\(\int x^2 (a+b x) (a c-b c x)^5 \, dx\) [29]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=-\frac {a^3 c^5 (a-b x)^6}{3 b^3}+\frac {5 a^2 c^5 (a-b x)^7}{7 b^3}-\frac {a c^5 (a-b x)^8}{2 b^3}+\frac {c^5 (a-b x)^9}{9 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=-\frac {a^3 c^5 (a-b x)^6}{3 b^3}+\frac {5 a^2 c^5 (a-b x)^7}{7 b^3}+\frac {c^5 (a-b x)^9}{9 b^3}-\frac {a c^5 (a-b x)^8}{2 b^3} \]

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

-1/3*(a^3*c^5*(a - b*x)^6)/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 76

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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=c^5 \left (\frac {a^6 x^3}{3}-a^5 b x^4+a^4 b^2 x^5-\frac {5}{7} a^2 b^4 x^7+\frac {1}{2} a b^5 x^8-\frac {b^6 x^9}{9}\right ) \]

[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)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {x^{3} \left (-14 b^{6} x^{6}+63 a \,x^{5} b^{5}-90 a^{2} x^{4} b^{4}+126 a^{4} x^{2} b^{2}-126 a^{5} x b +42 a^{6}\right ) c^{5}}{126}\)	\(61\)
default	\(-\frac {1}{9} b^{6} c^{5} x^{9}+\frac {1}{2} a \,b^{5} c^{5} x^{8}-\frac {5}{7} a^{2} c^{5} b^{4} x^{7}+a^{4} c^{5} b^{2} x^{5}-a^{5} c^{5} b \,x^{4}+\frac {1}{3} a^{6} c^{5} x^{3}\)	\(75\)
norman	\(-\frac {1}{9} b^{6} c^{5} x^{9}+\frac {1}{2} a \,b^{5} c^{5} x^{8}-\frac {5}{7} a^{2} c^{5} b^{4} x^{7}+a^{4} c^{5} b^{2} x^{5}-a^{5} c^{5} b \,x^{4}+\frac {1}{3} a^{6} c^{5} x^{3}\)	\(75\)
risch	\(-\frac {1}{9} b^{6} c^{5} x^{9}+\frac {1}{2} a \,b^{5} c^{5} x^{8}-\frac {5}{7} a^{2} c^{5} b^{4} x^{7}+a^{4} c^{5} b^{2} x^{5}-a^{5} c^{5} b \,x^{4}+\frac {1}{3} a^{6} c^{5} x^{3}\)	\(75\)
parallelrisch	\(-\frac {1}{9} b^{6} c^{5} x^{9}+\frac {1}{2} a \,b^{5} c^{5} x^{8}-\frac {5}{7} a^{2} c^{5} b^{4} x^{7}+a^{4} c^{5} b^{2} x^{5}-a^{5} c^{5} b \,x^{4}+\frac {1}{3} a^{6} c^{5} x^{3}\)	\(75\)

[In]

int(x^2*(b*x+a)*(-b*c*x+a*c)^5,x,method=_RETURNVERBOSE)

[Out]

1/126*x^3*(-14*b^6*x^6+63*a*b^5*x^5-90*a^2*b^4*x^4+126*a^4*b^2*x^2-126*a^5*b*x+42*a^6)*c^5

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{9} \, b^{6} c^{5} x^{9} + \frac {1}{2} \, a b^{5} c^{5} x^{8} - \frac {5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac {1}{3} \, a^{6} c^{5} x^{3} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=\frac {a^{6} c^{5} x^{3}}{3} - a^{5} b c^{5} x^{4} + a^{4} b^{2} c^{5} x^{5} - \frac {5 a^{2} b^{4} c^{5} x^{7}}{7} + \frac {a b^{5} c^{5} x^{8}}{2} - \frac {b^{6} c^{5} x^{9}}{9} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{9} \, b^{6} c^{5} x^{9} + \frac {1}{2} \, a b^{5} c^{5} x^{8} - \frac {5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac {1}{3} \, a^{6} c^{5} x^{3} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{9} \, b^{6} c^{5} x^{9} + \frac {1}{2} \, a b^{5} c^{5} x^{8} - \frac {5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac {1}{3} \, a^{6} c^{5} x^{3} \]

[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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int x^2 (a+b x) (a c-b c x)^5 \, dx=\frac {a^6\,c^5\,x^3}{3}-a^5\,b\,c^5\,x^4+a^4\,b^2\,c^5\,x^5-\frac {5\,a^2\,b^4\,c^5\,x^7}{7}+\frac {a\,b^5\,c^5\,x^8}{2}-\frac {b^6\,c^5\,x^9}{9} \]

[In]

int(x^2*(a*c - b*c*x)^5*(a + b*x),x)

[Out]

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