\(\int \frac {1}{(a+b x^2)^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [115]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\text {Int}\left (\frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx \]

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

Not integrable

Time = 10.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx \]

[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]

Maple [N/A]

Not integrable

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

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.74 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 7.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 9.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \]

[In]

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

[Out]

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