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

Optimal. Leaf size=55 \[ \frac {1}{2} a^4 c^3 x^2-\frac {2}{3} a^3 b c^3 x^3+\frac {2}{5} a b^3 c^3 x^5-\frac {1}{6} b^4 c^3 x^6 \]

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Rubi [A]  time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {75} \begin {gather*} -\frac {2}{3} a^3 b c^3 x^3+\frac {1}{2} a^4 c^3 x^2+\frac {2}{5} a b^3 c^3 x^5-\frac {1}{6} b^4 c^3 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 47, normalized size = 0.85 \begin {gather*} c^3 \left (\frac {a^4 x^2}{2}-\frac {2}{3} a^3 b x^3+\frac {2}{5} a b^3 x^5-\frac {1}{6} b^4 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 47, normalized size = 0.85 \begin {gather*} -\frac {1}{6} x^{6} c^{3} b^{4} + \frac {2}{5} x^{5} c^{3} b^{3} a - \frac {2}{3} x^{3} c^{3} b a^{3} + \frac {1}{2} x^{2} c^{3} a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.93, size = 47, normalized size = 0.85 \begin {gather*} -\frac {1}{6} \, b^{4} c^{3} x^{6} + \frac {2}{5} \, a b^{3} c^{3} x^{5} - \frac {2}{3} \, a^{3} b c^{3} x^{3} + \frac {1}{2} \, a^{4} c^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 48, normalized size = 0.87 \begin {gather*} -\frac {1}{6} b^{4} c^{3} x^{6}+\frac {2}{5} a \,b^{3} c^{3} x^{5}-\frac {2}{3} a^{3} b \,c^{3} x^{3}+\frac {1}{2} a^{4} c^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 47, normalized size = 0.85 \begin {gather*} -\frac {1}{6} \, b^{4} c^{3} x^{6} + \frac {2}{5} \, a b^{3} c^{3} x^{5} - \frac {2}{3} \, a^{3} b c^{3} x^{3} + \frac {1}{2} \, a^{4} c^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 47, normalized size = 0.85 \begin {gather*} \frac {a^4\,c^3\,x^2}{2}-\frac {2\,a^3\,b\,c^3\,x^3}{3}+\frac {2\,a\,b^3\,c^3\,x^5}{5}-\frac {b^4\,c^3\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a*c - b*c*x)^3*(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.08, size = 53, normalized size = 0.96 \begin {gather*} \frac {a^{4} c^{3} x^{2}}{2} - \frac {2 a^{3} b c^{3} x^{3}}{3} + \frac {2 a b^{3} c^{3} x^{5}}{5} - \frac {b^{4} c^{3} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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