*
b*a^5 + x*a^6

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Sympy [A]  time = 0.126362, size = 66, normalized size = 4.71 \[ a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac{b^{6} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*a**3*b**3*x**4 + 3*a**2*b**4*x**5
+ a*b**5*x**6 + b**6*x**7/7

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GIAC/XCAS [A]  time = 0.210594, size = 86, normalized size = 6.14 \[ \frac{1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 3*a^2*b^4*x^5 + 5*a^3*b^3*x^4 + 5*a^4*b^2*x^3 + 3*a^5*
b*x^2 + a^6*x