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

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

[Out]

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

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Rubi [A]  time = 0.0571239, 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 = {} \[ \int \frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.741494, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.071, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}\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((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

[Out]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

[Out]

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