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

Optimal. Leaf size=76 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0891844, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.1847, size = 65, normalized size = 0.86 \[ - \frac{3 a \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{b} - \frac{3 \sqrt{a^{2} - b^{2} x^{2}}}{b} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{b \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115509, size = 60, normalized size = 0.79 \[ -\frac{\frac{\sqrt{a^2-b^2 x^2} (5 a+b x)}{a+b x}+3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 206, normalized size = 2.7 \[ -{\frac{1}{a{b}^{4}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-3}}-2\,{\frac{1}{{b}^{3}{a}^{2}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{5/2} \left ( x+{\frac{a}{b}} \right ) ^{-2}}-2\,{\frac{1}{{a}^{2}b} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{3/2}}-3\,{\frac{x}{a}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}-3\,{\frac{a}{\sqrt{{b}^{2}}}\arctan \left ({\sqrt{{b}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/b^4/a/(x+a/b)^3*(-(x+a/b)^2*b^2+2*a*b*(x+a/b))^(5/2)-2/b^3/a^2/(x+a/b)^2*(-(x
+a/b)^2*b^2+2*a*b*(x+a/b))^(5/2)-2/b/a^2*(-(x+a/b)^2*b^2+2*a*b*(x+a/b))^(3/2)-3/
a*(-(x+a/b)^2*b^2+2*a*b*(x+a/b))^(1/2)*x-3*a/(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223586, size = 238, normalized size = 3.13 \[ \frac{b^{3} x^{3} + a b^{2} x^{2} + 8 \, a^{2} b x + 6 \,{\left (a b^{2} x^{2} - a^{2} b x - 2 \, a^{3} + \sqrt{-b^{2} x^{2} + a^{2}}{\left (a b x + 2 \, a^{2}\right )}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) -{\left (b^{2} x^{2} + 8 \, a b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{b^{3} x^{2} - a b^{2} x - 2 \, a^{2} b + \sqrt{-b^{2} x^{2} + a^{2}}{\left (b^{2} x + 2 \, a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^3*x^3 + a*b^2*x^2 + 8*a^2*b*x + 6*(a*b^2*x^2 - a^2*b*x - 2*a^3 + sqrt(-b^2*x^
2 + a^2)*(a*b*x + 2*a^2))*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) - (b^2*x^2 +
8*a*b*x)*sqrt(-b^2*x^2 + a^2))/(b^3*x^2 - a*b^2*x - 2*a^2*b + sqrt(-b^2*x^2 + a
^2)*(b^2*x + 2*a*b))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241266, size = 104, normalized size = 1.37 \[ -\frac{3 \, a \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{{\left | b \right |}} - \frac{\sqrt{-b^{2} x^{2} + a^{2}}}{b} + \frac{8 \, a}{{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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