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 = 0.0776877, 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 \[ -\frac{a^2 c^3 (a-b x)^4}{b}-\frac{c^3 (a-b x)^6}{6 b}+\frac{4 a c^3 (a-b x)^5}{5 b} \]

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)

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Rubi in Sympy [A]  time = 16.9795, size = 44, normalized size = 0.77 \[ - \frac{a^{2} c^{3} \left (a - b x\right )^{4}}{b} + \frac{4 a c^{3} \left (a - b x\right )^{5}}{5 b} - \frac{c^{3} \left (a - b x\right )^{6}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.00531588, size = 68, normalized size = 1.19 \[ c^3 \left (a^5 x-\frac{1}{2} a^4 b x^2-\frac{2}{3} a^3 b^2 x^3+\frac{1}{2} a^2 b^3 x^4+\frac{1}{5} a b^4 x^5-\frac{1}{6} b^5 x^6\right ) \]

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 \[ -{\frac{{b}^{5}{c}^{3}{x}^{6}}{6}}+{\frac{a{b}^{4}{c}^{3}{x}^{5}}{5}}+{\frac{{a}^{2}{b}^{3}{c}^{3}{x}^{4}}{2}}-{\frac{2\,{a}^{3}{c}^{3}{b}^{2}{x}^{3}}{3}}-{\frac{{a}^{4}{c}^{3}b{x}^{2}}{2}}+{a}^{5}{c}^{3}x \]

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 = 1.35376, size = 97, normalized size = 1.7 \[ -\frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{1}{5} \, a b^{4} c^{3} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{3} x^{4} - \frac{2}{3} \, a^{3} b^{2} c^{3} x^{3} - \frac{1}{2} \, a^{4} b c^{3} x^{2} + a^{5} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)^2,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 = 0.17653, size = 1, normalized size = 0.02 \[ -\frac{1}{6} x^{6} c^{3} b^{5} + \frac{1}{5} x^{5} c^{3} b^{4} a + \frac{1}{2} x^{4} c^{3} b^{3} a^{2} - \frac{2}{3} x^{3} c^{3} b^{2} a^{3} - \frac{1}{2} x^{2} c^{3} b a^{4} + x c^{3} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)^2,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 = 0.14738, size = 78, normalized size = 1.37 \[ a^{5} c^{3} x - \frac{a^{4} b c^{3} x^{2}}{2} - \frac{2 a^{3} b^{2} c^{3} x^{3}}{3} + \frac{a^{2} b^{3} c^{3} x^{4}}{2} + \frac{a b^{4} c^{3} x^{5}}{5} - \frac{b^{5} c^{3} x^{6}}{6} \]

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/XCAS [A]  time = 0.204992, size = 97, normalized size = 1.7 \[ -\frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{1}{5} \, a b^{4} c^{3} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{3} x^{4} - \frac{2}{3} \, a^{3} b^{2} c^{3} x^{3} - \frac{1}{2} \, a^{4} b c^{3} x^{2} + a^{5} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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