\(\int x^3 (a+b x) (a c-b c x)^5 \, dx\) [28]

Optimal result

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

[Out]

1/4*a^6*c^5*x^4-4/5*a^5*b*c^5*x^5+5/6*a^4*b^2*c^5*x^6-5/8*a^2*b^4*c^5*x^8+4/9*a*b^5*c^5*x^9-1/10*b^6*c^5*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} \[ \int x^3 (a+b x) (a c-b c x)^5 \, dx=\frac {1}{4} a^6 c^5 x^4-\frac {4}{5} a^5 b c^5 x^5+\frac {5}{6} a^4 b^2 c^5 x^6-\frac {5}{8} a^2 b^4 c^5 x^8+\frac {4}{9} a b^5 c^5 x^9-\frac {1}{10} b^6 c^5 x^{10} \]

[In]

Int[x^3*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int x^3 (a+b x) (a c-b c x)^5 \, dx=c^5 \left (\frac {a^6 x^4}{4}-\frac {4}{5} a^5 b x^5+\frac {5}{6} a^4 b^2 x^6-\frac {5}{8} a^2 b^4 x^8+\frac {4}{9} a b^5 x^9-\frac {b^6 x^{10}}{10}\right ) \]

[In]

Integrate[x^3*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70

method	result	size
gosper	\(\frac {x^{4} \left (-36 b^{6} x^{6}+160 a \,x^{5} b^{5}-225 a^{2} x^{4} b^{4}+300 a^{4} x^{2} b^{2}-288 a^{5} x b +90 a^{6}\right ) c^{5}}{360}\)	\(61\)
default	\(\frac {1}{4} a^{6} c^{5} x^{4}-\frac {4}{5} a^{5} b \,c^{5} x^{5}+\frac {5}{6} a^{4} b^{2} c^{5} x^{6}-\frac {5}{8} a^{2} b^{4} c^{5} x^{8}+\frac {4}{9} a \,b^{5} c^{5} x^{9}-\frac {1}{10} b^{6} c^{5} x^{10}\)	\(76\)
norman	\(\frac {1}{4} a^{6} c^{5} x^{4}-\frac {4}{5} a^{5} b \,c^{5} x^{5}+\frac {5}{6} a^{4} b^{2} c^{5} x^{6}-\frac {5}{8} a^{2} b^{4} c^{5} x^{8}+\frac {4}{9} a \,b^{5} c^{5} x^{9}-\frac {1}{10} b^{6} c^{5} x^{10}\)	\(76\)
risch	\(\frac {1}{4} a^{6} c^{5} x^{4}-\frac {4}{5} a^{5} b \,c^{5} x^{5}+\frac {5}{6} a^{4} b^{2} c^{5} x^{6}-\frac {5}{8} a^{2} b^{4} c^{5} x^{8}+\frac {4}{9} a \,b^{5} c^{5} x^{9}-\frac {1}{10} b^{6} c^{5} x^{10}\)	\(76\)
parallelrisch	\(\frac {1}{4} a^{6} c^{5} x^{4}-\frac {4}{5} a^{5} b \,c^{5} x^{5}+\frac {5}{6} a^{4} b^{2} c^{5} x^{6}-\frac {5}{8} a^{2} b^{4} c^{5} x^{8}+\frac {4}{9} a \,b^{5} c^{5} x^{9}-\frac {1}{10} b^{6} c^{5} x^{10}\)	\(76\)

[In]

int(x^3*(b*x+a)*(-b*c*x+a*c)^5,x,method=_RETURNVERBOSE)

[Out]

1/360*x^4*(-36*b^6*x^6+160*a*b^5*x^5-225*a^2*b^4*x^4+300*a^4*b^2*x^2-288*a^5*b*x+90*a^6)*c^5

Fricas [A] (verification not implemented)

none

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

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int x^3 (a+b x) (a c-b c x)^5 \, dx=\frac {a^{6} c^{5} x^{4}}{4} - \frac {4 a^{5} b c^{5} x^{5}}{5} + \frac {5 a^{4} b^{2} c^{5} x^{6}}{6} - \frac {5 a^{2} b^{4} c^{5} x^{8}}{8} + \frac {4 a b^{5} c^{5} x^{9}}{9} - \frac {b^{6} c^{5} x^{10}}{10} \]

[In]

integrate(x**3*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{10} \, b^{6} c^{5} x^{10} + \frac {4}{9} \, a b^{5} c^{5} x^{9} - \frac {5}{8} \, a^{2} b^{4} c^{5} x^{8} + \frac {5}{6} \, a^{4} b^{2} c^{5} x^{6} - \frac {4}{5} \, a^{5} b c^{5} x^{5} + \frac {1}{4} \, a^{6} c^{5} x^{4} \]

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{10} \, b^{6} c^{5} x^{10} + \frac {4}{9} \, a b^{5} c^{5} x^{9} - \frac {5}{8} \, a^{2} b^{4} c^{5} x^{8} + \frac {5}{6} \, a^{4} b^{2} c^{5} x^{6} - \frac {4}{5} \, a^{5} b c^{5} x^{5} + \frac {1}{4} \, a^{6} c^{5} x^{4} \]

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^5 \, dx=\frac {a^6\,c^5\,x^4}{4}-\frac {4\,a^5\,b\,c^5\,x^5}{5}+\frac {5\,a^4\,b^2\,c^5\,x^6}{6}-\frac {5\,a^2\,b^4\,c^5\,x^8}{8}+\frac {4\,a\,b^5\,c^5\,x^9}{9}-\frac {b^6\,c^5\,x^{10}}{10} \]

[In]

int(x^3*(a*c - b*c*x)^5*(a + b*x),x)

[Out]

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