3.1038 \(\int (a+b x)^2 (a c-b c x)^3 \, dx\)

Optimal. Leaf size=57 \[ -\frac{a^2 c^3 (a-b x)^4}{b}-\frac{c^3 (a-b x)^6}{6 b}+\frac{4 a c^3 (a-b x)^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0303072, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ -\frac{a^2 c^3 (a-b x)^4}{b}-\frac{c^3 (a-b x)^6}{6 b}+\frac{4 a c^3 (a-b x)^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(a*c - b*c*x)^3,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0029511, size = 68, normalized size = 1.19 \[ c^3 \left (\frac{1}{2} a^2 b^3 x^4-\frac{2}{3} a^3 b^2 x^3-\frac{1}{2} a^4 b x^2+a^5 x+\frac{1}{5} a b^4 x^5-\frac{1}{6} b^5 x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 73, normalized size = 1.3 \begin{align*} -{\frac{{b}^{5}{c}^{3}{x}^{6}}{6}}+{\frac{a{b}^{4}{c}^{3}{x}^{5}}{5}}+{\frac{{a}^{2}{b}^{3}{c}^{3}{x}^{4}}{2}}-{\frac{2\,{a}^{3}{c}^{3}{b}^{2}{x}^{3}}{3}}-{\frac{{a}^{4}{c}^{3}b{x}^{2}}{2}}+{a}^{5}{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.99351, size = 97, normalized size = 1.7 \begin{align*} -\frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{1}{5} \, a b^{4} c^{3} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{3} x^{4} - \frac{2}{3} \, a^{3} b^{2} c^{3} x^{3} - \frac{1}{2} \, a^{4} b c^{3} x^{2} + a^{5} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.34014, size = 154, normalized size = 2.7 \begin{align*} -\frac{1}{6} x^{6} c^{3} b^{5} + \frac{1}{5} x^{5} c^{3} b^{4} a + \frac{1}{2} x^{4} c^{3} b^{3} a^{2} - \frac{2}{3} x^{3} c^{3} b^{2} a^{3} - \frac{1}{2} x^{2} c^{3} b a^{4} + x c^{3} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.076897, size = 78, normalized size = 1.37 \begin{align*} a^{5} c^{3} x - \frac{a^{4} b c^{3} x^{2}}{2} - \frac{2 a^{3} b^{2} c^{3} x^{3}}{3} + \frac{a^{2} b^{3} c^{3} x^{4}}{2} + \frac{a b^{4} c^{3} x^{5}}{5} - \frac{b^{5} c^{3} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05994, size = 97, normalized size = 1.7 \begin{align*} -\frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{1}{5} \, a b^{4} c^{3} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{3} x^{4} - \frac{2}{3} \, a^{3} b^{2} c^{3} x^{3} - \frac{1}{2} \, a^{4} b c^{3} x^{2} + a^{5} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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