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

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

[Out]

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

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Rubi [A]  time = 0.0600382, 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 = {} \[ \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.967956, size = 0, normalized size = 0. \[ \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}} \left ( f{x}^{2}+e \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)), 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} f^{2} x^{8} + 2 \,{\left (b^{2} e f + a b f^{2}\right )} x^{6} +{\left (b^{2} e^{2} + 4 \, a b e f + a^{2} f^{2}\right )} x^{4} + a^{2} e^{2} + 2 \,{\left (a b e^{2} + a^{2} e f\right )} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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