3.1003 \(\int \frac{a+b x}{\left (\frac{a d}{b}+d x\right )^3} \, dx\)

Optimal. Leaf size=15 \[ -\frac{b^2}{d^3 (a+b x)} \]

[Out]

-(b^2/(d^3*(a + b*x)))

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Rubi [A]  time = 0.0113412, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{b^2}{d^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2/(d^3*(a + b*x)))

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Rubi in Sympy [A]  time = 4.26964, size = 12, normalized size = 0.8 \[ - \frac{b^{2}}{d^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2/(d**3*(a + b*x))

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Mathematica [A]  time = 0.00598944, size = 15, normalized size = 1. \[ -\frac{b^2}{d^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2/(d^3*(a + b*x)))

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Maple [A]  time = 0., size = 16, normalized size = 1.1 \[ -{\frac{{b}^{2}}{{d}^{3} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(a*d/b+d*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.34255, size = 26, normalized size = 1.73 \[ -\frac{b^{2}}{b d^{3} x + a d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2/(b*d^3*x + a*d^3)

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Fricas [A]  time = 0.198634, size = 26, normalized size = 1.73 \[ -\frac{b^{2}}{b d^{3} x + a d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2/(b*d^3*x + a*d^3)

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Sympy [A]  time = 1.24688, size = 19, normalized size = 1.27 \[ - \frac{b^{3}}{a b d^{3} + b^{2} d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**3/(a*b*d**3 + b**2*d**3*x)

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GIAC/XCAS [A]  time = 0.210464, size = 20, normalized size = 1.33 \[ -\frac{b^{2}}{{\left (b x + a\right )} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2/((b*x + a)*d^3)