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

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

[Out]

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

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Rubi [A]  time = 0.0462839, 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 \[ a^4 c^2 x-\frac{2}{3} a^2 b^2 c^2 x^3+\frac{1}{5} b^4 c^2 x^5 \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac{b^{4} c^{2} x^{5}}{5} + c^{2} \int a^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(-b*c*x+a*c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.00296528, size = 38, normalized size = 1. \[ a^4 c^2 x-\frac{2}{3} a^2 b^2 c^2 x^3+\frac{1}{5} b^4 c^2 x^5 \]

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

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

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

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Maxima [A]  time = 1.34315, size = 46, normalized size = 1.21 \[ \frac{1}{5} \, b^{4} c^{2} x^{5} - \frac{2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^2*(b*x + a)^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

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Fricas [A]  time = 0.200378, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} c^{2} b^{4} - \frac{2}{3} x^{3} c^{2} b^{2} a^{2} + x c^{2} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^2*(b*x + a)^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

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Sympy [A]  time = 0.120285, size = 36, normalized size = 0.95 \[ a^{4} c^{2} x - \frac{2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac{b^{4} c^{2} x^{5}}{5} \]

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

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GIAC/XCAS [A]  time = 0.201966, size = 46, normalized size = 1.21 \[ \frac{1}{5} \, b^{4} c^{2} x^{5} - \frac{2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^2*(b*x + a)^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