Optimal. Leaf size=214 \[ -\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {d x \sqrt {a+b x^2}}{b \sqrt {-c-d x^2}}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.10, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {422, 418, 492, 411} \[ -\frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {d x \sqrt {a+b x^2}}{b \sqrt {-c-d x^2}}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 422
Rule 492
Rubi steps
\begin {align*} \int \frac {\sqrt {-c-d x^2}}{\sqrt {a+b x^2}} \, dx &=-\left (c \int \frac {1}{\sqrt {a+b x^2} \sqrt {-c-d x^2}} \, dx\right )-d \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {-c-d x^2}} \, dx\\ &=-\frac {d x \sqrt {a+b x^2}}{b \sqrt {-c-d x^2}}-\frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {(c d) \int \frac {\sqrt {a+b x^2}}{\left (-c-d x^2\right )^{3/2}} \, dx}{b}\\ &=-\frac {d x \sqrt {a+b x^2}}{b \sqrt {-c-d x^2}}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.42 \[ \frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {\frac {c+d x^2}{c}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-d x^{2} - c}}{\sqrt {b x^{2} + a}}, x\right ) \]
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 \[ \int \frac {\sqrt {-d x^{2} - c}}{\sqrt {b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 161, normalized size = 0.75 \[ \frac {\left (a d \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {b c}{a d}}\right )-a d \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {b c}{a d}}\right )+b c \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {b c}{a d}}\right )\right ) \sqrt {-d \,x^{2}-c}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}}{\left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right ) \sqrt {-\frac {d}{c}}\, b} \]
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 \[ \int \frac {\sqrt {-d x^{2} - c}}{\sqrt {b x^{2} + a}}\,{d x} \]
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 \[ \int \frac {\sqrt {-d\,x^2-c}}{\sqrt {b\,x^2+a}} \,d x \]
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 \[ \int \frac {\sqrt {- c - d x^{2}}}{\sqrt {a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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