3.1039 \(\int (a+b x)^2 (a c-b c x)^2 \, dx\)

Optimal. Leaf size=38 \[ -\frac{2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac{1}{5} b^4 c^2 x^5 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0171346, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {41, 194} \[ -\frac{2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac{1}{5} b^4 c^2 x^5 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(a*c - b*c*x)^2,x]

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0016806, size = 38, normalized size = 1. \[ -\frac{2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac{1}{5} b^4 c^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 35, normalized size = 0.9 \begin{align*}{a}^{4}{c}^{2}x-{\frac{2\,{a}^{2}{b}^{2}{c}^{2}{x}^{3}}{3}}+{\frac{{b}^{4}{c}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02374, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{5} \, b^{4} c^{2} x^{5} - \frac{2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.28935, size = 69, normalized size = 1.82 \begin{align*} \frac{1}{5} x^{5} c^{2} b^{4} - \frac{2}{3} x^{3} c^{2} b^{2} a^{2} + x c^{2} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.067727, size = 36, normalized size = 0.95 \begin{align*} a^{4} c^{2} x - \frac{2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac{b^{4} c^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.0603, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{5} \, b^{4} c^{2} x^{5} - \frac{2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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