3.14 \(\int x^3 (a+b x) (a c-b c x)^4 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0422777, 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} \[ \frac{2}{7} a^2 b^3 c^4 x^7+\frac{1}{3} a^3 b^2 c^4 x^6-\frac{3}{5} a^4 b c^4 x^5+\frac{1}{4} a^5 c^4 x^4-\frac{3}{8} a b^4 c^4 x^8+\frac{1}{9} b^5 c^4 x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.00369, size = 87, normalized size = 1. \[ \frac{2}{7} a^2 b^3 c^4 x^7+\frac{1}{3} a^3 b^2 c^4 x^6-\frac{3}{5} a^4 b c^4 x^5+\frac{1}{4} a^5 c^4 x^4-\frac{3}{8} a b^4 c^4 x^8+\frac{1}{9} b^5 c^4 x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

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

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Maple [A]  time = 0.001, size = 76, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{c}^{4}{x}^{4}}{4}}-{\frac{3\,{a}^{4}b{c}^{4}{x}^{5}}{5}}+{\frac{{a}^{3}{b}^{2}{c}^{4}{x}^{6}}{3}}+{\frac{2\,{a}^{2}{b}^{3}{c}^{4}{x}^{7}}{7}}-{\frac{3\,a{b}^{4}{c}^{4}{x}^{8}}{8}}+{\frac{{b}^{5}{c}^{4}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)*(-b*c*x+a*c)^4,x)

[Out]

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

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Maxima [A]  time = 1.02448, size = 101, normalized size = 1.16 \begin{align*} \frac{1}{9} \, b^{5} c^{4} x^{9} - \frac{3}{8} \, a b^{4} c^{4} x^{8} + \frac{2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac{1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac{3}{5} \, a^{4} b c^{4} x^{5} + \frac{1}{4} \, a^{5} c^{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67918, size = 161, normalized size = 1.85 \begin{align*} \frac{1}{9} x^{9} c^{4} b^{5} - \frac{3}{8} x^{8} c^{4} b^{4} a + \frac{2}{7} x^{7} c^{4} b^{3} a^{2} + \frac{1}{3} x^{6} c^{4} b^{2} a^{3} - \frac{3}{5} x^{5} c^{4} b a^{4} + \frac{1}{4} x^{4} c^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.08092, size = 85, normalized size = 0.98 \begin{align*} \frac{a^{5} c^{4} x^{4}}{4} - \frac{3 a^{4} b c^{4} x^{5}}{5} + \frac{a^{3} b^{2} c^{4} x^{6}}{3} + \frac{2 a^{2} b^{3} c^{4} x^{7}}{7} - \frac{3 a b^{4} c^{4} x^{8}}{8} + \frac{b^{5} c^{4} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

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

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Giac [A]  time = 1.17521, size = 101, normalized size = 1.16 \begin{align*} \frac{1}{9} \, b^{5} c^{4} x^{9} - \frac{3}{8} \, a b^{4} c^{4} x^{8} + \frac{2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac{1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac{3}{5} \, a^{4} b c^{4} x^{5} + \frac{1}{4} \, a^{5} c^{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

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