3.1004 \(\int \frac{1}{(a+b x) \left (\frac{a d}{b}+d x\right )^3} \, dx\)

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

[Out]

-b^2/(3*d^3*(a + b*x)^3)

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Rubi [A]  time = 0.0110823, 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 \[ -\frac{b^2}{3 d^3 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-b^2/(3*d^3*(a + b*x)^3)

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Rubi in Sympy [A]  time = 4.40689, size = 15, normalized size = 0.88 \[ - \frac{b^{2}}{3 d^{3} \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2/(3*d**3*(a + b*x)**3)

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Mathematica [A]  time = 0.00754392, size = 17, normalized size = 1. \[ -\frac{b^2}{3 d^3 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-b^2/(3*d^3*(a + b*x)^3)

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Maple [A]  time = 0., size = 16, normalized size = 0.9 \[ -{\frac{{b}^{2}}{3\,{d}^{3} \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(a*d/b+d*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.34058, size = 63, normalized size = 3.71 \[ -\frac{b^{2}}{3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199678, size = 63, normalized size = 3.71 \[ -\frac{b^{2}}{3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.65115, size = 53, normalized size = 3.12 \[ - \frac{b^{3}}{3 a^{3} b d^{3} + 9 a^{2} b^{2} d^{3} x + 9 a b^{3} d^{3} x^{2} + 3 b^{4} d^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210949, size = 20, normalized size = 1.18 \[ -\frac{b^{2}}{3 \,{\left (b x + a\right )}^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b^2/((b*x + a)^3*d^3)