Optimal. Leaf size=87 \[ \frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {c \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.05, 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 = {507, 437, 435,
432, 430} \begin {gather*} \frac {\sqrt {c+d x^2} E\left (\text {ArcSin}\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 (\text {ArcSin}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 507
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 = 0.52, size = 59, normalized size = 0.68 \begin {gather*} \frac {c \sqrt {1+\frac {d x^2}{c}} \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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 59, normalized size = 0.68
method | result | size |
default | \(\frac {\left (-\EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )+\EllipticE \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) c \sqrt {\frac {d \,x^{2}+c}{c}}}{\sqrt {d \,x^{2}+c}\, d}\) | \(59\) |
elliptic | \(-\frac {\sqrt {-\left (d \,x^{2}+c \right ) \left (x^{2}-4\right )}\, c \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )-\EllipticE \left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}\, d}\) | \(111\) |
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 {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right )} \sqrt {c + d 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 = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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