3.784 \(\int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 a b (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.036616, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.82755, size = 26, normalized size = 0.79 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{5 a b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**5,x)

[Out]

-(a**2 - b**2*x**2)**(5/2)/(5*a*b*(a + b*x)**5)

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Mathematica [A]  time = 0.0409949, size = 41, normalized size = 1.24 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2}}{5 a b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^5,x]

[Out]

-((a - b*x)^2*Sqrt[a^2 - b^2*x^2])/(5*a*b*(a + b*x)^3)

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Maple [A]  time = 0.009, size = 36, normalized size = 1.1 \[ -{\frac{-bx+a}{5\, \left ( bx+a \right ) ^{4}ba} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^5,x)

[Out]

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

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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((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223256, size = 204, normalized size = 6.18 \[ -\frac{2 \,{\left (b^{4} x^{5} + 5 \, a^{2} b^{2} x^{3} - 10 \, a^{4} x + 10 \, \sqrt{-b^{2} x^{2} + a^{2}} a^{3} x\right )}}{5 \,{\left (a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} - 10 \, a^{5} b x - 4 \, a^{6} -{\left (a b^{4} x^{4} - 7 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x - 4 \, a^{5}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^5,x, algorithm="fricas")

[Out]

-2/5*(b^4*x^5 + 5*a^2*b^2*x^3 - 10*a^4*x + 10*sqrt(-b^2*x^2 + a^2)*a^3*x)/(a*b^5
*x^5 + 5*a^2*b^4*x^4 + 5*a^3*b^3*x^3 - 5*a^4*b^2*x^2 - 10*a^5*b*x - 4*a^6 - (a*b
^4*x^4 - 7*a^3*b^2*x^2 - 10*a^4*b*x - 4*a^5)*sqrt(-b^2*x^2 + a^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac{3}{2}}}{\left (a + b x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**5,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**5, x)

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GIAC/XCAS [A]  time = 0.243919, size = 139, normalized size = 4.21 \[ \frac{1}{15} \,{\left (\frac{3 \, i{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )}{a b^{2}} + \frac{5 \,{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{3}{2}}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right ) -{\left (3 \,{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{5}{2}} + 5 \,{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{3}{2}}\right )}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )}{a b^{2}}\right )}{\left | b \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^5,x, algorithm="giac")

[Out]

1/15*(3*i*sign(1/(b*x + a))*sign(b)/(a*b^2) + (5*(2*a/(b*x + a) - 1)^(3/2)*sign(
1/(b*x + a))*sign(b) - (3*(2*a/(b*x + a) - 1)^(5/2) + 5*(2*a/(b*x + a) - 1)^(3/2
))*sign(1/(b*x + a))*sign(b))/(a*b^2))*abs(b)