Optimal. Leaf size=545 \[ \frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}-\frac {\sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\sin ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b \sqrt {e} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {c \sqrt {e} (b d e-b c f-a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\sin ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.32, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {569, 568, 435,
567, 551, 566, 430} \begin {gather*} \frac {b \sqrt {e} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\text {ArcSin}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {e} \sqrt {a+b x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\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 )}{2 f \sqrt {e+f x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\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 )}} (-a d f-b c f+b d e) \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 )}{2 a d f \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 )}}}+\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 551
Rule 566
Rule 567
Rule 568
Rule 569
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx &=\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}-\frac {(c (d e-c f)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{2 f}+\frac {(b c (d e-c f)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 d f}-\frac {(b d e-b c f-a d f) \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \sqrt {e+f x^2}} \, dx}{2 d f}\\ &=\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}+\frac {\left (b (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {(b c-a d) x^2}{c}} \sqrt {1-\frac {(b e-a f) x^2}{e}}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 d f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {\left ((d e-c f) \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 {\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 )}{2 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (c (b d e-b c f-a d f) \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 )}{2 a d f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}\\ &=\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}-\frac {\sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\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 )}{2 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b \sqrt {e} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {c \sqrt {e} (b d e-b c f-a d f) \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 )}{2 a d f \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*}
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Mathematica [A]
time = 3.52, size = 503, normalized size = 0.92 \begin {gather*} \frac {\frac {x \sqrt {a+b x^2} \left (c+d x^2\right )}{\sqrt {e+f x^2}}-\frac {\sqrt {c} \sqrt {-d e+c f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\sin ^{-1}\left (\frac {\sqrt {-d e+c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|\frac {b c e-a c f}{a d e-a c f}\right )}{f \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b e-2 a f) (d e-c f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {b c e-a d e}{b c e-a c f}\right )}{\sqrt {e} f^2 \sqrt {b e-a f} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac {e (-b d e+b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \Pi \left (\frac {a f}{-b e+a f};\sin ^{-1}\left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a d e-a c f}{b c e-a c f}\right )}{\sqrt {a} f^2 \sqrt {-b e+a f} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}}{2 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{\sqrt {f \,x^{2}+e}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{\sqrt {e + f x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{\sqrt {f\,x^2+e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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