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

Optimal. Leaf size=55 \[ \frac {1}{3} a^4 c^3 x^3-\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7 \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \begin {gather*} -\frac {1}{2} a^3 b c^3 x^4+\frac {1}{3} a^4 c^3 x^3+\frac {1}{3} a b^3 c^3 x^6-\frac {1}{7} b^4 c^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[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 75

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*} \int x^2 (a+b x) (a c-b c x)^3 \, dx &=\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]  time = 0.00, size = 47, normalized size = 0.85 \begin {gather*} 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 {1}{7} b^4 x^7\right ) \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (a+b x) (a c-b c x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.22, size = 47, normalized size = 0.85 \begin {gather*} -\frac {1}{7} x^{7} c^{3} b^{4} + \frac {1}{3} x^{6} c^{3} b^{3} a - \frac {1}{2} x^{4} c^{3} b a^{3} + \frac {1}{3} x^{3} c^{3} a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.07, size = 47, normalized size = 0.85 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 48, normalized size = 0.87 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 47, normalized size = 0.85 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 0.31, size = 47, normalized size = 0.85 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 49, normalized size = 0.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________