3.15 \(\int x^2 (a+b x) (a c-b c x)^4 \, dx\)

Optimal. Leaf size=80 \[ \frac{5 a^2 c^4 (a-b x)^6}{6 b^3}-\frac{2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac{c^4 (a-b x)^8}{8 b^3}-\frac{4 a c^4 (a-b x)^7}{7 b^3} \]

[Out]

(-2*a^3*c^4*(a - b*x)^5)/(5*b^3) + (5*a^2*c^4*(a - b*x)^6)/(6*b^3) - (4*a*c^4*(a - b*x)^7)/(7*b^3) + (c^4*(a -
b*x)^8)/(8*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0387587, antiderivative size = 80, 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} \[ \frac{5 a^2 c^4 (a-b x)^6}{6 b^3}-\frac{2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac{c^4 (a-b x)^8}{8 b^3}-\frac{4 a c^4 (a-b x)^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(-2*a^3*c^4*(a - b*x)^5)/(5*b^3) + (5*a^2*c^4*(a - b*x)^6)/(6*b^3) - (4*a*c^4*(a - b*x)^7)/(7*b^3) + (c^4*(a -
b*x)^8)/(8*b^3)

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)^4 \, dx &=\int \left (\frac{2 a^3 (a c-b c x)^4}{b^2}-\frac{5 a^2 (a c-b c x)^5}{b^2 c}+\frac{4 a (a c-b c x)^6}{b^2 c^2}-\frac{(a c-b c x)^7}{b^2 c^3}\right ) \, dx\\ &=-\frac{2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac{5 a^2 c^4 (a-b x)^6}{6 b^3}-\frac{4 a c^4 (a-b x)^7}{7 b^3}+\frac{c^4 (a-b x)^8}{8 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0034771, size = 87, normalized size = 1.09 \[ \frac{1}{3} a^2 b^3 c^4 x^6+\frac{2}{5} a^3 b^2 c^4 x^5-\frac{3}{4} a^4 b c^4 x^4+\frac{1}{3} a^5 c^4 x^3-\frac{3}{7} a b^4 c^4 x^7+\frac{1}{8} b^5 c^4 x^8 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*x^3)/3 - (3*a^4*b*c^4*x^4)/4 + (2*a^3*b^2*c^4*x^5)/5 + (a^2*b^3*c^4*x^6)/3 - (3*a*b^4*c^4*x^7)/7 + (b
^5*c^4*x^8)/8

________________________________________________________________________________________

Maple [A]  time = 0., size = 76, normalized size = 1. \begin{align*}{\frac{{b}^{5}{c}^{4}{x}^{8}}{8}}-{\frac{3\,a{b}^{4}{c}^{4}{x}^{7}}{7}}+{\frac{{a}^{2}{c}^{4}{b}^{3}{x}^{6}}{3}}+{\frac{2\,{a}^{3}{c}^{4}{b}^{2}{x}^{5}}{5}}-{\frac{3\,{a}^{4}{c}^{4}b{x}^{4}}{4}}+{\frac{{a}^{5}{c}^{4}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)*(-b*c*x+a*c)^4,x)

[Out]

1/8*b^5*c^4*x^8-3/7*a*b^4*c^4*x^7+1/3*a^2*c^4*b^3*x^6+2/5*a^3*c^4*b^2*x^5-3/4*a^4*c^4*b*x^4+1/3*a^5*c^4*x^3

________________________________________________________________________________________

Maxima [A]  time = 1.06534, size = 101, normalized size = 1.26 \begin{align*} \frac{1}{8} \, b^{5} c^{4} x^{8} - \frac{3}{7} \, a b^{4} c^{4} x^{7} + \frac{1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac{2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac{3}{4} \, a^{4} b c^{4} x^{4} + \frac{1}{3} \, a^{5} c^{4} 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)^4,x, algorithm="maxima")

[Out]

1/8*b^5*c^4*x^8 - 3/7*a*b^4*c^4*x^7 + 1/3*a^2*b^3*c^4*x^6 + 2/5*a^3*b^2*c^4*x^5 - 3/4*a^4*b*c^4*x^4 + 1/3*a^5*
c^4*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.69045, size = 161, normalized size = 2.01 \begin{align*} \frac{1}{8} x^{8} c^{4} b^{5} - \frac{3}{7} x^{7} c^{4} b^{4} a + \frac{1}{3} x^{6} c^{4} b^{3} a^{2} + \frac{2}{5} x^{5} c^{4} b^{2} a^{3} - \frac{3}{4} x^{4} c^{4} b a^{4} + \frac{1}{3} x^{3} c^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

1/8*x^8*c^4*b^5 - 3/7*x^7*c^4*b^4*a + 1/3*x^6*c^4*b^3*a^2 + 2/5*x^5*c^4*b^2*a^3 - 3/4*x^4*c^4*b*a^4 + 1/3*x^3*
c^4*a^5

________________________________________________________________________________________

Sympy [A]  time = 0.078879, size = 85, normalized size = 1.06 \begin{align*} \frac{a^{5} c^{4} x^{3}}{3} - \frac{3 a^{4} b c^{4} x^{4}}{4} + \frac{2 a^{3} b^{2} c^{4} x^{5}}{5} + \frac{a^{2} b^{3} c^{4} x^{6}}{3} - \frac{3 a b^{4} c^{4} x^{7}}{7} + \frac{b^{5} c^{4} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

a**5*c**4*x**3/3 - 3*a**4*b*c**4*x**4/4 + 2*a**3*b**2*c**4*x**5/5 + a**2*b**3*c**4*x**6/3 - 3*a*b**4*c**4*x**7
/7 + b**5*c**4*x**8/8

________________________________________________________________________________________

Giac [A]  time = 1.18566, size = 101, normalized size = 1.26 \begin{align*} \frac{1}{8} \, b^{5} c^{4} x^{8} - \frac{3}{7} \, a b^{4} c^{4} x^{7} + \frac{1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac{2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac{3}{4} \, a^{4} b c^{4} x^{4} + \frac{1}{3} \, a^{5} c^{4} 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)^4,x, algorithm="giac")

[Out]

1/8*b^5*c^4*x^8 - 3/7*a*b^4*c^4*x^7 + 1/3*a^2*b^3*c^4*x^6 + 2/5*a^3*b^2*c^4*x^5 - 3/4*a^4*b*c^4*x^4 + 1/3*a^5*
c^4*x^3