3.1021 \(\int \frac{(a+b x)^5}{(a c+b c x)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac{(a+b x)^3}{3 b c^3} \]

[Out]

(a + b*x)^3/(3*b*c^3)

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Rubi [A]  time = 0.0120301, 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 \[ \frac{(a+b x)^3}{3 b c^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + b*c*x)^3,x]

[Out]

(a + b*x)^3/(3*b*c^3)

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Rubi in Sympy [A]  time = 4.14893, size = 12, normalized size = 0.71 \[ \frac{\left (a + b x\right )^{3}}{3 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(b*c*x+a*c)**3,x)

[Out]

(a + b*x)**3/(3*b*c**3)

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Mathematica [A]  time = 0.00204821, size = 17, normalized size = 1. \[ \frac{(a+b x)^3}{3 b c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + b*c*x)^3,x]

[Out]

(a + b*x)^3/(3*b*c^3)

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Maple [A]  time = 0., size = 16, normalized size = 0.9 \[{\frac{ \left ( bx+a \right ) ^{3}}{3\,b{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^3,x)

[Out]

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

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Maxima [A]  time = 1.3505, size = 35, normalized size = 2.06 \[ \frac{b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.188471, size = 35, normalized size = 2.06 \[ \frac{b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.214645, size = 29, normalized size = 1.71 \[ \frac{a^{2} x}{c^{3}} + \frac{a b x^{2}}{c^{3}} + \frac{b^{2} x^{3}}{3 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(b*c*x+a*c)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211752, size = 47, normalized size = 2.76 \[ \frac{b^{2} c^{6} x^{3} + 3 \, a b c^{6} x^{2} + 3 \, a^{2} c^{6} x}{3 \, c^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^3,x, algorithm="giac")

[Out]

1/3*(b^2*c^6*x^3 + 3*a*b*c^6*x^2 + 3*a^2*c^6*x)/c^9