3.1009 \(\int \frac{\left (\frac{b c}{d}+b x\right )^3}{(c+d x)^3} \, dx\)

Optimal. Leaf size=8 \[ \frac{b^3 x}{d^3} \]

[Out]

(b^3*x)/d^3

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Rubi [A]  time = 0.00696827, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b^3 x}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^3*x)/d^3

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Rubi in Sympy [A]  time = 4.00453, size = 7, normalized size = 0.88 \[ \frac{b^{3} x}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*x/d**3

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Mathematica [A]  time = 0.00057373, size = 8, normalized size = 1. \[ \frac{b^3 x}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^3*x)/d^3

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Maple [A]  time = 0., size = 9, normalized size = 1.1 \[{\frac{{b}^{3}x}{{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*c/d+b*x)^3/(d*x+c)^3,x)

[Out]

b^3*x/d^3

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Maxima [A]  time = 1.35011, size = 11, normalized size = 1.38 \[ \frac{b^{3} x}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*x/d^3

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Fricas [A]  time = 0.19321, size = 11, normalized size = 1.38 \[ \frac{b^{3} x}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*x/d^3

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Sympy [A]  time = 0.201996, size = 7, normalized size = 0.88 \[ \frac{b^{3} x}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*x/d**3

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GIAC/XCAS [A]  time = 0.207654, size = 11, normalized size = 1.38 \[ \frac{b^{3} x}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*x/d^3