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

Optimal. Leaf size=59 \[ -\frac{a^2 c^5 (a-b x)^6}{3 b^2}-\frac{c^5 (a-b x)^8}{8 b^2}+\frac{3 a c^5 (a-b x)^7}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0304457, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {75} \[ -\frac{a^2 c^5 (a-b x)^6}{3 b^2}-\frac{c^5 (a-b x)^8}{8 b^2}+\frac{3 a c^5 (a-b x)^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0028104, size = 73, normalized size = 1.24 \[ c^5 \left (-\frac{5}{6} a^2 b^4 x^6+\frac{5}{4} a^4 b^2 x^4-\frac{4}{3} a^5 b x^3+\frac{a^6 x^2}{2}+\frac{4}{7} a b^5 x^7-\frac{1}{8} b^6 x^8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 76, normalized size = 1.3 \begin{align*} -{\frac{{b}^{6}{c}^{5}{x}^{8}}{8}}+{\frac{4\,a{b}^{5}{c}^{5}{x}^{7}}{7}}-{\frac{5\,{a}^{2}{c}^{5}{b}^{4}{x}^{6}}{6}}+{\frac{5\,{a}^{4}{c}^{5}{b}^{2}{x}^{4}}{4}}-{\frac{4\,{a}^{5}{c}^{5}b{x}^{3}}{3}}+{\frac{{a}^{6}{c}^{5}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06523, size = 101, normalized size = 1.71 \begin{align*} -\frac{1}{8} \, b^{6} c^{5} x^{8} + \frac{4}{7} \, a b^{5} c^{5} x^{7} - \frac{5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac{5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac{4}{3} \, a^{5} b c^{5} x^{3} + \frac{1}{2} \, a^{6} c^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78517, size = 162, normalized size = 2.75 \begin{align*} -\frac{1}{8} x^{8} c^{5} b^{6} + \frac{4}{7} x^{7} c^{5} b^{5} a - \frac{5}{6} x^{6} c^{5} b^{4} a^{2} + \frac{5}{4} x^{4} c^{5} b^{2} a^{4} - \frac{4}{3} x^{3} c^{5} b a^{5} + \frac{1}{2} x^{2} c^{5} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.080291, size = 87, normalized size = 1.47 \begin{align*} \frac{a^{6} c^{5} x^{2}}{2} - \frac{4 a^{5} b c^{5} x^{3}}{3} + \frac{5 a^{4} b^{2} c^{5} x^{4}}{4} - \frac{5 a^{2} b^{4} c^{5} x^{6}}{6} + \frac{4 a b^{5} c^{5} x^{7}}{7} - \frac{b^{6} c^{5} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21633, size = 101, normalized size = 1.71 \begin{align*} -\frac{1}{8} \, b^{6} c^{5} x^{8} + \frac{4}{7} \, a b^{5} c^{5} x^{7} - \frac{5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac{5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac{4}{3} \, a^{5} b c^{5} x^{3} + \frac{1}{2} \, a^{6} c^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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