3.1900 \(\int (a+b x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=14 \[ \frac{(a+b x)^4}{4 b} \]

[Out]

(a + b*x)^4/(4*b)

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Rubi [A]  time = 0.00825652, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^4/(4*b)

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Rubi in Sympy [A]  time = 8.59923, size = 8, normalized size = 0.57 \[ \frac{\left (a + b x\right )^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**4/(4*b)

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Mathematica [A]  time = 0.00268882, size = 14, normalized size = 1. \[ \frac{(a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^4/(4*b)

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Maple [B]  time = 0.001, size = 32, normalized size = 2.3 \[{\frac{{b}^{3}{x}^{4}}{4}}+a{b}^{2}{x}^{3}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{a}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^3*x^4+a*b^2*x^3+3/2*a^2*b*x^2+a^3*x

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Maxima [A]  time = 0.707864, size = 31, normalized size = 2.21 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^2/b

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Fricas [A]  time = 0.256656, size = 1, normalized size = 0.07 \[ \frac{1}{4} x^{4} b^{3} + x^{3} b^{2} a + \frac{3}{2} x^{2} b a^{2} + x a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4*b^3 + x^3*b^2*a + 3/2*x^2*b*a^2 + x*a^3

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Sympy [A]  time = 0.093713, size = 32, normalized size = 2.29 \[ a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4

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GIAC/XCAS [A]  time = 0.283547, size = 42, normalized size = 3. \[ \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^3*x^4 + a*b^2*x^3 + 3/2*a^2*b*x^2 + a^3*x