3.760 \(\int \frac{(a+b x)^4}{\left (a^2-b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=10 \[ \frac{x}{(a-b x)^2} \]

[Out]

x/(a - b*x)^2

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Rubi [A]  time = 0.0187644, antiderivative size = 10, 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 \[ \frac{x}{(a-b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4/(a^2 - b^2*x^2)^3,x]

[Out]

x/(a - b*x)^2

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Rubi in Sympy [A]  time = 6.80813, size = 17, normalized size = 1.7 \[ \frac{\left (a + b x\right )^{2}}{4 a b \left (a - b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4/(-b**2*x**2+a**2)**3,x)

[Out]

(a + b*x)**2/(4*a*b*(a - b*x)**2)

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Mathematica [A]  time = 0.00647806, size = 10, normalized size = 1. \[ \frac{x}{(a-b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4/(a^2 - b^2*x^2)^3,x]

[Out]

x/(a - b*x)^2

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Maple [B]  time = 0.006, size = 29, normalized size = 2.9 \[{\frac{1}{b \left ( bx-a \right ) }}+{\frac{a}{b \left ( bx-a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4/(-b^2*x^2+a^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.681401, size = 27, normalized size = 2.7 \[ \frac{x}{b^{2} x^{2} - 2 \, a b x + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^4/(b^2*x^2 - a^2)^3,x, algorithm="maxima")

[Out]

x/(b^2*x^2 - 2*a*b*x + a^2)

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Fricas [A]  time = 0.207338, size = 27, normalized size = 2.7 \[ \frac{x}{b^{2} x^{2} - 2 \, a b x + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^4/(b^2*x^2 - a^2)^3,x, algorithm="fricas")

[Out]

x/(b^2*x^2 - 2*a*b*x + a^2)

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Sympy [A]  time = 1.46793, size = 17, normalized size = 1.7 \[ \frac{x}{a^{2} - 2 a b x + b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4/(-b**2*x**2+a**2)**3,x)

[Out]

x/(a**2 - 2*a*b*x + b**2*x**2)

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GIAC/XCAS [A]  time = 0.214285, size = 15, normalized size = 1.5 \[ \frac{x}{{\left (b x - a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^4/(b^2*x^2 - a^2)^3,x, algorithm="giac")

[Out]

x/(b*x - a)^2