3.27 \(\int x^4 (a+b x) (a c-b c x)^5 \, dx\)

Optimal. Leaf size=87 \[ -\frac{5}{9} a^2 b^4 c^5 x^9+\frac{5}{7} a^4 b^2 c^5 x^7-\frac{2}{3} a^5 b c^5 x^6+\frac{1}{5} a^6 c^5 x^5+\frac{2}{5} a b^5 c^5 x^{10}-\frac{1}{11} b^6 c^5 x^{11} \]

[Out]

(a^6*c^5*x^5)/5 - (2*a^5*b*c^5*x^6)/3 + (5*a^4*b^2*c^5*x^7)/7 - (5*a^2*b^4*c^5*x^9)/9 + (2*a*b^5*c^5*x^10)/5 -
(b^6*c^5*x^11)/11

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Rubi [A]  time = 0.0477253, 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} \[ -\frac{5}{9} a^2 b^4 c^5 x^9+\frac{5}{7} a^4 b^2 c^5 x^7-\frac{2}{3} a^5 b c^5 x^6+\frac{1}{5} a^6 c^5 x^5+\frac{2}{5} a b^5 c^5 x^{10}-\frac{1}{11} b^6 c^5 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

(a^6*c^5*x^5)/5 - (2*a^5*b*c^5*x^6)/3 + (5*a^4*b^2*c^5*x^7)/7 - (5*a^2*b^4*c^5*x^9)/9 + (2*a*b^5*c^5*x^10)/5 -
(b^6*c^5*x^11)/11

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)^5 \, dx &=\int \left (a^6 c^5 x^4-4 a^5 b c^5 x^5+5 a^4 b^2 c^5 x^6-5 a^2 b^4 c^5 x^8+4 a b^5 c^5 x^9-b^6 c^5 x^{10}\right ) \, dx\\ &=\frac{1}{5} a^6 c^5 x^5-\frac{2}{3} a^5 b c^5 x^6+\frac{5}{7} a^4 b^2 c^5 x^7-\frac{5}{9} a^2 b^4 c^5 x^9+\frac{2}{5} a b^5 c^5 x^{10}-\frac{1}{11} b^6 c^5 x^{11}\\ \end{align*}

Mathematica [A]  time = 0.003637, size = 73, normalized size = 0.84 \[ c^5 \left (-\frac{5}{9} a^2 b^4 x^9+\frac{5}{7} a^4 b^2 x^7-\frac{2}{3} a^5 b x^6+\frac{a^6 x^5}{5}+\frac{2}{5} a b^5 x^{10}-\frac{1}{11} b^6 x^{11}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

c^5*((a^6*x^5)/5 - (2*a^5*b*x^6)/3 + (5*a^4*b^2*x^7)/7 - (5*a^2*b^4*x^9)/9 + (2*a*b^5*x^10)/5 - (b^6*x^11)/11)

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Maple [A]  time = 0.003, size = 76, normalized size = 0.9 \begin{align*}{\frac{{a}^{6}{c}^{5}{x}^{5}}{5}}-{\frac{2\,{a}^{5}b{c}^{5}{x}^{6}}{3}}+{\frac{5\,{a}^{4}{b}^{2}{c}^{5}{x}^{7}}{7}}-{\frac{5\,{a}^{2}{b}^{4}{c}^{5}{x}^{9}}{9}}+{\frac{2\,a{b}^{5}{c}^{5}{x}^{10}}{5}}-{\frac{{b}^{6}{c}^{5}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

1/5*a^6*c^5*x^5-2/3*a^5*b*c^5*x^6+5/7*a^4*b^2*c^5*x^7-5/9*a^2*b^4*c^5*x^9+2/5*a*b^5*c^5*x^10-1/11*b^6*c^5*x^11

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Maxima [A]  time = 1.05091, size = 101, normalized size = 1.16 \begin{align*} -\frac{1}{11} \, b^{6} c^{5} x^{11} + \frac{2}{5} \, a b^{5} c^{5} x^{10} - \frac{5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac{5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac{2}{3} \, a^{5} b c^{5} x^{6} + \frac{1}{5} \, a^{6} c^{5} 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)^5,x, algorithm="maxima")

[Out]

-1/11*b^6*c^5*x^11 + 2/5*a*b^5*c^5*x^10 - 5/9*a^2*b^4*c^5*x^9 + 5/7*a^4*b^2*c^5*x^7 - 2/3*a^5*b*c^5*x^6 + 1/5*
a^6*c^5*x^5

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Fricas [A]  time = 1.6579, size = 166, normalized size = 1.91 \begin{align*} -\frac{1}{11} x^{11} c^{5} b^{6} + \frac{2}{5} x^{10} c^{5} b^{5} a - \frac{5}{9} x^{9} c^{5} b^{4} a^{2} + \frac{5}{7} x^{7} c^{5} b^{2} a^{4} - \frac{2}{3} x^{6} c^{5} b a^{5} + \frac{1}{5} x^{5} c^{5} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

-1/11*x^11*c^5*b^6 + 2/5*x^10*c^5*b^5*a - 5/9*x^9*c^5*b^4*a^2 + 5/7*x^7*c^5*b^2*a^4 - 2/3*x^6*c^5*b*a^5 + 1/5*
x^5*c^5*a^6

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Sympy [A]  time = 0.081655, size = 87, normalized size = 1. \begin{align*} \frac{a^{6} c^{5} x^{5}}{5} - \frac{2 a^{5} b c^{5} x^{6}}{3} + \frac{5 a^{4} b^{2} c^{5} x^{7}}{7} - \frac{5 a^{2} b^{4} c^{5} x^{9}}{9} + \frac{2 a b^{5} c^{5} x^{10}}{5} - \frac{b^{6} c^{5} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

a**6*c**5*x**5/5 - 2*a**5*b*c**5*x**6/3 + 5*a**4*b**2*c**5*x**7/7 - 5*a**2*b**4*c**5*x**9/9 + 2*a*b**5*c**5*x*
*10/5 - b**6*c**5*x**11/11

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Giac [A]  time = 1.26259, size = 101, normalized size = 1.16 \begin{align*} -\frac{1}{11} \, b^{6} c^{5} x^{11} + \frac{2}{5} \, a b^{5} c^{5} x^{10} - \frac{5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac{5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac{2}{3} \, a^{5} b c^{5} x^{6} + \frac{1}{5} \, a^{6} c^{5} 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)^5,x, algorithm="giac")

[Out]

-1/11*b^6*c^5*x^11 + 2/5*a*b^5*c^5*x^10 - 5/9*a^2*b^4*c^5*x^9 + 5/7*a^4*b^2*c^5*x^7 - 2/3*a^5*b*c^5*x^6 + 1/5*
a^6*c^5*x^5