Optimal. Leaf size=232 \[ -\frac {d \sqrt {a+\frac {b}{x^2}}}{c x \sqrt {c+\frac {d}{x^2}}}+\frac {x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}}{c}-\frac {b \sqrt {c} \sqrt {a+\frac {b}{x^2}} F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}+\frac {\sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}} \]
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Rubi [A] time = 0.20, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {375, 475, 21, 422, 418, 492, 411} \[ -\frac {d \sqrt {a+\frac {b}{x^2}}}{c x \sqrt {c+\frac {d}{x^2}}}+\frac {x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}}{c}-\frac {b \sqrt {c} \sqrt {a+\frac {b}{x^2}} F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}+\frac {\sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 375
Rule 411
Rule 418
Rule 422
Rule 475
Rule 492
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^2 \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c}-\frac {\operatorname {Subst}\left (\int \frac {b c+b d x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c}-\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c}-b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )-\frac {(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 )}{c}\\ &=-\frac {d \sqrt {a+\frac {b}{x^2}}}{c \sqrt {c+\frac {d}{x^2}} x}+\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c}-\frac {b \sqrt {c} \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}}}+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 {d \sqrt {a+\frac {b}{x^2}}}{c \sqrt {c+\frac {d}{x^2}} x}+\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c}+\frac {\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} \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 {b \sqrt {c} \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 [A] time = 0.07, size = 86, normalized size = 0.37 \[ \frac {\sqrt {a+\frac {b}{x^2}} \sqrt {\frac {c x^2+d}{d}} E\left (\sin ^{-1}\left (\sqrt {-\frac {c}{d}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {c}{d}} \sqrt {\frac {a x^2+b}{b}} \sqrt {c+\frac {d}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}, x\right ) \]
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 \[ \int \frac {\sqrt {a + \frac {b}{x^{2}}}}{\sqrt {c + \frac {d}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 94, normalized size = 0.41 \[ \frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, \sqrt {\frac {c \,x^{2}+d}{d}}\, b \EllipticE \left (\sqrt {-\frac {c}{d}}\, x , \sqrt {\frac {a d}{b c}}\right )}{\left (a \,x^{2}+b \right ) \sqrt {-\frac {c}{d}}\, \sqrt {\frac {c \,x^{2}+d}{x^{2}}}} \]
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 \[ \int \frac {\sqrt {a + \frac {b}{x^{2}}}}{\sqrt {c + \frac {d}{x^{2}}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}}} \,d x \]
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 \[ \int \frac {\sqrt {a + \frac {b}{x^{2}}}}{\sqrt {c + \frac {d}{x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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