3.1014 \(\int \frac{1}{\left (\frac{b c}{d}+b x\right )^3 (c+d x)^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{d^2}{5 b^3 (c+d x)^5} \]

[Out]

-d^2/(5*b^3*(c + d*x)^5)

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Rubi [A]  time = 0.0117367, antiderivative size = 17, 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 \[ -\frac{d^2}{5 b^3 (c+d x)^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(((b*c)/d + b*x)^3*(c + d*x)^3),x]

[Out]

-d^2/(5*b^3*(c + d*x)^5)

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Rubi in Sympy [A]  time = 4.731, size = 15, normalized size = 0.88 \[ - \frac{d^{2}}{5 b^{3} \left (c + d x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*c/d+b*x)**3/(d*x+c)**3,x)

[Out]

-d**2/(5*b**3*(c + d*x)**5)

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Mathematica [A]  time = 0.00910416, size = 17, normalized size = 1. \[ -\frac{d^2}{5 b^3 (c+d x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(((b*c)/d + b*x)^3*(c + d*x)^3),x]

[Out]

-d^2/(5*b^3*(c + d*x)^5)

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[ -{\frac{{d}^{2}}{5\,{b}^{3} \left ( dx+c \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*c/d+b*x)^3/(d*x+c)^3,x)

[Out]

-1/5*d^2/b^3/(d*x+c)^5

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Maxima [A]  time = 1.43506, size = 101, normalized size = 5.94 \[ -\frac{d^{2}}{5 \,{\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + b*c/d)^3*(d*x + c)^3),x, algorithm="maxima")

[Out]

-1/5*d^2/(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^
2 + 5*b^3*c^4*d*x + b^3*c^5)

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Fricas [A]  time = 0.211353, size = 101, normalized size = 5.94 \[ -\frac{d^{2}}{5 \,{\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + b*c/d)^3*(d*x + c)^3),x, algorithm="fricas")

[Out]

-1/5*d^2/(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^
2 + 5*b^3*c^4*d*x + b^3*c^5)

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Sympy [A]  time = 2.15747, size = 83, normalized size = 4.88 \[ - \frac{d^{3}}{5 b^{3} c^{5} d + 25 b^{3} c^{4} d^{2} x + 50 b^{3} c^{3} d^{3} x^{2} + 50 b^{3} c^{2} d^{4} x^{3} + 25 b^{3} c d^{5} x^{4} + 5 b^{3} d^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*c/d+b*x)**3/(d*x+c)**3,x)

[Out]

-d**3/(5*b**3*c**5*d + 25*b**3*c**4*d**2*x + 50*b**3*c**3*d**3*x**2 + 50*b**3*c*
*2*d**4*x**3 + 25*b**3*c*d**5*x**4 + 5*b**3*d**6*x**5)

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GIAC/XCAS [A]  time = 0.206191, size = 20, normalized size = 1.18 \[ -\frac{d^{2}}{5 \,{\left (d x + c\right )}^{5} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + b*c/d)^3*(d*x + c)^3),x, algorithm="giac")

[Out]

-1/5*d^2/((d*x + c)^5*b^3)