\(\int x (a+b x) (a c-b c x)^4 \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 59 \[ \int x (a+b x) (a c-b c x)^4 \, dx=-\frac {2 a^2 c^4 (a-b x)^5}{5 b^2}+\frac {a c^4 (a-b x)^6}{2 b^2}-\frac {c^4 (a-b x)^7}{7 b^2} \]

[Out]

-2/5*a^2*c^4*(-b*x+a)^5/b^2+1/2*a*c^4*(-b*x+a)^6/b^2-1/7*c^4*(-b*x+a)^7/b^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {76} \[ \int x (a+b x) (a c-b c x)^4 \, dx=-\frac {2 a^2 c^4 (a-b x)^5}{5 b^2}-\frac {c^4 (a-b x)^7}{7 b^2}+\frac {a c^4 (a-b x)^6}{2 b^2} \]

[In]

Int[x*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.44 \[ \int x (a+b x) (a c-b c x)^4 \, dx=\frac {1}{2} a^5 c^4 x^2-a^4 b c^4 x^3+\frac {1}{2} a^3 b^2 c^4 x^4+\frac {2}{5} a^2 b^3 c^4 x^5-\frac {1}{2} a b^4 c^4 x^6+\frac {1}{7} b^5 c^4 x^7 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03

method	result	size
gosper	\(\frac {x^{2} \left (10 b^{5} x^{5}-35 a \,b^{4} x^{4}+28 a^{2} b^{3} x^{3}+35 a^{3} b^{2} x^{2}-70 a^{4} b x +35 a^{5}\right ) c^{4}}{70}\)	\(61\)
default	\(\frac {1}{7} b^{5} c^{4} x^{7}-\frac {1}{2} a \,b^{4} c^{4} x^{6}+\frac {2}{5} a^{2} c^{4} b^{3} x^{5}+\frac {1}{2} a^{3} c^{4} b^{2} x^{4}-a^{4} c^{4} b \,x^{3}+\frac {1}{2} a^{5} c^{4} x^{2}\)	\(76\)
norman	\(\frac {1}{7} b^{5} c^{4} x^{7}-\frac {1}{2} a \,b^{4} c^{4} x^{6}+\frac {2}{5} a^{2} c^{4} b^{3} x^{5}+\frac {1}{2} a^{3} c^{4} b^{2} x^{4}-a^{4} c^{4} b \,x^{3}+\frac {1}{2} a^{5} c^{4} x^{2}\)	\(76\)
risch	\(\frac {1}{7} b^{5} c^{4} x^{7}-\frac {1}{2} a \,b^{4} c^{4} x^{6}+\frac {2}{5} a^{2} c^{4} b^{3} x^{5}+\frac {1}{2} a^{3} c^{4} b^{2} x^{4}-a^{4} c^{4} b \,x^{3}+\frac {1}{2} a^{5} c^{4} x^{2}\)	\(76\)
parallelrisch	\(\frac {1}{7} b^{5} c^{4} x^{7}-\frac {1}{2} a \,b^{4} c^{4} x^{6}+\frac {2}{5} a^{2} c^{4} b^{3} x^{5}+\frac {1}{2} a^{3} c^{4} b^{2} x^{4}-a^{4} c^{4} b \,x^{3}+\frac {1}{2} a^{5} c^{4} x^{2}\)	\(76\)

[In]

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

[Out]

1/70*x^2*(10*b^5*x^5-35*a*b^4*x^4+28*a^2*b^3*x^3+35*a^3*b^2*x^2-70*a^4*b*x+35*a^5)*c^4

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^4 \, dx=\frac {1}{7} \, b^{5} c^{4} x^{7} - \frac {1}{2} \, a b^{4} c^{4} x^{6} + \frac {2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac {1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac {1}{2} \, a^{5} c^{4} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int x (a+b x) (a c-b c x)^4 \, dx=\frac {a^{5} c^{4} x^{2}}{2} - a^{4} b c^{4} x^{3} + \frac {a^{3} b^{2} c^{4} x^{4}}{2} + \frac {2 a^{2} b^{3} c^{4} x^{5}}{5} - \frac {a b^{4} c^{4} x^{6}}{2} + \frac {b^{5} c^{4} x^{7}}{7} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^4 \, dx=\frac {1}{7} \, b^{5} c^{4} x^{7} - \frac {1}{2} \, a b^{4} c^{4} x^{6} + \frac {2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac {1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac {1}{2} \, a^{5} c^{4} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^4 \, dx=\frac {1}{7} \, b^{5} c^{4} x^{7} - \frac {1}{2} \, a b^{4} c^{4} x^{6} + \frac {2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac {1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac {1}{2} \, a^{5} c^{4} x^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int x (a+b x) (a c-b c x)^4 \, dx=\frac {a^5\,c^4\,x^2}{2}-a^4\,b\,c^4\,x^3+\frac {a^3\,b^2\,c^4\,x^4}{2}+\frac {2\,a^2\,b^3\,c^4\,x^5}{5}-\frac {a\,b^4\,c^4\,x^6}{2}+\frac {b^5\,c^4\,x^7}{7} \]

[In]

int(x*(a*c - b*c*x)^4*(a + b*x),x)

[Out]

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