Optimal. Leaf size=87 \[ \frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}}-\frac {c \sqrt {\frac {d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {493, 426, 424, 421, 419} \[ \frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}}-\frac {c \sqrt {\frac {d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rule 424
Rule 426
Rule 493
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx &=\frac {\int \frac {\sqrt {c+d x^2}}{\sqrt {4-x^2}} \, dx}{d}-\frac {c \int \frac {1}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx}{d}\\ &=\frac {\sqrt {c+d x^2} \int \frac {\sqrt {1+\frac {d x^2}{c}}}{\sqrt {4-x^2}} \, dx}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {\left (c \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {4-x^2} \sqrt {1+\frac {d x^2}{c}}} \, dx}{d \sqrt {c+d x^2}}\\ &=\frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {c \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.68 \[ \frac {c \sqrt {\frac {d x^2}{c}+1} \left (E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )-F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )\right )}{d \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4} x^{2}}{d x^{4} + {\left (c - 4 \, d\right )} x^{2} - 4 \, c}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 0.68 \[ \frac {\left (\EllipticE \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )-\EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, c}{\sqrt {d \,x^{2}+c}\, d} \]
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 \[ \int \frac {x^{2}}{\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right )} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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