3.1029 \(\int (a+b x) (a c-b c x)^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{c^2 (a-b x)^4}{4 b}-\frac{2 a c^2 (a-b x)^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0170622, antiderivative size = 38, 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} \[ \frac{c^2 (a-b x)^4}{4 b}-\frac{2 a c^2 (a-b x)^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0023617, size = 42, normalized size = 1.11 \[ c^2 \left (-\frac{1}{2} a^2 b x^2+a^3 x-\frac{1}{3} a b^2 x^3+\frac{b^3 x^4}{4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^2*(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 = 0.001, size = 45, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}{c}^{2}{x}^{4}}{4}}-{\frac{a{b}^{2}{c}^{2}{x}^{3}}{3}}-{\frac{{a}^{2}{c}^{2}b{x}^{2}}{2}}+{a}^{3}{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00264, size = 59, normalized size = 1.55 \begin{align*} \frac{1}{4} \, b^{3} c^{2} x^{4} - \frac{1}{3} \, a b^{2} c^{2} x^{3} - \frac{1}{2} \, a^{2} b c^{2} x^{2} + a^{3} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.31821, size = 93, normalized size = 2.45 \begin{align*} \frac{1}{4} x^{4} c^{2} b^{3} - \frac{1}{3} x^{3} c^{2} b^{2} a - \frac{1}{2} x^{2} c^{2} b a^{2} + x c^{2} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.065695, size = 46, normalized size = 1.21 \begin{align*} a^{3} c^{2} x - \frac{a^{2} b c^{2} x^{2}}{2} - \frac{a b^{2} c^{2} x^{3}}{3} + \frac{b^{3} c^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05925, size = 59, normalized size = 1.55 \begin{align*} \frac{1}{4} \, b^{3} c^{2} x^{4} - \frac{1}{3} \, a b^{2} c^{2} x^{3} - \frac{1}{2} \, a^{2} b c^{2} x^{2} + a^{3} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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