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

Optimal. Leaf size=85 \[ \frac{3 a \sqrt{a^2-b^2 x^2}}{2 b}+\frac{\left (a^2-b^2 x^2\right )^{3/2}}{2 b (a+b x)}+\frac{3 a^2 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0915286, antiderivative size = 85, 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 \[ \frac{3 a \sqrt{a^2-b^2 x^2}}{2 b}+\frac{\left (a^2-b^2 x^2\right )^{3/2}}{2 b (a+b x)}+\frac{3 a^2 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.7608, size = 68, normalized size = 0.8 \[ \frac{3 a^{2} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{2 b} + \frac{3 a \sqrt{a^{2} - b^{2} x^{2}}}{2 b} + \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{2 b \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0537399, size = 60, normalized size = 0.71 \[ \left (\frac{2 a}{b}-\frac{x}{2}\right ) \sqrt{a^2-b^2 x^2}+\frac{3 a^2 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 158, normalized size = 1.9 \[{\frac{1}{a{b}^{3}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-2}}+{\frac{1}{ab} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x}{2}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}+{\frac{3\,{a}^{2}}{2}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b^3/a/(x+a/b)^2*(-(x+a/b)^2*b^2+2*a*b*(x+a/b))^(5/2)+1/b/a*(-(x+a/b)^2*b^2+2*a
*b*(x+a/b))^(3/2)+3/2*(-(x+a/b)^2*b^2+2*a*b*(x+a/b))^(1/2)*x+3/2*a^2/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221882, size = 227, normalized size = 2.67 \[ \frac{2 \, a b^{3} x^{3} - 4 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x - 6 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{4} + 2 \, \sqrt{-b^{2} x^{2} + a^{2}} a^{3}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) -{\left (b^{3} x^{3} - 4 \, a b^{2} x^{2} - 2 \, a^{2} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{2 \,{\left (b^{3} x^{2} - 2 \, a^{2} b + 2 \, \sqrt{-b^{2} x^{2} + a^{2}} a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a*b^3*x^3 - 4*a^2*b^2*x^2 - 2*a^3*b*x - 6*(a^2*b^2*x^2 - 2*a^4 + 2*sqrt(-
b^2*x^2 + a^2)*a^3)*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) - (b^3*x^3 - 4*a*b
^2*x^2 - 2*a^2*b*x)*sqrt(-b^2*x^2 + a^2))/(b^3*x^2 - 2*a^2*b + 2*sqrt(-b^2*x^2 +
a^2)*a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24475, size = 163, normalized size = 1.92 \[ -\frac{{\left (12 \, a^{3} b^{3} \arctan \left (\sqrt{\frac{2 \, a}{b x + a} - 1}\right ){\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right ) - \frac{{\left (5 \, a^{3} b^{3}{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{3}{2}}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right ) + 3 \, a^{3} b^{3} \sqrt{\frac{2 \, a}{b x + a} - 1}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )\right )}{\left (b x + a\right )}^{2}}{a^{2}}\right )}{\left | b \right |}}{4 \, a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(12*a^3*b^3*arctan(sqrt(2*a/(b*x + a) - 1))*sign(1/(b*x + a))*sign(b) - (5*
a^3*b^3*(2*a/(b*x + a) - 1)^(3/2)*sign(1/(b*x + a))*sign(b) + 3*a^3*b^3*sqrt(2*a
/(b*x + a) - 1)*sign(1/(b*x + a))*sign(b))*(b*x + a)^2/a^2)*abs(b)/(a*b^5)