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

Optimal. Leaf size=109 \[ \frac{3 a \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b} \sqrt{a^2-b^2 x^4}}-\frac{x \sqrt{a-b x^2} \left (a+b x^2\right )}{2 \sqrt{a^2-b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.101543, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{3 a \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b} \sqrt{a^2-b^2 x^4}}-\frac{x \sqrt{a-b x^2} \left (a+b x^2\right )}{2 \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.3618, size = 88, normalized size = 0.81 \[ \frac{3 a \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} - \frac{x \sqrt{a^{2} - b^{2} x^{4}}}{2 \sqrt{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(3/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

3*a*sqrt(a**2 - b**2*x**4)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2*sqrt(b)*sqrt(a -
b*x**2)*sqrt(a + b*x**2)) - x*sqrt(a**2 - b**2*x**4)/(2*sqrt(a - b*x**2))

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Mathematica [A]  time = 0.152985, size = 110, normalized size = 1.01 \[ \frac{1}{2} \left (-\frac{x \sqrt{a^2-b^2 x^4}}{\sqrt{a-b x^2}}+\frac{3 a \log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )}{\sqrt{b}}-\frac{3 a \log \left (b x^2-a\right )}{\sqrt{b}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 85, normalized size = 0.8 \[ -{\frac{1}{2\,b{x}^{2}-2\,a}\sqrt{-b{x}^{2}+a}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( -x\sqrt{b{x}^{2}+a}\sqrt{b}+3\,a\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(3/2)/(-b^2*x^4+a^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286728, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{b} x + 3 \,{\left (a b x^{2} - a^{2}\right )} \log \left (-\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x -{\left (2 \, b^{2} x^{4} - a b x^{2} - a^{2}\right )} \sqrt{b}}{b x^{2} - a}\right )}{4 \,{\left (b x^{2} - a\right )} \sqrt{b}}, \frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b} x - 3 \,{\left (a b x^{2} - a^{2}\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right )}{2 \,{\left (b x^{2} - a\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(b)*x + 3*(a*b*x^2 - a^2)*log(
-(2*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*b*x - (2*b^2*x^4 - a*b*x^2 - a^2)*sqrt
(b))/(b*x^2 - a)))/((b*x^2 - a)*sqrt(b)), 1/2*(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2
+ a)*sqrt(-b)*x - 3*(a*b*x^2 - a^2)*arctan(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)
*sqrt(-b)/(b^2*x^3 - a*b*x)))/((b*x^2 - a)*sqrt(-b))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(3/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(3/2)/sqrt(-b^2*x^4 + a^2), x)