3.969 \(\int \frac{x^5}{\sqrt{a+b x^2-c x^4}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{16 c^{5/2}}-\frac{3 b \sqrt{a+b x^2-c x^4}}{8 c^2}-\frac{x^2 \sqrt{a+b x^2-c x^4}}{4 c} \]

[Out]

(-3*b*Sqrt[a + b*x^2 - c*x^4])/(8*c^2) - (x^2*Sqrt[a + b*x^2 - c*x^4])/(4*c) - (
(3*b^2 + 4*a*c)*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(16*c
^(5/2))

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Rubi [A]  time = 0.202522, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{16 c^{5/2}}-\frac{3 b \sqrt{a+b x^2-c x^4}}{8 c^2}-\frac{x^2 \sqrt{a+b x^2-c x^4}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*Sqrt[a + b*x^2 - c*x^4])/(8*c^2) - (x^2*Sqrt[a + b*x^2 - c*x^4])/(4*c) - (
(3*b^2 + 4*a*c)*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(16*c
^(5/2))

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Rubi in Sympy [A]  time = 22.7034, size = 95, normalized size = 0.89 \[ - \frac{3 b \sqrt{a + b x^{2} - c x^{4}}}{8 c^{2}} - \frac{x^{2} \sqrt{a + b x^{2} - c x^{4}}}{4 c} - \frac{\left (4 a c + 3 b^{2}\right ) \operatorname{atan}{\left (\frac{b - 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} \right )}}{16 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b*sqrt(a + b*x**2 - c*x**4)/(8*c**2) - x**2*sqrt(a + b*x**2 - c*x**4)/(4*c) -
(4*a*c + 3*b**2)*atan((b - 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 - c*x**4)))/(16
*c**(5/2))

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Mathematica [C]  time = 0.129357, size = 94, normalized size = 0.88 \[ -\frac{\left (3 b+2 c x^2\right ) \sqrt{a+b x^2-c x^4}}{8 c^2}+\frac{i \left (4 a c+3 b^2\right ) \log \left (2 \sqrt{a+b x^2-c x^4}+\frac{i \left (b-2 c x^2\right )}{\sqrt{c}}\right )}{16 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((3*b + 2*c*x^2)*Sqrt[a + b*x^2 - c*x^4])/(8*c^2) + ((I/16)*(3*b^2 + 4*a*c)*Log
[(I*(b - 2*c*x^2))/Sqrt[c] + 2*Sqrt[a + b*x^2 - c*x^4]])/c^(5/2)

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Maple [A]  time = 0.021, size = 120, normalized size = 1.1 \[ -{\frac{{x}^{2}}{4\,c}\sqrt{-c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,b}{8\,{c}^{2}}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{2}}{16}\arctan \left ({1\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){c}^{-{\frac{5}{2}}}}+{\frac{a}{4}\arctan \left ({1\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x^2*(-c*x^4+b*x^2+a)^(1/2)/c-3/8*b*(-c*x^4+b*x^2+a)^(1/2)/c^2+3/16*b^2/c^(5
/2)*arctan(c^(1/2)*(x^2-1/2*b/c)/(-c*x^4+b*x^2+a)^(1/2))+1/4*a/c^(3/2)*arctan(c^
(1/2)*(x^2-1/2*b/c)/(-c*x^4+b*x^2+a)^(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(x^5/sqrt(-c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294223, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + 3 \, b\right )} \sqrt{-c} -{\left (3 \, b^{2} + 4 \, a c\right )} \log \left (4 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} - b c\right )} +{\left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, a c\right )} \sqrt{-c}\right )}{32 \, \sqrt{-c} c^{2}}, -\frac{2 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + 3 \, b\right )} \sqrt{c} -{\left (3 \, b^{2} + 4 \, a c\right )} \arctan \left (\frac{2 \, c x^{2} - b}{2 \, \sqrt{-c x^{4} + b x^{2} + a} \sqrt{c}}\right )}{16 \, c^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/32*(4*sqrt(-c*x^4 + b*x^2 + a)*(2*c*x^2 + 3*b)*sqrt(-c) - (3*b^2 + 4*a*c)*lo
g(4*sqrt(-c*x^4 + b*x^2 + a)*(2*c^2*x^2 - b*c) + (8*c^2*x^4 - 8*b*c*x^2 + b^2 -
4*a*c)*sqrt(-c)))/(sqrt(-c)*c^2), -1/16*(2*sqrt(-c*x^4 + b*x^2 + a)*(2*c*x^2 + 3
*b)*sqrt(c) - (3*b^2 + 4*a*c)*arctan(1/2*(2*c*x^2 - b)/(sqrt(-c*x^4 + b*x^2 + a)
*sqrt(c))))/c^(5/2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{a + b x^{2} - c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**5/sqrt(a + b*x**2 - c*x**4), x)

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GIAC/XCAS [A]  time = 0.306066, size = 123, normalized size = 1.15 \[ -\frac{1}{8} \, \sqrt{-c x^{4} + b x^{2} + a}{\left (\frac{2 \, x^{2}}{c} + \frac{3 \, b}{c^{2}}\right )} - \frac{{\left (3 \, b^{2} + 4 \, a c\right )}{\rm ln}\left ({\left | 2 \,{\left (\sqrt{-c} x^{2} - \sqrt{-c x^{4} + b x^{2} + a}\right )} \sqrt{-c} + b \right |}\right )}{16 \, \sqrt{-c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(-c*x^4 + b*x^2 + a)*(2*x^2/c + 3*b/c^2) - 1/16*(3*b^2 + 4*a*c)*ln(abs(
2*(sqrt(-c)*x^2 - sqrt(-c*x^4 + b*x^2 + a))*sqrt(-c) + b))/(sqrt(-c)*c^2)