Optimal. Leaf size=87 \[ \frac {\sqrt {c} \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {432, 430}
\begin {gather*} \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} F\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 432
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx &=\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {c-d x^2}}\\ &=\frac {\left (\sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {a+b x^2} \sqrt {c-d x^2}}\\ &=\frac {\sqrt {c} \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 89, normalized size = 1.02 \begin {gather*} \frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {\frac {c-d x^2}{c}} F\left (\sin ^{-1}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 103, normalized size = 1.18
method | result | size |
default | \(\frac {\EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}{\sqrt {\frac {d}{c}}\, \left (-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c \right )}\) | \(103\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}\, \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}}\) | \(127\) |
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 [A]
time = 0.13, size = 39, normalized size = 0.45 \begin {gather*} \frac {\sqrt {a c} \sqrt {\frac {d}{c}} {\rm ellipticF}\left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}\right )}{a d} \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 {1}{\sqrt {a + b x^{2}} \sqrt {c - d 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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {c-d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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