3.67 \(\int x (a+b x)^3 \, dx\)

Optimal. Leaf size=30 \[ \frac{(a+b x)^5}{5 b^2}-\frac{a (a+b x)^4}{4 b^2} \]

[Out]

-(a*(a + b*x)^4)/(4*b^2) + (a + b*x)^5/(5*b^2)

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Rubi [A]  time = 0.026278, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^5}{5 b^2}-\frac{a (a+b x)^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^3,x]

[Out]

-(a*(a + b*x)^4)/(4*b^2) + (a + b*x)^5/(5*b^2)

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Rubi in Sympy [A]  time = 5.84566, size = 24, normalized size = 0.8 \[ - \frac{a \left (a + b x\right )^{4}}{4 b^{2}} + \frac{\left (a + b x\right )^{5}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**3,x)

[Out]

-a*(a + b*x)**4/(4*b**2) + (a + b*x)**5/(5*b**2)

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Mathematica [A]  time = 0.00223828, size = 40, normalized size = 1.33 \[ \frac{a^3 x^2}{2}+a^2 b x^3+\frac{3}{4} a b^2 x^4+\frac{b^3 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^3,x]

[Out]

(a^3*x^2)/2 + a^2*b*x^3 + (3*a*b^2*x^4)/4 + (b^3*x^5)/5

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Maple [A]  time = 0., size = 35, normalized size = 1.2 \[{\frac{{b}^{3}{x}^{5}}{5}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{a}^{2}b{x}^{3}+{\frac{{a}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.33258, size = 46, normalized size = 1.53 \[ \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.184825, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} b^{3} + \frac{3}{4} x^{4} b^{2} a + x^{3} b a^{2} + \frac{1}{2} x^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.092957, size = 36, normalized size = 1.2 \[ \frac{a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207005, size = 46, normalized size = 1.53 \[ \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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