3.761 \(\int \frac{(a+b x)^3}{\left (a^2-b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{2 b (a-b x)^2} \]

[Out]

1/(2*b*(a - b*x)^2)

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Rubi [A]  time = 0.0189244, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{2 b (a-b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/(a^2 - b^2*x^2)^3,x]

[Out]

1/(2*b*(a - b*x)^2)

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Rubi in Sympy [A]  time = 4.50872, size = 10, normalized size = 0.67 \[ \frac{1}{2 b \left (a - b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(-b**2*x**2+a**2)**3,x)

[Out]

1/(2*b*(a - b*x)**2)

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Mathematica [A]  time = 0.00268562, size = 15, normalized size = 1. \[ \frac{1}{2 b (a-b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/(a^2 - b^2*x^2)^3,x]

[Out]

1/(2*b*(a - b*x)^2)

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Maple [A]  time = 0.001, size = 15, normalized size = 1. \[{\frac{1}{2\,b \left ( bx-a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(-b^2*x^2+a^2)^3,x)

[Out]

1/2/b/(b*x-a)^2

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Maxima [A]  time = 0.6794, size = 32, normalized size = 2.13 \[ \frac{1}{2 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^3/(b^2*x^2 - a^2)^3,x, algorithm="maxima")

[Out]

1/2/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Fricas [A]  time = 0.196722, size = 32, normalized size = 2.13 \[ \frac{1}{2 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^3/(b^2*x^2 - a^2)^3,x, algorithm="fricas")

[Out]

1/2/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Sympy [A]  time = 1.3628, size = 24, normalized size = 1.6 \[ \frac{1}{2 a^{2} b - 4 a b^{2} x + 2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/(-b**2*x**2+a**2)**3,x)

[Out]

1/(2*a**2*b - 4*a*b**2*x + 2*b**3*x**2)

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GIAC/XCAS [A]  time = 0.21706, size = 19, normalized size = 1.27 \[ \frac{1}{2 \,{\left (b x - a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^3/(b^2*x^2 - a^2)^3,x, algorithm="giac")

[Out]

1/2/((b*x - a)^2*b)