3.206 \(\int (a+b x) \left (1+\left (a x+\frac{b x^2}{2}\right )^4\right ) \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{160} x^5 (2 a+b x)^5+a x+\frac{b x^2}{2} \]

[Out]

a*x + (b*x^2)/2 + (x^5*(2*a + b*x)^5)/160

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Rubi [A]  time = 0.0263356, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{1}{160} x^5 (2 a+b x)^5+a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(1 + (a*x + (b*x^2)/2)^4),x]

[Out]

a*x + (b*x^2)/2 + (x^5*(2*a + b*x)^5)/160

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Rubi in Sympy [A]  time = 2.4626, size = 22, normalized size = 0.79 \[ a x + \frac{b x^{2}}{2} + \frac{\left (a x + \frac{b x^{2}}{2}\right )^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(1+(a*x+1/2*b*x**2)**4),x)

[Out]

a*x + b*x**2/2 + (a*x + b*x**2/2)**5/5

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Mathematica [B]  time = 0.00773175, size = 80, normalized size = 2.86 \[ \frac{a^5 x^5}{5}+\frac{1}{2} a^4 b x^6+\frac{1}{2} a^3 b^2 x^7+\frac{1}{4} a^2 b^3 x^8+\frac{1}{16} a b^4 x^9+a x+\frac{b^5 x^{10}}{160}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(1 + (a*x + (b*x^2)/2)^4),x]

[Out]

a*x + (b*x^2)/2 + (a^5*x^5)/5 + (a^4*b*x^6)/2 + (a^3*b^2*x^7)/2 + (a^2*b^3*x^8)/
4 + (a*b^4*x^9)/16 + (b^5*x^10)/160

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Maple [B]  time = 0.001, size = 67, normalized size = 2.4 \[{\frac{{b}^{5}{x}^{10}}{160}}+{\frac{a{b}^{4}{x}^{9}}{16}}+{\frac{{a}^{2}{b}^{3}{x}^{8}}{4}}+{\frac{{a}^{3}{b}^{2}{x}^{7}}{2}}+{\frac{{a}^{4}b{x}^{6}}{2}}+{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{b{x}^{2}}{2}}+ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(1+(a*x+1/2*b*x^2)^4),x)

[Out]

1/160*b^5*x^10+1/16*a*b^4*x^9+1/4*a^2*b^3*x^8+1/2*a^3*b^2*x^7+1/2*a^4*b*x^6+1/5*
a^5*x^5+1/2*b*x^2+a*x

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Maxima [A]  time = 0.815985, size = 89, normalized size = 3.18 \[ \frac{1}{160} \, b^{5} x^{10} + \frac{1}{16} \, a b^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} x^{8} + \frac{1}{2} \, a^{3} b^{2} x^{7} + \frac{1}{2} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/16*((b*x^2 + 2*a*x)^4 + 16)*(b*x + a),x, algorithm="maxima")

[Out]

1/160*b^5*x^10 + 1/16*a*b^4*x^9 + 1/4*a^2*b^3*x^8 + 1/2*a^3*b^2*x^7 + 1/2*a^4*b*
x^6 + 1/5*a^5*x^5 + 1/2*b*x^2 + a*x

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Fricas [A]  time = 0.256412, size = 89, normalized size = 3.18 \[ \frac{1}{160} \, b^{5} x^{10} + \frac{1}{16} \, a b^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} x^{8} + \frac{1}{2} \, a^{3} b^{2} x^{7} + \frac{1}{2} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/16*((b*x^2 + 2*a*x)^4 + 16)*(b*x + a),x, algorithm="fricas")

[Out]

1/160*b^5*x^10 + 1/16*a*b^4*x^9 + 1/4*a^2*b^3*x^8 + 1/2*a^3*b^2*x^7 + 1/2*a^4*b*
x^6 + 1/5*a^5*x^5 + 1/2*b*x^2 + a*x

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Sympy [A]  time = 0.160112, size = 70, normalized size = 2.5 \[ \frac{a^{5} x^{5}}{5} + \frac{a^{4} b x^{6}}{2} + \frac{a^{3} b^{2} x^{7}}{2} + \frac{a^{2} b^{3} x^{8}}{4} + \frac{a b^{4} x^{9}}{16} + a x + \frac{b^{5} x^{10}}{160} + \frac{b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(1+(a*x+1/2*b*x**2)**4),x)

[Out]

a**5*x**5/5 + a**4*b*x**6/2 + a**3*b**2*x**7/2 + a**2*b**3*x**8/4 + a*b**4*x**9/
16 + a*x + b**5*x**10/160 + b*x**2/2

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GIAC/XCAS [A]  time = 0.258726, size = 89, normalized size = 3.18 \[ \frac{1}{160} \, b^{5} x^{10} + \frac{1}{16} \, a b^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} x^{8} + \frac{1}{2} \, a^{3} b^{2} x^{7} + \frac{1}{2} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/16*((b*x^2 + 2*a*x)^4 + 16)*(b*x + a),x, algorithm="giac")

[Out]

1/160*b^5*x^10 + 1/16*a*b^4*x^9 + 1/4*a^2*b^3*x^8 + 1/2*a^3*b^2*x^7 + 1/2*a^4*b*
x^6 + 1/5*a^5*x^5 + 1/2*b*x^2 + a*x