3.773 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^4} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{15 a^2 b (a+b x)^3}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4} \]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(5*a*b*(a + b*x)^4) - (a^2 - b^2*x^2)^(3/2)/(15*a^2*b*(a
+ b*x)^3)

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Rubi [A]  time = 0.0739798, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{15 a^2 b (a+b x)^3}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2 - b^2*x^2)^(3/2)/(5*a*b*(a + b*x)^4) - (a^2 - b^2*x^2)^(3/2)/(15*a^2*b*(a
+ b*x)^3)

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Rubi in Sympy [A]  time = 8.79601, size = 53, normalized size = 0.79 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{5 a b \left (a + b x\right )^{4}} - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{15 a^{2} b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**4,x)

[Out]

-(a**2 - b**2*x**2)**(3/2)/(5*a*b*(a + b*x)**4) - (a**2 - b**2*x**2)**(3/2)/(15*
a**2*b*(a + b*x)**3)

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Mathematica [A]  time = 0.0362704, size = 51, normalized size = 0.76 \[ \frac{\sqrt{a^2-b^2 x^2} \left (-4 a^2+3 a b x+b^2 x^2\right )}{15 a^2 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-4*a^2 + 3*a*b*x + b^2*x^2))/(15*a^2*b*(a + b*x)^3)

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Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[ -{\frac{ \left ( bx+4\,a \right ) \left ( -bx+a \right ) }{15\, \left ( bx+a \right ) ^{3}{a}^{2}b}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^4,x)

[Out]

-1/15*(-b*x+a)*(b*x+4*a)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^3/a^2/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218581, size = 271, normalized size = 4.04 \[ -\frac{3 \, b^{4} x^{5} + 20 \, a b^{3} x^{4} + 35 \, a^{2} b^{2} x^{3} - 30 \, a^{3} b x^{2} - 60 \, a^{4} x - 5 \,{\left (b^{3} x^{4} + a b^{2} x^{3} - 6 \, a^{2} b x^{2} - 12 \, a^{3} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{15 \,{\left (a^{2} b^{5} x^{5} + 5 \, a^{3} b^{4} x^{4} + 5 \, a^{4} b^{3} x^{3} - 5 \, a^{5} b^{2} x^{2} - 10 \, a^{6} b x - 4 \, a^{7} -{\left (a^{2} b^{4} x^{4} - 7 \, a^{4} b^{2} x^{2} - 10 \, a^{5} b x - 4 \, a^{6}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(3*b^4*x^5 + 20*a*b^3*x^4 + 35*a^2*b^2*x^3 - 30*a^3*b*x^2 - 60*a^4*x - 5*(
b^3*x^4 + a*b^2*x^3 - 6*a^2*b*x^2 - 12*a^3*x)*sqrt(-b^2*x^2 + a^2))/(a^2*b^5*x^5
+ 5*a^3*b^4*x^4 + 5*a^4*b^3*x^3 - 5*a^5*b^2*x^2 - 10*a^6*b*x - 4*a^7 - (a^2*b^4
*x^4 - 7*a^4*b^2*x^2 - 10*a^5*b*x - 4*a^6)*sqrt(-b^2*x^2 + a^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**4,x)

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**4, x)

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GIAC/XCAS [A]  time = 0.239261, size = 223, normalized size = 3.33 \[ \frac{2 \,{\left (\frac{5 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{25 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{15 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{15 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + 4\right )}}{15 \, a^{2}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{5}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 25*(a*b + sqrt(-b^2*x^2 +
a^2)*abs(b))^2/(b^4*x^2) + 15*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) +
15*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 4)/(a^2*((a*b + sqrt(-b^2*x
^2 + a^2)*abs(b))/(b^2*x) + 1)^5*abs(b))