3.1006 \(\int \frac{1}{(a+b x)^3 \left (\frac{a d}{b}+d x\right )^3} \, dx\)

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

[Out]

-b^2/(5*d^3*(a + b*x)^5)

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Rubi [A]  time = 0.0107054, 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 \[ -\frac{b^2}{5 d^3 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-b^2/(5*d^3*(a + b*x)^5)

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Rubi in Sympy [A]  time = 4.50113, size = 15, normalized size = 0.88 \[ - \frac{b^{2}}{5 d^{3} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2/(5*d**3*(a + b*x)**5)

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Mathematica [A]  time = 0.00959981, size = 17, normalized size = 1. \[ -\frac{b^2}{5 d^3 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-b^2/(5*d^3*(a + b*x)^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34774, size = 101, normalized size = 5.94 \[ -\frac{b^{2}}{5 \,{\left (b^{5} d^{3} x^{5} + 5 \, a b^{4} d^{3} x^{4} + 10 \, a^{2} b^{3} d^{3} x^{3} + 10 \, a^{3} b^{2} d^{3} x^{2} + 5 \, a^{4} b d^{3} x + a^{5} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.194533, size = 101, normalized size = 5.94 \[ -\frac{b^{2}}{5 \,{\left (b^{5} d^{3} x^{5} + 5 \, a b^{4} d^{3} x^{4} + 10 \, a^{2} b^{3} d^{3} x^{3} + 10 \, a^{3} b^{2} d^{3} x^{2} + 5 \, a^{4} b d^{3} x + a^{5} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.14561, size = 83, normalized size = 4.88 \[ - \frac{b^{3}}{5 a^{5} b d^{3} + 25 a^{4} b^{2} d^{3} x + 50 a^{3} b^{3} d^{3} x^{2} + 50 a^{2} b^{4} d^{3} x^{3} + 25 a b^{5} d^{3} x^{4} + 5 b^{6} d^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213264, size = 20, normalized size = 1.18 \[ -\frac{b^{2}}{5 \,{\left (b x + a\right )}^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*b^2/((b*x + a)^5*d^3)