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

Optimal. Leaf size=153 \[ -\frac{9 a x \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

(-9*a*x*(a - b*x^2)*Sqrt[a + b*x^2])/(8*Sqrt[a^2 - b^2*x^4]) - (x*(a - b*x^2)*(a
+ b*x^2)^(3/2))/(4*Sqrt[a^2 - b^2*x^4]) + (19*a^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^
2]*ArcTan[(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 = 0.157747, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{9 a x \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-9*a*x*(a - b*x^2)*Sqrt[a + b*x^2])/(8*Sqrt[a^2 - b^2*x^4]) - (x*(a - b*x^2)*(a
+ b*x^2)^(3/2))/(4*Sqrt[a^2 - b^2*x^4]) + (19*a^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^
2]*ArcTan[(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 = 28.0154, size = 121, normalized size = 0.79 \[ \frac{19 a^{2} \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a - b x^{2}}} \right )}}{8 \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} - \frac{9 a x \sqrt{a^{2} - b^{2} x^{4}}}{8 \sqrt{a + b x^{2}}} - \frac{x \sqrt{a + b x^{2}} \sqrt{a^{2} - b^{2} x^{4}}}{4} \]

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)*atan(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 [C]  time = 0.228307, size = 98, normalized size = 0.64 \[ -\frac{\left (11 a x+2 b x^3\right ) \sqrt{a^2-b^2 x^4}}{8 \sqrt{a+b x^2}}+\frac{19 i a^2 \log \left (\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right )}{8 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.079, size = 132, normalized size = 0.9 \[ -{\frac{1}{8}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( 2\,{x}^{3}{b}^{3/2}\sqrt{-b{x}^{2}+a}+11\,ax\sqrt{-b{x}^{2}+a}\sqrt{b}+13\,{a}^{2}\arctan \left ({\frac{x\sqrt{b}}{\sqrt{-b{x}^{2}+a}}} \right ) -32\,{a}^{2}\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{ \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }{b}}}}}} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \]

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^2*x^4+a^2)^(1/2)*(2*x^3*b^(3/2)*(-b*x^2+a)^(1/2)+11*a*x*(-b*x^2+a)^(1/2
)*b^(1/2)+13*a^2*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))-32*a^2*arctan(b^(1/2)*x/(1/b
*(-b*x+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2)))/(b*x^2+a)^(1/2)/(-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 = 0.305802, 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)*ar
ctan(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)