Optimal. Leaf size=215 \[ \frac {\sqrt {2} b \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),-\frac {b-\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\sqrt {b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}} \]
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Rubi [A] time = 0.21, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {423, 424, 419} \[ \frac {\sqrt {2} b F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\sqrt {b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 423
Rule 424
Rubi steps
\begin {align*} \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx &=\frac {(2 b) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx}{b-\sqrt {b^2-4 a c}}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) \int \frac {\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}} \, dx}{b-\sqrt {b^2-4 a c}}\\ &=-\frac {\left (b+\sqrt {b^2-4 a c}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} b F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 102, normalized size = 0.47 \[ \frac {\sqrt {-\sqrt {b^2-4 a c}-b} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}-b}\right )}{\sqrt {2} \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a\right )} \sqrt {-\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}} \sqrt {\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}}}{4 \, {\left (c x^{4} + b x^{2} + a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}+1}}{\sqrt {\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+1}}\, dx \]
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 \[ \int \frac {\sqrt {-\frac {2 \, c x^{2}}{b - \sqrt {b^{2} - 4 \, a c}} + 1}}{\sqrt {\frac {2 \, c x^{2}}{b + \sqrt {b^{2} - 4 \, a c}} + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}}}{\sqrt {\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}+1}} \,d x \]
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 \[ \int \frac {\sqrt {- \frac {- b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{b - \sqrt {- 4 a c + b^{2}}}}}{\sqrt {\frac {b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{b + \sqrt {- 4 a c + b^{2}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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