Optimal. Leaf size=95 \[ \frac {\sqrt {b+\sqrt {b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \]
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Rubi [A]
time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {435}
\begin {gather*} \frac {\sqrt {\sqrt {b^2-4 a c}+b} E\left (\text {ArcSin}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 435
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 {\sqrt {b+\sqrt {b^2-4 a c}} 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}}\\ \end {align*}
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Mathematica [A]
time = 2.30, size = 95, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b+\sqrt {b^2-4 a c}} 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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(808\) vs.
\(2(80)=160\).
time = 0.33, size = 809, normalized size = 8.52
method | result | size |
elliptic | \(\frac {\sqrt {\frac {-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}}\, \left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {-\frac {\left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right ) \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{a c}}\, \left (\frac {\sqrt {2}\, \sqrt {1-\frac {2 c \,x^{2}}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (x \sqrt {2}\, \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4+\frac {2 \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}\right ) \left (b -\sqrt {-4 a c +b^{2}}\right )}{c}}}{2}\right )}{2 \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}-\frac {4 c^{2} x^{4}}{\left (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 c \sqrt {2}\, \sqrt {1-\frac {2 c \,x^{2}}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\EllipticF \left (x \sqrt {2}\, \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4+\frac {2 \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}\right ) \left (b -\sqrt {-4 a c +b^{2}}\right )}{c}}}{2}\right )-\EllipticE \left (x \sqrt {2}\, \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4+\frac {2 \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}\right ) \left (b -\sqrt {-4 a c +b^{2}}\right )}{c}}}{2}\right )\right )}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}-\frac {4 c^{2} x^{4}}{\left (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}}\, \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}-\frac {b}{a}\right )}\right )}{2 \sqrt {\frac {-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}\) | \(809\) |
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 {\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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}+1}}{\sqrt {1-\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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