\(\int x^2 (a+b x) (a c-b c x)^4 \, dx\) [15]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=-\frac {2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac {5 a^2 c^4 (a-b x)^6}{6 b^3}-\frac {4 a c^4 (a-b x)^7}{7 b^3}+\frac {c^4 (a-b x)^8}{8 b^3} \]

[Out]

-2/5*a^3*c^4*(-b*x+a)^5/b^3+5/6*a^2*c^4*(-b*x+a)^6/b^3-4/7*a*c^4*(-b*x+a)^7/b^3+1/8*c^4*(-b*x+a)^8/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} \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=-\frac {2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac {5 a^2 c^4 (a-b x)^6}{6 b^3}+\frac {c^4 (a-b x)^8}{8 b^3}-\frac {4 a c^4 (a-b x)^7}{7 b^3} \]

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{3} a^5 c^4 x^3-\frac {3}{4} a^4 b c^4 x^4+\frac {2}{5} a^3 b^2 c^4 x^5+\frac {1}{3} a^2 b^3 c^4 x^6-\frac {3}{7} a b^4 c^4 x^7+\frac {1}{8} b^5 c^4 x^8 \]

[In]

Integrate[x^2*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {x^{3} \left (105 b^{5} x^{5}-360 a \,b^{4} x^{4}+280 a^{2} b^{3} x^{3}+336 a^{3} b^{2} x^{2}-630 a^{4} b x +280 a^{5}\right ) c^{4}}{840}\)	\(61\)
default	\(\frac {1}{8} b^{5} c^{4} x^{8}-\frac {3}{7} a \,b^{4} c^{4} x^{7}+\frac {1}{3} a^{2} c^{4} b^{3} x^{6}+\frac {2}{5} a^{3} c^{4} b^{2} x^{5}-\frac {3}{4} a^{4} c^{4} b \,x^{4}+\frac {1}{3} a^{5} c^{4} x^{3}\)	\(76\)
norman	\(\frac {1}{8} b^{5} c^{4} x^{8}-\frac {3}{7} a \,b^{4} c^{4} x^{7}+\frac {1}{3} a^{2} c^{4} b^{3} x^{6}+\frac {2}{5} a^{3} c^{4} b^{2} x^{5}-\frac {3}{4} a^{4} c^{4} b \,x^{4}+\frac {1}{3} a^{5} c^{4} x^{3}\)	\(76\)
risch	\(\frac {1}{8} b^{5} c^{4} x^{8}-\frac {3}{7} a \,b^{4} c^{4} x^{7}+\frac {1}{3} a^{2} c^{4} b^{3} x^{6}+\frac {2}{5} a^{3} c^{4} b^{2} x^{5}-\frac {3}{4} a^{4} c^{4} b \,x^{4}+\frac {1}{3} a^{5} c^{4} x^{3}\)	\(76\)
parallelrisch	\(\frac {1}{8} b^{5} c^{4} x^{8}-\frac {3}{7} a \,b^{4} c^{4} x^{7}+\frac {1}{3} a^{2} c^{4} b^{3} x^{6}+\frac {2}{5} a^{3} c^{4} b^{2} x^{5}-\frac {3}{4} a^{4} c^{4} b \,x^{4}+\frac {1}{3} a^{5} c^{4} x^{3}\)	\(76\)

[In]

int(x^2*(b*x+a)*(-b*c*x+a*c)^4,x,method=_RETURNVERBOSE)

[Out]

1/840*x^3*(105*b^5*x^5-360*a*b^4*x^4+280*a^2*b^3*x^3+336*a^3*b^2*x^2-630*a^4*b*x+280*a^5)*c^4

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{8} \, b^{5} c^{4} x^{8} - \frac {3}{7} \, a b^{4} c^{4} x^{7} + \frac {1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac {2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac {3}{4} \, a^{4} b c^{4} x^{4} + \frac {1}{3} \, a^{5} c^{4} x^{3} \]

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=\frac {a^{5} c^{4} x^{3}}{3} - \frac {3 a^{4} b c^{4} x^{4}}{4} + \frac {2 a^{3} b^{2} c^{4} x^{5}}{5} + \frac {a^{2} b^{3} c^{4} x^{6}}{3} - \frac {3 a b^{4} c^{4} x^{7}}{7} + \frac {b^{5} c^{4} x^{8}}{8} \]

[In]

integrate(x**2*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{8} \, b^{5} c^{4} x^{8} - \frac {3}{7} \, a b^{4} c^{4} x^{7} + \frac {1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac {2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac {3}{4} \, a^{4} b c^{4} x^{4} + \frac {1}{3} \, a^{5} c^{4} x^{3} \]

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{8} \, b^{5} c^{4} x^{8} - \frac {3}{7} \, a b^{4} c^{4} x^{7} + \frac {1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac {2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac {3}{4} \, a^{4} b c^{4} x^{4} + \frac {1}{3} \, a^{5} c^{4} x^{3} \]

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int x^2 (a+b x) (a c-b c x)^4 \, dx=\frac {a^5\,c^4\,x^3}{3}-\frac {3\,a^4\,b\,c^4\,x^4}{4}+\frac {2\,a^3\,b^2\,c^4\,x^5}{5}+\frac {a^2\,b^3\,c^4\,x^6}{3}-\frac {3\,a\,b^4\,c^4\,x^7}{7}+\frac {b^5\,c^4\,x^8}{8} \]

[In]

int(x^2*(a*c - b*c*x)^4*(a + b*x),x)

[Out]

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