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

Optimal. Leaf size=83 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}+\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0790259, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}+\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8999, size = 66, normalized size = 0.8 \[ \frac{\operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{b} + \frac{2 \sqrt{a^{2} - b^{2} x^{2}}}{b \left (a + b x\right )} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(b*x/sqrt(a**2 - b**2*x**2))/b + 2*sqrt(a**2 - b**2*x**2)/(b*(a + b*x)) - 2*
(a**2 - b**2*x**2)**(3/2)/(3*b*(a + b*x)**3)

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Mathematica [A]  time = 0.0836548, size = 61, normalized size = 0.73 \[ \frac{\frac{4 \sqrt{a^2-b^2 x^2} (a+2 b x)}{(a+b x)^2}+3 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 248, normalized size = 3. \[ -{\frac{1}{3\,{b}^{5}a} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-4}}+{\frac{1}{3\,{a}^{2}{b}^{4}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-3}}+{\frac{2}{3\,{a}^{3}{b}^{3}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-2}}+{\frac{2}{3\,{a}^{3}b} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{x}{{a}^{2}}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}+{1\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224857, size = 250, normalized size = 3.01 \[ -\frac{2 \,{\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 6 \, \sqrt{-b^{2} x^{2} + a^{2}} b^{2} x^{2} + 3 \,{\left (b^{3} x^{3} - 3 \, a^{2} b x - 2 \, a^{3} +{\left (b^{2} x^{2} + 3 \, a b x + 2 \, a^{2}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right )\right )}}{3 \,{\left (b^{4} x^{3} - 3 \, a^{2} b^{2} x - 2 \, a^{3} b +{\left (b^{3} x^{2} + 3 \, a b^{2} x + 2 \, a^{2} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(2*b^3*x^3 + 6*a*b^2*x^2 - 6*sqrt(-b^2*x^2 + a^2)*b^2*x^2 + 3*(b^3*x^3 - 3*
a^2*b*x - 2*a^3 + (b^2*x^2 + 3*a*b*x + 2*a^2)*sqrt(-b^2*x^2 + a^2))*arctan(-(a -
sqrt(-b^2*x^2 + a^2))/(b*x)))/(b^4*x^3 - 3*a^2*b^2*x - 2*a^3*b + (b^3*x^2 + 3*a
*b^2*x + 2*a^2*b)*sqrt(-b^2*x^2 + a^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac{3}{2}}}{\left (a + b x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.245082, size = 116, normalized size = 1.4 \[ \frac{\arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{{\left | b \right |}} - \frac{8 \,{\left (\frac{3 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + 1\right )}}{3 \,{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 8/3*(3*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b)
)/(b^2*x) + 1)/(((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^3*abs(b))