3.1012 \(\int \frac{1}{\left (\frac{b c}{d}+b x\right ) (c+d x)^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{3 b (c+d x)^3} \]

[Out]

-1/(3*b*(c + d*x)^3)

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Rubi [A]  time = 0.00909648, antiderivative size = 14, 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 \[ -\frac{1}{3 b (c+d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*b*(c + d*x)^3)

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Rubi in Sympy [A]  time = 4.22088, size = 12, normalized size = 0.86 \[ - \frac{1}{3 b \left (c + d x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*b*(c + d*x)**3)

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Mathematica [A]  time = 0.00721082, size = 14, normalized size = 1. \[ -\frac{1}{3 b (c+d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*b*(c + d*x)^3)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \[ -{\frac{1}{3\,b \left ( dx+c \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*c/d+b*x)/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.34919, size = 49, normalized size = 3.5 \[ -\frac{1}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.197369, size = 49, normalized size = 3.5 \[ -\frac{1}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.58611, size = 44, normalized size = 3.14 \[ - \frac{d}{3 b c^{3} d + 9 b c^{2} d^{2} x + 9 b c d^{3} x^{2} + 3 b d^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(3*b*c**3*d + 9*b*c**2*d**2*x + 9*b*c*d**3*x**2 + 3*b*d**4*x**3)

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GIAC/XCAS [A]  time = 0.205704, size = 16, normalized size = 1.14 \[ -\frac{1}{3 \,{\left (d x + c\right )}^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/((d*x + c)^3*b)