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

Optimal. Leaf size=36 \[ \text{Unintegrable}\left (\frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}},x\right ) \]

[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 [A]  time = 0.0552669, 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., Rules used = {} \[ \int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx &=\int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.873079, 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 = 0.07, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{b^{2} d f x^{8} +{\left (b^{2} d e +{\left (b^{2} c + 2 \, a b d\right )} f\right )} x^{6} +{\left ({\left (b^{2} c + 2 \, a b d\right )} e +{\left (2 \, a b c + a^{2} d\right )} f\right )} x^{4} + a^{2} c e +{\left (a^{2} c f +{\left (2 \, a b c + a^{2} d\right )} e\right )} x^{2}}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^2*d*f*x^8 + (b^2*d*e + (b^2*c + 2*a*b*d)*f)*x^6 +
((b^2*c + 2*a*b*d)*e + (2*a*b*c + a^2*d)*f)*x^4 + a^2*c*e + (a^2*c*f + (2*a*b*c + a^2*d)*e)*x^2), x)

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

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

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, algorithm="giac")

[Out]

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