3.174 \(\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx\)

Optimal. Leaf size=112 \[ \frac {a \sqrt {1-x^2} \sqrt {\frac {a \left (x^2+1\right )}{a+b x^2}} \Pi \left (\frac {b}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b} x}{\sqrt {b x^2+a}}\right )|-\frac {a-b}{a+b}\right )}{\sqrt {x^2+1} \sqrt {a+b} \sqrt {\frac {a \left (1-x^2\right )}{a+b x^2}}} \]

[Out]

a*EllipticPi(x*(a+b)^(1/2)/(b*x^2+a)^(1/2),b/(a+b),((-a+b)/(a+b))^(1/2))*(-x^2+1)^(1/2)*(a*(x^2+1)/(b*x^2+a))^
(1/2)/(a+b)^(1/2)/(x^2+1)^(1/2)/(a*(-x^2+1)/(b*x^2+a))^(1/2)

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Rubi [F]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[a + b*x^2]/Sqrt[1 - x^4],x]

[Out]

Defer[Int][Sqrt[a + b*x^2]/Sqrt[1 - x^4], x]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx &=\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1} \sqrt {b x^{2} + a}}{x^{4} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{2} + a}}{\sqrt {-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \,x^{2}+a}}{\sqrt {-x^{4}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x)

[Out]

int((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{2} + a}}{\sqrt {-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,x^2+a}}{\sqrt {1-x^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(1/2)/(1 - x^4)^(1/2),x)

[Out]

int((a + b*x^2)^(1/2)/(1 - x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x^{2}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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