3.209 \(\int \frac{\left (a-b x^2\right )^{5/2}}{\sqrt{a^2-b^2 x^4}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

(-9*a*x*Sqrt[a - b*x^2]*(a + b*x^2))/(8*Sqrt[a^2 - b^2*x^4]) - (x*(a - b*x^2)^(3
/2)*(a + b*x^2))/(4*Sqrt[a^2 - b^2*x^4]) + (19*a^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^
2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*Sqrt[b]*Sqrt[a^2 - b^2*x^4])

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Rubi [A]  time = 0.1507, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(5/2)/Sqrt[a^2 - b^2*x^4],x]

[Out]

(-9*a*x*Sqrt[a - b*x^2]*(a + b*x^2))/(8*Sqrt[a^2 - b^2*x^4]) - (x*(a - b*x^2)^(3
/2)*(a + b*x^2))/(4*Sqrt[a^2 - b^2*x^4]) + (19*a^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^
2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*Sqrt[b]*Sqrt[a^2 - b^2*x^4])

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Rubi in Sympy [A]  time = 28.7303, size = 121, normalized size = 0.8 \[ \frac{19 a^{2} \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} - \frac{9 a x \sqrt{a^{2} - b^{2} x^{4}}}{8 \sqrt{a - b x^{2}}} - \frac{x \sqrt{a - b x^{2}} \sqrt{a^{2} - b^{2} x^{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(5/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

19*a**2*sqrt(a**2 - b**2*x**4)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(8*sqrt(b)*sqrt
(a - b*x**2)*sqrt(a + b*x**2)) - 9*a*x*sqrt(a**2 - b**2*x**4)/(8*sqrt(a - b*x**2
)) - x*sqrt(a - b*x**2)*sqrt(a**2 - b**2*x**4)/4

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Mathematica [A]  time = 0.297825, size = 123, normalized size = 0.81 \[ \frac{1}{8} \left (\frac{x \left (2 b x^2-11 a\right ) \sqrt{a^2-b^2 x^4}}{\sqrt{a-b x^2}}+\frac{19 a^2 \log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )}{\sqrt{b}}-\frac{19 a^2 \log \left (b x^2-a\right )}{\sqrt{b}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(5/2)/Sqrt[a^2 - b^2*x^4],x]

[Out]

((x*(-11*a + 2*b*x^2)*Sqrt[a^2 - b^2*x^4])/Sqrt[a - b*x^2] - (19*a^2*Log[-a + b*
x^2])/Sqrt[b] + (19*a^2*Log[a*b*x - b^2*x^3 + Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a^2 -
b^2*x^4]])/Sqrt[b])/8

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Maple [A]  time = 0.022, size = 105, normalized size = 0.7 \[ -{\frac{1}{8\,b{x}^{2}-8\,a}\sqrt{-b{x}^{2}+a}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( 2\,{b}^{3/2}{x}^{3}\sqrt{b{x}^{2}+a}-11\,ax\sqrt{b{x}^{2}+a}\sqrt{b}+19\,{a}^{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(5/2)/(-b^2*x^4+a^2)^(1/2),x)

[Out]

-1/8*(-b*x^2+a)^(1/2)*(-b^2*x^4+a^2)^(1/2)*(2*b^(3/2)*x^3*(b*x^2+a)^(1/2)-11*a*x
*(b*x^2+a)^(1/2)*b^(1/2)+19*a^2*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)^(5/2)/sqrt(-b^2*x^4 + a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282586, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}}{\left (2 \, b x^{3} - 11 \, a x\right )} \sqrt{-b x^{2} + a} \sqrt{b} - 19 \,{\left (a^{2} b x^{2} - a^{3}\right )} \log \left (-\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x -{\left (2 \, b^{2} x^{4} - a b x^{2} - a^{2}\right )} \sqrt{b}}{b x^{2} - a}\right )}{16 \,{\left (b x^{2} - a\right )} \sqrt{b}}, -\frac{\sqrt{-b^{2} x^{4} + a^{2}}{\left (2 \, b x^{3} - 11 \, a x\right )} \sqrt{-b x^{2} + a} \sqrt{-b} + 19 \,{\left (a^{2} b x^{2} - a^{3}\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right )}{8 \,{\left (b x^{2} - a\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(5/2)/sqrt(-b^2*x^4 + a^2),x, algorithm="fricas")

[Out]

[-1/16*(2*sqrt(-b^2*x^4 + a^2)*(2*b*x^3 - 11*a*x)*sqrt(-b*x^2 + a)*sqrt(b) - 19*
(a^2*b*x^2 - a^3)*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)), -1/8*(sqrt(-b^2*
x^4 + a^2)*(2*b*x^3 - 11*a*x)*sqrt(-b*x^2 + a)*sqrt(-b) + 19*(a^2*b*x^2 - a^3)*a
rctan(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. \[ \int \frac{\left (a - b x^{2}\right )^{\frac{5}{2}}}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(5/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(5/2)/sqrt(-b^2*x^4 + a^2),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(5/2)/sqrt(-b^2*x^4 + a^2), x)