Optimal. Leaf size=88 \[ \frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2-c}} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {427, 426, 424} \[ \frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2-c}} \]
Antiderivative was successfully verified.
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Rule 424
Rule 426
Rule 427
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{\sqrt {-c+d x^2}} \, dx &=\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {a+b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {-c+d x^2}}\\ &=\frac {\left (\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {1+\frac {b x^2}{a}} \sqrt {-c+d x^2}}\\ &=\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {-c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 1.00 \[ \frac {\sqrt {a+b x^2} \sqrt {\frac {c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {\frac {d}{c}} \sqrt {\frac {a+b x^2}{a}} \sqrt {d x^2-c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} - c}}, x\right ) \]
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 \[ \int \frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} - c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 168, normalized size = 1.91 \[ \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}-c}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {-\frac {d \,x^{2}-c}{c}}\, \left (a d \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {-\frac {a d}{b c}}\right )-b c \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {-\frac {a d}{b c}}\right )+b c \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {-\frac {a d}{b c}}\right )\right )}{\left (b d \,x^{4}+a d \,x^{2}-b c \,x^{2}-a c \right ) \sqrt {-\frac {b}{a}}\, d} \]
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 \[ \int \frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} - c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,x^2+a}}{\sqrt {d\,x^2-c}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x^{2}}}{\sqrt {- c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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