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

Optimal. Leaf size=87 \[ \frac{1}{4} a^2 b^3 c^4 x^8+\frac{2}{7} a^3 b^2 c^4 x^7-\frac{1}{2} a^4 b c^4 x^6+\frac{1}{5} a^5 c^4 x^5-\frac{1}{3} a b^4 c^4 x^9+\frac{1}{10} b^5 c^4 x^{10} \]

[Out]

(a^5*c^4*x^5)/5 - (a^4*b*c^4*x^6)/2 + (2*a^3*b^2*c^4*x^7)/7 + (a^2*b^3*c^4*x^8)/4 - (a*b^4*c^4*x^9)/3 + (b^5*c
^4*x^10)/10

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Rubi [A]  time = 0.0469877, antiderivative size = 87, 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} \[ \frac{1}{4} a^2 b^3 c^4 x^8+\frac{2}{7} a^3 b^2 c^4 x^7-\frac{1}{2} a^4 b c^4 x^6+\frac{1}{5} a^5 c^4 x^5-\frac{1}{3} a b^4 c^4 x^9+\frac{1}{10} b^5 c^4 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*x^5)/5 - (a^4*b*c^4*x^6)/2 + (2*a^3*b^2*c^4*x^7)/7 + (a^2*b^3*c^4*x^8)/4 - (a*b^4*c^4*x^9)/3 + (b^5*c
^4*x^10)/10

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^4 (a+b x) (a c-b c x)^4 \, dx &=\int \left (a^5 c^4 x^4-3 a^4 b c^4 x^5+2 a^3 b^2 c^4 x^6+2 a^2 b^3 c^4 x^7-3 a b^4 c^4 x^8+b^5 c^4 x^9\right ) \, dx\\ &=\frac{1}{5} a^5 c^4 x^5-\frac{1}{2} a^4 b c^4 x^6+\frac{2}{7} a^3 b^2 c^4 x^7+\frac{1}{4} a^2 b^3 c^4 x^8-\frac{1}{3} a b^4 c^4 x^9+\frac{1}{10} b^5 c^4 x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0052567, size = 87, normalized size = 1. \[ \frac{1}{4} a^2 b^3 c^4 x^8+\frac{2}{7} a^3 b^2 c^4 x^7-\frac{1}{2} a^4 b c^4 x^6+\frac{1}{5} a^5 c^4 x^5-\frac{1}{3} a b^4 c^4 x^9+\frac{1}{10} b^5 c^4 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*c^4*x^5)/5 - (a^4*b*c^4*x^6)/2 + (2*a^3*b^2*c^4*x^7)/7 + (a^2*b^3*c^4*x^8)/4 - (a*b^4*c^4*x^9)/3 + (b^5*c
^4*x^10)/10

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Maple [A]  time = 0.002, size = 76, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{c}^{4}{x}^{5}}{5}}-{\frac{{a}^{4}b{c}^{4}{x}^{6}}{2}}+{\frac{2\,{a}^{3}{b}^{2}{c}^{4}{x}^{7}}{7}}+{\frac{{a}^{2}{b}^{3}{c}^{4}{x}^{8}}{4}}-{\frac{a{b}^{4}{c}^{4}{x}^{9}}{3}}+{\frac{{b}^{5}{c}^{4}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^5*c^4*x^5-1/2*a^4*b*c^4*x^6+2/7*a^3*b^2*c^4*x^7+1/4*a^2*b^3*c^4*x^8-1/3*a*b^4*c^4*x^9+1/10*b^5*c^4*x^10

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Maxima [A]  time = 1.03647, size = 101, normalized size = 1.16 \begin{align*} \frac{1}{10} \, b^{5} c^{4} x^{10} - \frac{1}{3} \, a b^{4} c^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac{2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac{1}{2} \, a^{4} b c^{4} x^{6} + \frac{1}{5} \, a^{5} c^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*c^4*x^10 - 1/3*a*b^4*c^4*x^9 + 1/4*a^2*b^3*c^4*x^8 + 2/7*a^3*b^2*c^4*x^7 - 1/2*a^4*b*c^4*x^6 + 1/5*a^
5*c^4*x^5

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Fricas [A]  time = 1.73967, size = 163, normalized size = 1.87 \begin{align*} \frac{1}{10} x^{10} c^{4} b^{5} - \frac{1}{3} x^{9} c^{4} b^{4} a + \frac{1}{4} x^{8} c^{4} b^{3} a^{2} + \frac{2}{7} x^{7} c^{4} b^{2} a^{3} - \frac{1}{2} x^{6} c^{4} b a^{4} + \frac{1}{5} x^{5} c^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*c^4*b^5 - 1/3*x^9*c^4*b^4*a + 1/4*x^8*c^4*b^3*a^2 + 2/7*x^7*c^4*b^2*a^3 - 1/2*x^6*c^4*b*a^4 + 1/5*x^
5*c^4*a^5

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Sympy [A]  time = 0.077899, size = 82, normalized size = 0.94 \begin{align*} \frac{a^{5} c^{4} x^{5}}{5} - \frac{a^{4} b c^{4} x^{6}}{2} + \frac{2 a^{3} b^{2} c^{4} x^{7}}{7} + \frac{a^{2} b^{3} c^{4} x^{8}}{4} - \frac{a b^{4} c^{4} x^{9}}{3} + \frac{b^{5} c^{4} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*c**4*x**5/5 - a**4*b*c**4*x**6/2 + 2*a**3*b**2*c**4*x**7/7 + a**2*b**3*c**4*x**8/4 - a*b**4*c**4*x**9/3 +
b**5*c**4*x**10/10

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Giac [A]  time = 1.178, size = 101, normalized size = 1.16 \begin{align*} \frac{1}{10} \, b^{5} c^{4} x^{10} - \frac{1}{3} \, a b^{4} c^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac{2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac{1}{2} \, a^{4} b c^{4} x^{6} + \frac{1}{5} \, a^{5} c^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*c^4*x^10 - 1/3*a*b^4*c^4*x^9 + 1/4*a^2*b^3*c^4*x^8 + 2/7*a^3*b^2*c^4*x^7 - 1/2*a^4*b*c^4*x^6 + 1/5*a^
5*c^4*x^5