3.2.10 \(\int \frac {\sqrt {c+d x^2}}{(a+b x^2)^{3/2} (e+f x^2)^{3/2}} \, dx\) [110]

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

[Out]

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

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \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 {gather*}

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*}

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Mathematica [A]
time = 18.55, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \,x^{2}+c}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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*a*b*f^2*x^6 + a^2*f^2*x^4 + (b^2*x^4
+ 2*a*b*x^2 + a^2)*e^2 + 2*(b^2*f*x^6 + 2*a*b*f*x^4 + a^2*f*x^2)*e), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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