3.3 \(\int x (a+b x) (a c-b c x)^3 \, dx\)

Optimal. Leaf size=55 \[ -\frac{2}{3} a^3 b c^3 x^3+\frac{1}{2} a^4 c^3 x^2+\frac{2}{5} a b^3 c^3 x^5-\frac{1}{6} b^4 c^3 x^6 \]

[Out]

(a^4*c^3*x^2)/2 - (2*a^3*b*c^3*x^3)/3 + (2*a*b^3*c^3*x^5)/5 - (b^4*c^3*x^6)/6

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Rubi [A]  time = 0.0231921, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {75} \[ -\frac{2}{3} a^3 b c^3 x^3+\frac{1}{2} a^4 c^3 x^2+\frac{2}{5} a b^3 c^3 x^5-\frac{1}{6} b^4 c^3 x^6 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

(a^4*c^3*x^2)/2 - (2*a^3*b*c^3*x^3)/3 + (2*a*b^3*c^3*x^5)/5 - (b^4*c^3*x^6)/6

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int x (a+b x) (a c-b c x)^3 \, dx &=\int \left (a^4 c^3 x-2 a^3 b c^3 x^2+2 a b^3 c^3 x^4-b^4 c^3 x^5\right ) \, dx\\ &=\frac{1}{2} a^4 c^3 x^2-\frac{2}{3} a^3 b c^3 x^3+\frac{2}{5} a b^3 c^3 x^5-\frac{1}{6} b^4 c^3 x^6\\ \end{align*}

Mathematica [A]  time = 0.0025646, size = 47, normalized size = 0.85 \[ c^3 \left (-\frac{2}{3} a^3 b x^3+\frac{a^4 x^2}{2}+\frac{2}{5} a b^3 x^5-\frac{1}{6} b^4 x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

c^3*((a^4*x^2)/2 - (2*a^3*b*x^3)/3 + (2*a*b^3*x^5)/5 - (b^4*x^6)/6)

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Maple [A]  time = 0., size = 48, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}{c}^{3}{x}^{2}}{2}}-{\frac{2\,{a}^{3}b{c}^{3}{x}^{3}}{3}}+{\frac{2\,a{b}^{3}{c}^{3}{x}^{5}}{5}}-{\frac{{b}^{4}{c}^{3}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)*(-b*c*x+a*c)^3,x)

[Out]

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

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Maxima [A]  time = 1.01843, size = 63, normalized size = 1.15 \begin{align*} -\frac{1}{6} \, b^{4} c^{3} x^{6} + \frac{2}{5} \, a b^{3} c^{3} x^{5} - \frac{2}{3} \, a^{3} b c^{3} x^{3} + \frac{1}{2} \, a^{4} c^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64652, size = 103, normalized size = 1.87 \begin{align*} -\frac{1}{6} x^{6} c^{3} b^{4} + \frac{2}{5} x^{5} c^{3} b^{3} a - \frac{2}{3} x^{3} c^{3} b a^{3} + \frac{1}{2} x^{2} c^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.071222, size = 53, normalized size = 0.96 \begin{align*} \frac{a^{4} c^{3} x^{2}}{2} - \frac{2 a^{3} b c^{3} x^{3}}{3} + \frac{2 a b^{3} c^{3} x^{5}}{5} - \frac{b^{4} c^{3} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

a**4*c**3*x**2/2 - 2*a**3*b*c**3*x**3/3 + 2*a*b**3*c**3*x**5/5 - b**4*c**3*x**6/6

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Giac [A]  time = 1.19074, size = 63, normalized size = 1.15 \begin{align*} -\frac{1}{6} \, b^{4} c^{3} x^{6} + \frac{2}{5} \, a b^{3} c^{3} x^{5} - \frac{2}{3} \, a^{3} b c^{3} x^{3} + \frac{1}{2} \, a^{4} c^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

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