3.28 \(\int x^2 (a+b x) (a c-b c x)^5 \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 c^5 (a-b x)^6}{3 b^3}+\frac{5 a^2 c^5 (a-b x)^7}{7 b^3}+\frac{c^5 (a-b x)^9}{9 b^3}-\frac{a c^5 (a-b x)^8}{2 b^3} \]

[Out]

-(a^3*c^5*(a - b*x)^6)/(3*b^3) + (5*a^2*c^5*(a - b*x)^7)/(7*b^3) - (a*c^5*(a - b
*x)^8)/(2*b^3) + (c^5*(a - b*x)^9)/(9*b^3)

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Rubi [A]  time = 0.132133, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^3 c^5 (a-b x)^6}{3 b^3}+\frac{5 a^2 c^5 (a-b x)^7}{7 b^3}+\frac{c^5 (a-b x)^9}{9 b^3}-\frac{a c^5 (a-b x)^8}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

-(a^3*c^5*(a - b*x)^6)/(3*b^3) + (5*a^2*c^5*(a - b*x)^7)/(7*b^3) - (a*c^5*(a - b
*x)^8)/(2*b^3) + (c^5*(a - b*x)^9)/(9*b^3)

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Rubi in Sympy [A]  time = 34.0316, size = 78, normalized size = 0.98 \[ \frac{a^{6} c^{5} x^{3}}{3} - a^{5} b c^{5} x^{4} + a^{4} b^{2} c^{5} x^{5} - \frac{5 a^{2} b^{4} c^{5} x^{7}}{7} + \frac{a b^{5} c^{5} x^{8}}{2} - \frac{b^{6} c^{5} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

a**6*c**5*x**3/3 - a**5*b*c**5*x**4 + a**4*b**2*c**5*x**5 - 5*a**2*b**4*c**5*x**
7/7 + a*b**5*c**5*x**8/2 - b**6*c**5*x**9/9

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Mathematica [A]  time = 0.00512101, size = 68, normalized size = 0.85 \[ c^5 \left (\frac{a^6 x^3}{3}-a^5 b x^4+a^4 b^2 x^5-\frac{5}{7} a^2 b^4 x^7+\frac{1}{2} a b^5 x^8-\frac{1}{9} b^6 x^9\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

c^5*((a^6*x^3)/3 - a^5*b*x^4 + a^4*b^2*x^5 - (5*a^2*b^4*x^7)/7 + (a*b^5*x^8)/2 -
(b^6*x^9)/9)

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Maple [A]  time = 0.001, size = 75, normalized size = 0.9 \[ -{\frac{{b}^{6}{c}^{5}{x}^{9}}{9}}+{\frac{a{b}^{5}{c}^{5}{x}^{8}}{2}}-{\frac{5\,{a}^{2}{c}^{5}{b}^{4}{x}^{7}}{7}}+{a}^{4}{c}^{5}{b}^{2}{x}^{5}-{a}^{5}b{c}^{5}{x}^{4}+{\frac{{a}^{6}{c}^{5}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

-1/9*b^6*c^5*x^9+1/2*a*b^5*c^5*x^8-5/7*a^2*c^5*b^4*x^7+a^4*c^5*b^2*x^5-a^5*b*c^5
*x^4+1/3*a^6*c^5*x^3

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Maxima [A]  time = 1.3483, size = 100, normalized size = 1.25 \[ -\frac{1}{9} \, b^{6} c^{5} x^{9} + \frac{1}{2} \, a b^{5} c^{5} x^{8} - \frac{5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac{1}{3} \, a^{6} c^{5} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)*x^2,x, algorithm="maxima")

[Out]

-1/9*b^6*c^5*x^9 + 1/2*a*b^5*c^5*x^8 - 5/7*a^2*b^4*c^5*x^7 + a^4*b^2*c^5*x^5 - a
^5*b*c^5*x^4 + 1/3*a^6*c^5*x^3

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Fricas [A]  time = 0.181328, size = 1, normalized size = 0.01 \[ -\frac{1}{9} x^{9} c^{5} b^{6} + \frac{1}{2} x^{8} c^{5} b^{5} a - \frac{5}{7} x^{7} c^{5} b^{4} a^{2} + x^{5} c^{5} b^{2} a^{4} - x^{4} c^{5} b a^{5} + \frac{1}{3} x^{3} c^{5} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)*x^2,x, algorithm="fricas")

[Out]

-1/9*x^9*c^5*b^6 + 1/2*x^8*c^5*b^5*a - 5/7*x^7*c^5*b^4*a^2 + x^5*c^5*b^2*a^4 - x
^4*c^5*b*a^5 + 1/3*x^3*c^5*a^6

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Sympy [A]  time = 0.158462, size = 78, normalized size = 0.98 \[ \frac{a^{6} c^{5} x^{3}}{3} - a^{5} b c^{5} x^{4} + a^{4} b^{2} c^{5} x^{5} - \frac{5 a^{2} b^{4} c^{5} x^{7}}{7} + \frac{a b^{5} c^{5} x^{8}}{2} - \frac{b^{6} c^{5} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

a**6*c**5*x**3/3 - a**5*b*c**5*x**4 + a**4*b**2*c**5*x**5 - 5*a**2*b**4*c**5*x**
7/7 + a*b**5*c**5*x**8/2 - b**6*c**5*x**9/9

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GIAC/XCAS [A]  time = 0.255371, size = 100, normalized size = 1.25 \[ -\frac{1}{9} \, b^{6} c^{5} x^{9} + \frac{1}{2} \, a b^{5} c^{5} x^{8} - \frac{5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac{1}{3} \, a^{6} c^{5} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)*x^2,x, algorithm="giac")

[Out]

-1/9*b^6*c^5*x^9 + 1/2*a*b^5*c^5*x^8 - 5/7*a^2*b^4*c^5*x^7 + a^4*b^2*c^5*x^5 - a
^5*b*c^5*x^4 + 1/3*a^6*c^5*x^3