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

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 21.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx \]

[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]

Maple [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82

\[\int \frac {\sqrt {d \,x^{2}+c}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.26 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 11.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 6.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]

[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)