3.29 \(\int x^2 (a+b x) (a c-b c x)^5 \, dx\)

Optimal. Leaf size=80 \[ \frac{5 a^2 c^5 (a-b x)^7}{7 b^3}-\frac{a^3 c^5 (a-b x)^6}{3 b^3}+\frac{c^5 (a-b x)^9}{9 b^3}-\frac{a c^5 (a-b x)^8}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0411217, 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} \[ \frac{5 a^2 c^5 (a-b x)^7}{7 b^3}-\frac{a^3 c^5 (a-b x)^6}{3 b^3}+\frac{c^5 (a-b x)^9}{9 b^3}-\frac{a c^5 (a-b x)^8}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.0032214, size = 68, normalized size = 0.85 \[ c^5 \left (-\frac{5}{7} a^2 b^4 x^7+a^4 b^2 x^5-a^5 b x^4+\frac{a^6 x^3}{3}+\frac{1}{2} a b^5 x^8-\frac{1}{9} b^6 x^9\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

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Maple [A]  time = 0.002, size = 75, normalized size = 0.9 \begin{align*} -{\frac{{b}^{6}{c}^{5}{x}^{9}}{9}}+{\frac{a{b}^{5}{c}^{5}{x}^{8}}{2}}-{\frac{5\,{a}^{2}{c}^{5}{b}^{4}{x}^{7}}{7}}+{a}^{4}{c}^{5}{b}^{2}{x}^{5}-{a}^{5}{c}^{5}b{x}^{4}+{\frac{{a}^{6}{c}^{5}{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)^5,x)

[Out]

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

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Maxima [A]  time = 1.06779, size = 100, normalized size = 1.25 \begin{align*} -\frac{1}{9} \, b^{6} c^{5} x^{9} + \frac{1}{2} \, a b^{5} c^{5} x^{8} - \frac{5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac{1}{3} \, a^{6} c^{5} 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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.82654, size = 151, normalized size = 1.89 \begin{align*} -\frac{1}{9} x^{9} c^{5} b^{6} + \frac{1}{2} x^{8} c^{5} b^{5} a - \frac{5}{7} x^{7} c^{5} b^{4} a^{2} + x^{5} c^{5} b^{2} a^{4} - x^{4} c^{5} b a^{5} + \frac{1}{3} x^{3} c^{5} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.08012, size = 78, normalized size = 0.98 \begin{align*} \frac{a^{6} c^{5} x^{3}}{3} - a^{5} b c^{5} x^{4} + a^{4} b^{2} c^{5} x^{5} - \frac{5 a^{2} b^{4} c^{5} x^{7}}{7} + \frac{a b^{5} c^{5} x^{8}}{2} - \frac{b^{6} c^{5} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

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

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Giac [A]  time = 1.27819, size = 100, normalized size = 1.25 \begin{align*} -\frac{1}{9} \, b^{6} c^{5} x^{9} + \frac{1}{2} \, a b^{5} c^{5} x^{8} - \frac{5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac{1}{3} \, a^{6} c^{5} 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)^5,x, algorithm="giac")

[Out]

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