3.28 \(\int x^3 (a+b x) (a c-b c x)^5 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0401473, 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} \[ -\frac{5}{8} a^2 b^4 c^5 x^8+\frac{5}{6} a^4 b^2 c^5 x^6-\frac{4}{5} a^5 b c^5 x^5+\frac{1}{4} a^6 c^5 x^4+\frac{4}{9} a b^5 c^5 x^9-\frac{1}{10} b^6 c^5 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.0032305, size = 73, normalized size = 0.84 \[ c^5 \left (-\frac{5}{8} a^2 b^4 x^8+\frac{5}{6} a^4 b^2 x^6-\frac{4}{5} a^5 b x^5+\frac{a^6 x^4}{4}+\frac{4}{9} a b^5 x^9-\frac{1}{10} b^6 x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

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Maple [A]  time = 0.001, size = 76, normalized size = 0.9 \begin{align*}{\frac{{a}^{6}{c}^{5}{x}^{4}}{4}}-{\frac{4\,{a}^{5}b{c}^{5}{x}^{5}}{5}}+{\frac{5\,{a}^{4}{b}^{2}{c}^{5}{x}^{6}}{6}}-{\frac{5\,{a}^{2}{b}^{4}{c}^{5}{x}^{8}}{8}}+{\frac{4\,a{b}^{5}{c}^{5}{x}^{9}}{9}}-{\frac{{b}^{6}{c}^{5}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

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

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Maxima [A]  time = 1.14579, size = 101, normalized size = 1.16 \begin{align*} -\frac{1}{10} \, b^{6} c^{5} x^{10} + \frac{4}{9} \, a b^{5} c^{5} x^{9} - \frac{5}{8} \, a^{2} b^{4} c^{5} x^{8} + \frac{5}{6} \, a^{4} b^{2} c^{5} x^{6} - \frac{4}{5} \, a^{5} b c^{5} x^{5} + \frac{1}{4} \, a^{6} c^{5} 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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71193, size = 165, normalized size = 1.9 \begin{align*} -\frac{1}{10} x^{10} c^{5} b^{6} + \frac{4}{9} x^{9} c^{5} b^{5} a - \frac{5}{8} x^{8} c^{5} b^{4} a^{2} + \frac{5}{6} x^{6} c^{5} b^{2} a^{4} - \frac{4}{5} x^{5} c^{5} b a^{5} + \frac{1}{4} x^{4} c^{5} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.081261, size = 87, normalized size = 1. \begin{align*} \frac{a^{6} c^{5} x^{4}}{4} - \frac{4 a^{5} b c^{5} x^{5}}{5} + \frac{5 a^{4} b^{2} c^{5} x^{6}}{6} - \frac{5 a^{2} b^{4} c^{5} x^{8}}{8} + \frac{4 a b^{5} c^{5} x^{9}}{9} - \frac{b^{6} c^{5} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

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

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Giac [A]  time = 1.25258, size = 101, normalized size = 1.16 \begin{align*} -\frac{1}{10} \, b^{6} c^{5} x^{10} + \frac{4}{9} \, a b^{5} c^{5} x^{9} - \frac{5}{8} \, a^{2} b^{4} c^{5} x^{8} + \frac{5}{6} \, a^{4} b^{2} c^{5} x^{6} - \frac{4}{5} \, a^{5} b c^{5} x^{5} + \frac{1}{4} \, a^{6} c^{5} 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)^5,x, algorithm="giac")

[Out]

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