Optimal. Leaf size=214 \[ -\frac {d x \sqrt {a+b x^2}}{b \sqrt {-c-d x^2}}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.07, 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 = {433, 429, 506,
422} \begin {gather*} -\frac {c^{3/2} \sqrt {a+b x^2} F\left (\text {ArcTan}\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 {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\text {ArcTan}\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 {d x \sqrt {a+b x^2}}{b \sqrt {-c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 433
Rule 506
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 = 0.87, size = 89, normalized size = 0.42 \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 = 0.08, size = 161, normalized size = 0.75
method | result | size |
default | \(\frac {\left (-a d \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )+c \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) b +a d \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )\right ) \sqrt {-d \,x^{2}-c}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}}{\left (b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c \right ) \sqrt {-\frac {d}{c}}\, b}\) | \(161\) |
elliptic | \(\frac {\sqrt {-\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {c \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}-c \,x^{2} b -a c}}+\frac {d a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}-c \,x^{2} b -a c}\, b}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}}\) | \(270\) |
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 = -0.00 \begin {gather*} \int \frac {\sqrt {-d\,x^2-c}}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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