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

Optimal. Leaf size=59 \[ -\frac{a^2 c^5 (a-b x)^6}{3 b^2}-\frac{c^5 (a-b x)^8}{8 b^2}+\frac{3 a c^5 (a-b x)^7}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.100716, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a^2 c^5 (a-b x)^6}{3 b^2}-\frac{c^5 (a-b x)^8}{8 b^2}+\frac{3 a c^5 (a-b x)^7}{7 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.4809, size = 51, normalized size = 0.86 \[ - \frac{a^{2} c^{5} \left (a - b x\right )^{6}}{3 b^{2}} + \frac{3 a c^{5} \left (a - b x\right )^{7}}{7 b^{2}} - \frac{c^{5} \left (a - b x\right )^{8}}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00435593, size = 73, normalized size = 1.24 \[ c^5 \left (\frac{a^6 x^2}{2}-\frac{4}{3} a^5 b x^3+\frac{5}{4} a^4 b^2 x^4-\frac{5}{6} a^2 b^4 x^6+\frac{4}{7} a b^5 x^7-\frac{1}{8} b^6 x^8\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 76, normalized size = 1.3 \[ -{\frac{{b}^{6}{c}^{5}{x}^{8}}{8}}+{\frac{4\,a{b}^{5}{c}^{5}{x}^{7}}{7}}-{\frac{5\,{a}^{2}{c}^{5}{b}^{4}{x}^{6}}{6}}+{\frac{5\,{a}^{4}{c}^{5}{b}^{2}{x}^{4}}{4}}-{\frac{4\,{a}^{5}b{c}^{5}{x}^{3}}{3}}+{\frac{{a}^{6}{c}^{5}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

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

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Maxima [A]  time = 1.34909, size = 101, normalized size = 1.71 \[ -\frac{1}{8} \, b^{6} c^{5} x^{8} + \frac{4}{7} \, a b^{5} c^{5} x^{7} - \frac{5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac{5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac{4}{3} \, a^{5} b c^{5} x^{3} + \frac{1}{2} \, a^{6} c^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.184253, size = 1, normalized size = 0.02 \[ -\frac{1}{8} x^{8} c^{5} b^{6} + \frac{4}{7} x^{7} c^{5} b^{5} a - \frac{5}{6} x^{6} c^{5} b^{4} a^{2} + \frac{5}{4} x^{4} c^{5} b^{2} a^{4} - \frac{4}{3} x^{3} c^{5} b a^{5} + \frac{1}{2} x^{2} c^{5} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.156608, size = 87, normalized size = 1.47 \[ \frac{a^{6} c^{5} x^{2}}{2} - \frac{4 a^{5} b c^{5} x^{3}}{3} + \frac{5 a^{4} b^{2} c^{5} x^{4}}{4} - \frac{5 a^{2} b^{4} c^{5} x^{6}}{6} + \frac{4 a b^{5} c^{5} x^{7}}{7} - \frac{b^{6} c^{5} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234386, size = 101, normalized size = 1.71 \[ -\frac{1}{8} \, b^{6} c^{5} x^{8} + \frac{4}{7} \, a b^{5} c^{5} x^{7} - \frac{5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac{5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac{4}{3} \, a^{5} b c^{5} x^{3} + \frac{1}{2} \, a^{6} c^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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