3.2 \(\int x^2 (a+b x) (a c-b c x)^3 \, dx\)

Optimal. Leaf size=55 \[ -\frac{1}{2} a^3 b c^3 x^4+\frac{1}{3} a^4 c^3 x^3+\frac{1}{3} a b^3 c^3 x^6-\frac{1}{7} b^4 c^3 x^7 \]

[Out]

(a^4*c^3*x^3)/3 - (a^3*b*c^3*x^4)/2 + (a*b^3*c^3*x^6)/3 - (b^4*c^3*x^7)/7

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Rubi [A]  time = 0.028009, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {75} \[ -\frac{1}{2} a^3 b c^3 x^4+\frac{1}{3} a^4 c^3 x^3+\frac{1}{3} a b^3 c^3 x^6-\frac{1}{7} b^4 c^3 x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

(a^4*c^3*x^3)/3 - (a^3*b*c^3*x^4)/2 + (a*b^3*c^3*x^6)/3 - (b^4*c^3*x^7)/7

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^2 (a+b x) (a c-b c x)^3 \, dx &=\int \left (a^4 c^3 x^2-2 a^3 b c^3 x^3+2 a b^3 c^3 x^5-b^4 c^3 x^6\right ) \, dx\\ &=\frac{1}{3} a^4 c^3 x^3-\frac{1}{2} a^3 b c^3 x^4+\frac{1}{3} a b^3 c^3 x^6-\frac{1}{7} b^4 c^3 x^7\\ \end{align*}

Mathematica [A]  time = 0.00358, size = 47, normalized size = 0.85 \[ c^3 \left (-\frac{1}{2} a^3 b x^4+\frac{a^4 x^3}{3}+\frac{1}{3} a b^3 x^6-\frac{1}{7} b^4 x^7\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

c^3*((a^4*x^3)/3 - (a^3*b*x^4)/2 + (a*b^3*x^6)/3 - (b^4*x^7)/7)

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Maple [A]  time = 0.001, size = 48, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}{c}^{3}{x}^{3}}{3}}-{\frac{{a}^{3}b{c}^{3}{x}^{4}}{2}}+{\frac{a{b}^{3}{c}^{3}{x}^{6}}{3}}-{\frac{{b}^{4}{c}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)*(-b*c*x+a*c)^3,x)

[Out]

1/3*a^4*c^3*x^3-1/2*a^3*b*c^3*x^4+1/3*a*b^3*c^3*x^6-1/7*b^4*c^3*x^7

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Maxima [A]  time = 1.01132, size = 63, normalized size = 1.15 \begin{align*} -\frac{1}{7} \, b^{4} c^{3} x^{7} + \frac{1}{3} \, a b^{3} c^{3} x^{6} - \frac{1}{2} \, a^{3} b c^{3} x^{4} + \frac{1}{3} \, a^{4} c^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

-1/7*b^4*c^3*x^7 + 1/3*a*b^3*c^3*x^6 - 1/2*a^3*b*c^3*x^4 + 1/3*a^4*c^3*x^3

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Fricas [A]  time = 1.75207, size = 103, normalized size = 1.87 \begin{align*} -\frac{1}{7} x^{7} c^{3} b^{4} + \frac{1}{3} x^{6} c^{3} b^{3} a - \frac{1}{2} x^{4} c^{3} b a^{3} + \frac{1}{3} x^{3} c^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

-1/7*x^7*c^3*b^4 + 1/3*x^6*c^3*b^3*a - 1/2*x^4*c^3*b*a^3 + 1/3*x^3*c^3*a^4

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Sympy [A]  time = 0.072819, size = 49, normalized size = 0.89 \begin{align*} \frac{a^{4} c^{3} x^{3}}{3} - \frac{a^{3} b c^{3} x^{4}}{2} + \frac{a b^{3} c^{3} x^{6}}{3} - \frac{b^{4} c^{3} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

a**4*c**3*x**3/3 - a**3*b*c**3*x**4/2 + a*b**3*c**3*x**6/3 - b**4*c**3*x**7/7

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Giac [A]  time = 1.18596, size = 63, normalized size = 1.15 \begin{align*} -\frac{1}{7} \, b^{4} c^{3} x^{7} + \frac{1}{3} \, a b^{3} c^{3} x^{6} - \frac{1}{2} \, a^{3} b c^{3} x^{4} + \frac{1}{3} \, a^{4} c^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

-1/7*b^4*c^3*x^7 + 1/3*a*b^3*c^3*x^6 - 1/2*a^3*b*c^3*x^4 + 1/3*a^4*c^3*x^3