3.208 \(\int \frac{1}{\left (a+b x^2\right )^{5/2} \sqrt{a^2-b^2 x^4}} \, dx\)

Optimal. Leaf size=168 \[ \frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

(x*(a - b*x^2))/(8*a^2*(a + b*x^2)^(3/2)*Sqrt[a^2 - b^2*x^4]) + (9*x*(a - b*x^2)
)/(32*a^3*Sqrt[a + b*x^2]*Sqrt[a^2 - b^2*x^4]) + (19*Sqrt[a - b*x^2]*Sqrt[a + b*
x^2]*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]])/(32*Sqrt[2]*a^3*Sqrt[b]*Sqrt[a
^2 - b^2*x^4])

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Rubi [A]  time = 0.247323, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a - b*x^2))/(8*a^2*(a + b*x^2)^(3/2)*Sqrt[a^2 - b^2*x^4]) + (9*x*(a - b*x^2)
)/(32*a^3*Sqrt[a + b*x^2]*Sqrt[a^2 - b^2*x^4]) + (19*Sqrt[a - b*x^2]*Sqrt[a + b*
x^2]*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]])/(32*Sqrt[2]*a^3*Sqrt[b]*Sqrt[a
^2 - b^2*x^4])

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Rubi in Sympy [A]  time = 43.8884, size = 136, normalized size = 0.81 \[ \frac{x \sqrt{a^{2} - b^{2} x^{4}}}{8 a^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{9 x \sqrt{a^{2} - b^{2} x^{4}}}{32 a^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{19 \sqrt{2} \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a - b x^{2}}} \right )}}{64 a^{3} \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a**2 - b**2*x**4)/(8*a**2*(a + b*x**2)**(5/2)) + 9*x*sqrt(a**2 - b**2*x**
4)/(32*a**3*(a + b*x**2)**(3/2)) + 19*sqrt(2)*sqrt(a**2 - b**2*x**4)*atan(sqrt(2
)*sqrt(b)*x/sqrt(a - b*x**2))/(64*a**3*sqrt(b)*sqrt(a - b*x**2)*sqrt(a + b*x**2)
)

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Mathematica [A]  time = 0.123945, size = 123, normalized size = 0.73 \[ \frac{\sqrt{a^2-b^2 x^4} \left (2 \sqrt{b} x \sqrt{a-b x^2} \left (13 a+9 b x^2\right )+19 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )\right )}{64 a^3 \sqrt{b} \sqrt{a-b x^2} \left (a+b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a - b*x^2]*(13*a + 9*b*x^2) + 19*Sqrt[2]*
(a + b*x^2)^2*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]]))/(64*a^3*Sqrt[b]*Sqrt
[a - b*x^2]*(a + b*x^2)^(5/2))

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Maple [B]  time = 0.072, size = 729, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(-b^2*x^4+a^2)^(1/2)*b^4*(19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1
/2)-(-a*b)^(1/2)*x+a)/(b*x-(-a*b)^(1/2)))*x^4*a*b^3-19*2^(1/2)*ln(2*b*(2^(1/2)*a
^(1/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x+(-a*b)^(1/2)))*x^4*a*b^3-16*arcta
n(b^(1/2)*x/(1/b*(-b*x+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2))*x^4*b^(5/2)*a^(1/2
)*(-a*b)^(1/2)+16*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))*x^4*b^(5/2)*a^(1/2)*(-a*b)^
(1/2)+38*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)-(-a*b)^(1/2)*x+a)/(b*x
-(-a*b)^(1/2)))*x^2*a^2*b^2-38*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)+
(-a*b)^(1/2)*x+a)/(b*x+(-a*b)^(1/2)))*x^2*a^2*b^2-32*arctan(b^(1/2)*x/(1/b*(-b*x
+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2))*x^2*a^(3/2)*b^(3/2)*(-a*b)^(1/2)+32*arct
an(b^(1/2)*x/(-b*x^2+a)^(1/2))*x^2*a^(3/2)*b^(3/2)*(-a*b)^(1/2)-36*a^(1/2)*b^2*(
-a*b)^(1/2)*(-b*x^2+a)^(1/2)*x^3+19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(
1/2)-(-a*b)^(1/2)*x+a)/(b*x-(-a*b)^(1/2)))*a^3*b-19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1
/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x+(-a*b)^(1/2)))*a^3*b-16*arctan(b^(1/
2)*x/(1/b*(-b*x+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2))*a^(5/2)*b^(1/2)*(-a*b)^(1
/2)+16*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))*a^(5/2)*b^(1/2)*(-a*b)^(1/2)-52*a^(3/2
)*b*(-a*b)^(1/2)*(-b*x^2+a)^(1/2)*x)/(b*x^2+a)^(1/2)/(-b*x^2+a)^(1/2)/((a*b)^(1/
2)+(-a*b)^(1/2))^3/((a*b)^(1/2)-(-a*b)^(1/2))^3/a^(1/2)/(b*x-(-a*b)^(1/2))^2/(b*
x+(-a*b)^(1/2))^2/(-a*b)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b^{2} x^{4} + a^{2}}{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)), x)

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Fricas [A]  time = 0.279335, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}}{\left (9 \, b x^{3} + 13 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 19 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (-\frac{4 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} b x + \sqrt{2}{\left (3 \, b^{2} x^{4} + 2 \, a b x^{2} - a^{2}\right )} \sqrt{-b}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )\right )}}{128 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \sqrt{-b}}, \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}}{\left (9 \, b x^{3} + 13 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 19 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{b}}{2 \,{\left (b^{2} x^{3} + a b x\right )}}\right )\right )}}{64 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)),x, algorithm="fricas")

[Out]

[1/128*sqrt(2)*(2*sqrt(2)*sqrt(-b^2*x^4 + a^2)*(9*b*x^3 + 13*a*x)*sqrt(b*x^2 + a
)*sqrt(-b) + 19*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*log(-(4*sqrt(-b^2*x^
4 + a^2)*sqrt(b*x^2 + a)*b*x + sqrt(2)*(3*b^2*x^4 + 2*a*b*x^2 - a^2)*sqrt(-b))/(
b^2*x^4 + 2*a*b*x^2 + a^2)))/((a^3*b^3*x^6 + 3*a^4*b^2*x^4 + 3*a^5*b*x^2 + a^6)*
sqrt(-b)), 1/64*sqrt(2)*(sqrt(2)*sqrt(-b^2*x^4 + a^2)*(9*b*x^3 + 13*a*x)*sqrt(b*
x^2 + a)*sqrt(b) - 19*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*arctan(1/2*sqr
t(2)*sqrt(-b^2*x^4 + a^2)*sqrt(b*x^2 + a)*sqrt(b)/(b^2*x^3 + a*b*x)))/((a^3*b^3*
x^6 + 3*a^4*b^2*x^4 + 3*a^5*b*x^2 + a^6)*sqrt(b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a + b x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-(-a + b*x**2)*(a + b*x**2))*(a + b*x**2)**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b^{2} x^{4} + a^{2}}{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)), x)