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

Optimal. Leaf size=38 \[ \frac{c^5 (a-b x)^7}{7 b}-\frac{a c^5 (a-b x)^6}{3 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0387781, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{c^5 (a-b x)^7}{7 b}-\frac{a c^5 (a-b x)^6}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.9502, size = 27, normalized size = 0.71 \[ - \frac{a c^{5} \left (a - b x\right )^{6}}{3 b} + \frac{c^{5} \left (a - b x\right )^{7}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00396523, size = 64, normalized size = 1.68 \[ c^5 \left (a^6 x-2 a^5 b x^2+\frac{5}{3} a^4 b^2 x^3-a^2 b^4 x^5+\frac{2}{3} a b^5 x^6-\frac{1}{7} b^6 x^7\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.001, size = 73, normalized size = 1.9 \[ -{\frac{{b}^{6}{c}^{5}{x}^{7}}{7}}+{\frac{2\,a{b}^{5}{c}^{5}{x}^{6}}{3}}-{a}^{2}{b}^{4}{c}^{5}{x}^{5}+{\frac{5\,{a}^{4}{b}^{2}{c}^{5}{x}^{3}}{3}}-2\,{a}^{5}b{c}^{5}{x}^{2}+{a}^{6}{c}^{5}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34342, size = 97, normalized size = 2.55 \[ -\frac{1}{7} \, b^{6} c^{5} x^{7} + \frac{2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac{5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.185656, size = 1, normalized size = 0.03 \[ -\frac{1}{7} x^{7} c^{5} b^{6} + \frac{2}{3} x^{6} c^{5} b^{5} a - x^{5} c^{5} b^{4} a^{2} + \frac{5}{3} x^{3} c^{5} b^{2} a^{4} - 2 x^{2} c^{5} b a^{5} + x c^{5} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.157425, size = 78, normalized size = 2.05 \[ a^{6} c^{5} x - 2 a^{5} b c^{5} x^{2} + \frac{5 a^{4} b^{2} c^{5} x^{3}}{3} - a^{2} b^{4} c^{5} x^{5} + \frac{2 a b^{5} c^{5} x^{6}}{3} - \frac{b^{6} c^{5} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.24946, size = 97, normalized size = 2.55 \[ -\frac{1}{7} \, b^{6} c^{5} x^{7} + \frac{2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac{5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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