3.1022 \(\int \frac{(a+b x)^5}{(a c+b c x)^4} \, dx\)

Optimal. Leaf size=18 \[ \frac{a x}{c^4}+\frac{b x^2}{2 c^4} \]

[Out]

(a*x)/c^4 + (b*x^2)/(2*c^4)

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Rubi [A]  time = 0.0129734, antiderivative size = 18, 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 \[ \frac{a x}{c^4}+\frac{b x^2}{2 c^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

(a*x)/c^4 + (b*x^2)/(2*c^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \int x\, dx}{c^{4}} + \frac{\int a\, dx}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(b*c*x+a*c)**4,x)

[Out]

b*Integral(x, x)/c**4 + Integral(a, x)/c**4

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Mathematica [A]  time = 0.00101563, size = 16, normalized size = 0.89 \[ \frac{a x+\frac{b x^2}{2}}{c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

(a*x + (b*x^2)/2)/c^4

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Maple [A]  time = 0., size = 15, normalized size = 0.8 \[{\frac{1}{{c}^{4}} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^4,x)

[Out]

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

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Maxima [A]  time = 1.32772, size = 20, normalized size = 1.11 \[ \frac{b x^{2} + 2 \, a x}{2 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^4,x, algorithm="maxima")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

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Fricas [A]  time = 0.191292, size = 20, normalized size = 1.11 \[ \frac{b x^{2} + 2 \, a x}{2 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^4,x, algorithm="fricas")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

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Sympy [A]  time = 0.202368, size = 15, normalized size = 0.83 \[ \frac{a x}{c^{4}} + \frac{b x^{2}}{2 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(b*c*x+a*c)**4,x)

[Out]

a*x/c**4 + b*x**2/(2*c**4)

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GIAC/XCAS [A]  time = 0.208434, size = 28, normalized size = 1.56 \[ \frac{b c^{4} x^{2} + 2 \, a c^{4} x}{2 \, c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^4,x, algorithm="giac")

[Out]

1/2*(b*c^4*x^2 + 2*a*c^4*x)/c^8