3.115 \(\int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=37 \[ \text{Int}\left (\frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}},x\right ) \]

[Out]

Unintegrable[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]

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Rubi [A]  time = 0.172056, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]

[Out]

Defer[Int][1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.11097, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]

[Out]

Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]

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Maple [A]  time = 0.101, size = 0, normalized size = 0. \[ \int{1 \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

[Out]

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral(1/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)