Optimal. Leaf size=189 \[ -\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {434, 438, 437,
435, 432, 430} \begin {gather*} \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) F\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 432
Rule 434
Rule 435
Rule 437
Rule 438
Rubi steps
\begin {align*} \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx &=-\frac {d \int \frac {\sqrt {a+b x^2}}{\sqrt {c-d x^2}} \, dx}{b}+\frac {(b c+a d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx}{b}\\ &=-\frac {\left (d \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {a+b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{b \sqrt {c-d x^2}}+\frac {\left ((b c+a d) \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}} \, dx}{b \sqrt {c-d x^2}}\\ &=-\frac {\left (d \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}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\left ((b c+a d) \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}{b \sqrt {a+b x^2} \sqrt {c-d x^2}}\\ &=-\frac {\sqrt {c} \sqrt {d} \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 )}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b c+a d) \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 )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 89, normalized size = 0.47 \begin {gather*} \frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {c-d x^2} E\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 {\frac {c-d x^2}{c}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 161, normalized size = 0.85
method | result | size |
default | \(\frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \left (a d \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )+c \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b -a d \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )\right )}{\left (-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c \right ) \sqrt {\frac {d}{c}}\, b}\) | \(161\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (\frac {c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}}+\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}\, b}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\) | \(257\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {c - d x^{2}}}{\sqrt {a + b 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 {\sqrt {c-d\,x^2}}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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