3.1911 \(\int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=14 \[ \frac{(a+b x)^6}{6 b} \]

[Out]

(a + b*x)^6/(6*b)

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Rubi [A]  time = 0.00869202, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^6}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^6/(6*b)

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Rubi in Sympy [A]  time = 9.47501, size = 8, normalized size = 0.57 \[ \frac{\left (a + b x\right )^{6}}{6 b} \]

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)**2,x)

[Out]

(a + b*x)**6/(6*b)

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Mathematica [A]  time = 0.0034795, size = 14, normalized size = 1. \[ \frac{(a+b x)^6}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^6/(6*b)

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Maple [B]  time = 0.002, size = 54, normalized size = 3.9 \[{\frac{{b}^{5}{x}^{6}}{6}}+a{b}^{4}{x}^{5}+{\frac{5\,{a}^{2}{b}^{3}{x}^{4}}{2}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{3}}{3}}+{\frac{5\,{a}^{4}b{x}^{2}}{2}}+{a}^{5}x \]

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)^2,x)

[Out]

1/6*b^5*x^6+a*b^4*x^5+5/2*a^2*b^3*x^4+10/3*a^3*b^2*x^3+5/2*a^4*b*x^2+a^5*x

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Maxima [A]  time = 0.710033, size = 31, normalized size = 2.21 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^3/b

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Fricas [A]  time = 0.254799, size = 1, normalized size = 0.07 \[ \frac{1}{6} x^{6} b^{5} + x^{5} b^{4} a + \frac{5}{2} x^{4} b^{3} a^{2} + \frac{10}{3} x^{3} b^{2} a^{3} + \frac{5}{2} x^{2} b a^{4} + x a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*b^5 + x^5*b^4*a + 5/2*x^4*b^3*a^2 + 10/3*x^3*b^2*a^3 + 5/2*x^2*b*a^4 + x
*a^5

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Sympy [A]  time = 0.1257, size = 60, normalized size = 4.29 \[ a^{5} x + \frac{5 a^{4} b x^{2}}{2} + \frac{10 a^{3} b^{2} x^{3}}{3} + \frac{5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac{b^{5} x^{6}}{6} \]

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)**2,x)

[Out]

a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**
5 + b**5*x**6/6

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GIAC/XCAS [A]  time = 0.271114, size = 72, normalized size = 5.14 \[ \frac{1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac{5}{2} \, a^{2} b^{3} x^{4} + \frac{10}{3} \, a^{3} b^{2} x^{3} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^5*x^6 + a*b^4*x^5 + 5/2*a^2*b^3*x^4 + 10/3*a^3*b^2*x^3 + 5/2*a^4*b*x^2 + a
^5*x