3.1040 \(\int (a+b x)^2 (a c-b c x) \, dx\)

Optimal. Leaf size=32 \[ \frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0149207, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(a*c - b*c*x),x]

[Out]

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

Mathematica [A]  time = 0.001673, size = 40, normalized size = 1.25 \[ c \left (\frac{1}{2} a^2 b x^2+a^3 x-\frac{1}{3} a b^2 x^3-\frac{1}{4} b^3 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 37, normalized size = 1.2 \begin{align*} -{\frac{{b}^{3}c{x}^{4}}{4}}-{\frac{a{b}^{2}c{x}^{3}}{3}}+{\frac{{a}^{2}bc{x}^{2}}{2}}+{a}^{3}cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01329, size = 49, normalized size = 1.53 \begin{align*} -\frac{1}{4} \, b^{3} c x^{4} - \frac{1}{3} \, a b^{2} c x^{3} + \frac{1}{2} \, a^{2} b c x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.35025, size = 84, normalized size = 2.62 \begin{align*} -\frac{1}{4} x^{4} c b^{3} - \frac{1}{3} x^{3} c b^{2} a + \frac{1}{2} x^{2} c b a^{2} + x c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.066783, size = 39, normalized size = 1.22 \begin{align*} a^{3} c x + \frac{a^{2} b c x^{2}}{2} - \frac{a b^{2} c x^{3}}{3} - \frac{b^{3} c x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05419, size = 49, normalized size = 1.53 \begin{align*} -\frac{1}{4} \, b^{3} c x^{4} - \frac{1}{3} \, a b^{2} c x^{3} + \frac{1}{2} \, a^{2} b c x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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