\(\int x^2 (a+b x) (a c-b c x)^3 \, dx\) [2]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=\frac {1}{3} a^4 c^3 x^3-\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7 \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, 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} \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=\frac {1}{3} a^4 c^3 x^3-\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7 \]

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=c^3 \left (\frac {a^4 x^3}{3}-\frac {1}{2} a^3 b x^4+\frac {1}{3} a b^3 x^6-\frac {b^4 x^7}{7}\right ) \]

[In]

Integrate[x^2*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71

method	result	size
gosper	\(\frac {x^{3} \left (-6 b^{4} x^{4}+14 a \,b^{3} x^{3}-21 a^{3} b x +14 a^{4}\right ) c^{3}}{42}\)	\(39\)
default	\(\frac {1}{3} a^{4} c^{3} x^{3}-\frac {1}{2} a^{3} b \,c^{3} x^{4}+\frac {1}{3} a \,b^{3} c^{3} x^{6}-\frac {1}{7} b^{4} c^{3} x^{7}\)	\(48\)
norman	\(\frac {1}{3} a^{4} c^{3} x^{3}-\frac {1}{2} a^{3} b \,c^{3} x^{4}+\frac {1}{3} a \,b^{3} c^{3} x^{6}-\frac {1}{7} b^{4} c^{3} x^{7}\)	\(48\)
risch	\(\frac {1}{3} a^{4} c^{3} x^{3}-\frac {1}{2} a^{3} b \,c^{3} x^{4}+\frac {1}{3} a \,b^{3} c^{3} x^{6}-\frac {1}{7} b^{4} c^{3} x^{7}\)	\(48\)
parallelrisch	\(\frac {1}{3} a^{4} c^{3} x^{3}-\frac {1}{2} a^{3} b \,c^{3} x^{4}+\frac {1}{3} a \,b^{3} c^{3} x^{6}-\frac {1}{7} b^{4} c^{3} x^{7}\)	\(48\)

[In]

int(x^2*(b*x+a)*(-b*c*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

1/42*x^3*(-6*b^4*x^4+14*a*b^3*x^3-21*a^3*b*x+14*a^4)*c^3

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{7} \, b^{4} c^{3} x^{7} + \frac {1}{3} \, a b^{3} c^{3} x^{6} - \frac {1}{2} \, a^{3} b c^{3} x^{4} + \frac {1}{3} \, a^{4} c^{3} x^{3} \]

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=\frac {a^{4} c^{3} x^{3}}{3} - \frac {a^{3} b c^{3} x^{4}}{2} + \frac {a b^{3} c^{3} x^{6}}{3} - \frac {b^{4} c^{3} x^{7}}{7} \]

[In]

integrate(x**2*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{7} \, b^{4} c^{3} x^{7} + \frac {1}{3} \, a b^{3} c^{3} x^{6} - \frac {1}{2} \, a^{3} b c^{3} x^{4} + \frac {1}{3} \, a^{4} c^{3} x^{3} \]

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=-\frac {1}{7} \, b^{4} c^{3} x^{7} + \frac {1}{3} \, a b^{3} c^{3} x^{6} - \frac {1}{2} \, a^{3} b c^{3} x^{4} + \frac {1}{3} \, a^{4} c^{3} x^{3} \]

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int x^2 (a+b x) (a c-b c x)^3 \, dx=\frac {a^4\,c^3\,x^3}{3}-\frac {a^3\,b\,c^3\,x^4}{2}+\frac {a\,b^3\,c^3\,x^6}{3}-\frac {b^4\,c^3\,x^7}{7} \]

[In]

int(x^2*(a*c - b*c*x)^3*(a + b*x),x)

[Out]

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