Optimal. Leaf size=163 \[ \frac {c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\sin ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {567, 551}
\begin {gather*} \frac {c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\text {ArcSin}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a \sqrt {e+f x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 551
Rule 567
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \sqrt {e+f x^2}} \, dx &=\frac {\left (c \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-d x^2\right ) \sqrt {1-\frac {(-b c+a d) x^2}{a}} \sqrt {1-\frac {(d e-c f) x^2}{e}}} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}\\ &=\frac {c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\sin ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.82, size = 162, normalized size = 0.99 \begin {gather*} \frac {c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\sin ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|\frac {(-b c+a d) e}{a (d e-c f)}\right )}{a \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \,x^{2}+c}}{\sqrt {b \,x^{2}+a}\, \sqrt {f \,x^{2}+e}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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]
[Out]
________________________________________________________________________________________
Fricas [F]
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]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{2}}}{\sqrt {a + b x^{2}} \sqrt {e + f x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {d\,x^2+c}}{\sqrt {b\,x^2+a}\,\sqrt {f\,x^2+e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________