3.1020 \(\int \frac{(a+b x)^5}{(a c+b c x)^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{(a+b x)^4}{4 b c^2} \]

[Out]

(a + b*x)^4/(4*b*c^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0116, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^4}{4 b c^2} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^4/(4*b*c^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.35229, size = 12, normalized size = 0.71 \[ \frac{\left (a + b x\right )^{4}}{4 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**4/(4*b*c**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00229364, size = 17, normalized size = 1. \[ \frac{(a+b x)^4}{4 b c^2} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^4/(4*b*c^2)

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[{\frac{ \left ( bx+a \right ) ^{4}}{4\,b{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^2,x)

[Out]

1/4*(b*x+a)^4/b/c^2

_______________________________________________________________________________________

Maxima [A]  time = 1.33756, size = 50, normalized size = 2.94 \[ \frac{b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

_______________________________________________________________________________________

Fricas [A]  time = 0.188796, size = 50, normalized size = 2.94 \[ \frac{b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

_______________________________________________________________________________________

Sympy [A]  time = 0.204478, size = 46, normalized size = 2.71 \[ \frac{a^{3} x}{c^{2}} + \frac{3 a^{2} b x^{2}}{2 c^{2}} + \frac{a b^{2} x^{3}}{c^{2}} + \frac{b^{3} x^{4}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205819, size = 24, normalized size = 1.41 \[ \frac{{\left (b c x + a c\right )}^{4}}{4 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*c*x + a*c)^4/(b*c^6)