Optimal. Leaf size=233 \[ -\frac {2 d \sqrt {a+\frac {b}{x^2}}}{x \sqrt {c+\frac {d}{x^2}}}+x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}-\frac {\sqrt {c} \sqrt {a+\frac {b}{x^2}} (a d+b c) F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}+\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}} \]
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Rubi [A] time = 0.22, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {375, 473, 531, 418, 492, 411} \[ -\frac {2 d \sqrt {a+\frac {b}{x^2}}}{x \sqrt {c+\frac {d}{x^2}}}+x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}-\frac {\sqrt {c} \sqrt {a+\frac {b}{x^2}} (a d+b c) F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}+\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}} \]
Antiderivative was successfully verified.
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Rule 375
Rule 411
Rule 418
Rule 473
Rule 492
Rule 531
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x-2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c+a d)+b d x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x-(2 b d) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )-(b c+a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 d \sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}} x}+\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x-\frac {\sqrt {c} (b c+a d) \sqrt {a+\frac {b}{x^2}} F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}+(2 c d) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 d \sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}} x}+\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x+\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}-\frac {\sqrt {c} (b c+a d) \sqrt {a+\frac {b}{x^2}} F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 205, normalized size = 0.88 \[ -\frac {x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \left (\sqrt {\frac {a}{b}} \left (a x^2+b\right ) \left (c x^2+d\right )+i x \sqrt {\frac {a x^2}{b}+1} \sqrt {\frac {c x^2}{d}+1} (b c-a d) F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )+2 i a d x \sqrt {\frac {a x^2}{b}+1} \sqrt {\frac {c x^2}{d}+1} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )\right )}{\sqrt {\frac {a}{b}} \left (a x^2+b\right ) \left (c x^2+d\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\frac {a x^{2} + b}{x^{2}}} \sqrt {\frac {c x^{2} + d}{x^{2}}}, x\right ) \]
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 \[ \int \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 277, normalized size = 1.19 \[ \frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (-\sqrt {-\frac {c}{d}}\, a c \,x^{4}-\sqrt {-\frac {c}{d}}\, a d \,x^{2}+\sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, a d x \EllipticF \left (\sqrt {-\frac {c}{d}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {c}{d}}\, b c \,x^{2}+2 \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, b c x \EllipticE \left (\sqrt {-\frac {c}{d}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, b c x \EllipticF \left (\sqrt {-\frac {c}{d}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {c}{d}}\, b d \right ) x}{\left (a c \,x^{4}+a d \,x^{2}+b c \,x^{2}+b d \right ) \sqrt {-\frac {c}{d}}} \]
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 \[ \int \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}}\,{d x} \]
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 \[ \int \sqrt {a+\frac {b}{x^2}}\,\sqrt {c+\frac {d}{x^2}} \,d x \]
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 \[ \int \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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