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 = 0.008693, 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, Rules used = {43} \[ \frac{(a+b x)^5}{5 b^2}-\frac{a (a+b x)^4}{4 b^2} \]

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)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x (a+b x)^3 \, dx &=\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx\\ &=-\frac{a (a+b x)^4}{4 b^2}+\frac{(a+b x)^5}{5 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0015622, size = 40, normalized size = 1.33 \[ a^2 b x^3+\frac{a^3 x^2}{2}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^5}{5} \]

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 \begin{align*}{\frac{{b}^{3}{x}^{5}}{5}}+{\frac{3\,{b}^{2}a{x}^{4}}{4}}+{a}^{2}b{x}^{3}+{\frac{{a}^{3}{x}^{2}}{2}} \end{align*}

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

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Maxima [A]  time = 1.03847, size = 46, normalized size = 1.53 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^3,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 = 1.36675, size = 74, normalized size = 2.47 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^3,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 = 0.065796, size = 36, normalized size = 1.2 \begin{align*} \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} \end{align*}

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 [A]  time = 1.14779, size = 46, normalized size = 1.53 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^3,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