\(\int x^4 (a+b x) (a c-b c x)^4 \, dx\) [13]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{5} a^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10} \]

[Out]

1/5*a^5*c^4*x^5-1/2*a^4*b*c^4*x^6+2/7*a^3*b^2*c^4*x^7+1/4*a^2*b^3*c^4*x^8-1/3*a*b^4*c^4*x^9+1/10*b^5*c^4*x^10

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, 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} \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{5} a^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10} \]

[In]

Int[x^4*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*x^5)/5 - (a^4*b*c^4*x^6)/2 + (2*a^3*b^2*c^4*x^7)/7 + (a^2*b^3*c^4*x^8)/4 - (a*b^4*c^4*x^9)/3 + (b^5*c
^4*x^10)/10

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{5} a^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10} \]

[In]

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

[Out]

(a^5*c^4*x^5)/5 - (a^4*b*c^4*x^6)/2 + (2*a^3*b^2*c^4*x^7)/7 + (a^2*b^3*c^4*x^8)/4 - (a*b^4*c^4*x^9)/3 + (b^5*c
^4*x^10)/10

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70

method	result	size
gosper	\(\frac {x^{5} \left (42 b^{5} x^{5}-140 a \,b^{4} x^{4}+105 a^{2} b^{3} x^{3}+120 a^{3} b^{2} x^{2}-210 a^{4} b x +84 a^{5}\right ) c^{4}}{420}\)	\(61\)
default	\(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\)	\(76\)
norman	\(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\)	\(76\)
risch	\(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\)	\(76\)
parallelrisch	\(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\)	\(76\)

[In]

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

[Out]

1/420*x^5*(42*b^5*x^5-140*a*b^4*x^4+105*a^2*b^3*x^3+120*a^3*b^2*x^2-210*a^4*b*x+84*a^5)*c^4

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{10} \, b^{5} c^{4} x^{10} - \frac {1}{3} \, a b^{4} c^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac {2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac {1}{2} \, a^{4} b c^{4} x^{6} + \frac {1}{5} \, a^{5} c^{4} x^{5} \]

[In]

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

[Out]

1/10*b^5*c^4*x^10 - 1/3*a*b^4*c^4*x^9 + 1/4*a^2*b^3*c^4*x^8 + 2/7*a^3*b^2*c^4*x^7 - 1/2*a^4*b*c^4*x^6 + 1/5*a^
5*c^4*x^5

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {a^{5} c^{4} x^{5}}{5} - \frac {a^{4} b c^{4} x^{6}}{2} + \frac {2 a^{3} b^{2} c^{4} x^{7}}{7} + \frac {a^{2} b^{3} c^{4} x^{8}}{4} - \frac {a b^{4} c^{4} x^{9}}{3} + \frac {b^{5} c^{4} x^{10}}{10} \]

[In]

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

[Out]

a**5*c**4*x**5/5 - a**4*b*c**4*x**6/2 + 2*a**3*b**2*c**4*x**7/7 + a**2*b**3*c**4*x**8/4 - a*b**4*c**4*x**9/3 +
b**5*c**4*x**10/10

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{10} \, b^{5} c^{4} x^{10} - \frac {1}{3} \, a b^{4} c^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac {2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac {1}{2} \, a^{4} b c^{4} x^{6} + \frac {1}{5} \, a^{5} c^{4} x^{5} \]

[In]

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

[Out]

1/10*b^5*c^4*x^10 - 1/3*a*b^4*c^4*x^9 + 1/4*a^2*b^3*c^4*x^8 + 2/7*a^3*b^2*c^4*x^7 - 1/2*a^4*b*c^4*x^6 + 1/5*a^
5*c^4*x^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{10} \, b^{5} c^{4} x^{10} - \frac {1}{3} \, a b^{4} c^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac {2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac {1}{2} \, a^{4} b c^{4} x^{6} + \frac {1}{5} \, a^{5} c^{4} x^{5} \]

[In]

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

[Out]

1/10*b^5*c^4*x^10 - 1/3*a*b^4*c^4*x^9 + 1/4*a^2*b^3*c^4*x^8 + 2/7*a^3*b^2*c^4*x^7 - 1/2*a^4*b*c^4*x^6 + 1/5*a^
5*c^4*x^5

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^4 \, dx=\frac {a^5\,c^4\,x^5}{5}-\frac {a^4\,b\,c^4\,x^6}{2}+\frac {2\,a^3\,b^2\,c^4\,x^7}{7}+\frac {a^2\,b^3\,c^4\,x^8}{4}-\frac {a\,b^4\,c^4\,x^9}{3}+\frac {b^5\,c^4\,x^{10}}{10} \]

[In]

int(x^4*(a*c - b*c*x)^4*(a + b*x),x)

[Out]

(a^5*c^4*x^5)/5 + (b^5*c^4*x^10)/10 - (a^4*b*c^4*x^6)/2 - (a*b^4*c^4*x^9)/3 + (2*a^3*b^2*c^4*x^7)/7 + (a^2*b^3
*c^4*x^8)/4