Optimal. Leaf size=88 \[ \frac {x \sqrt {c+d x^2}}{d \sqrt {x^2+4}}-\frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{d \sqrt {x^2+4} \sqrt {\frac {c+d x^2}{c \left (x^2+4\right )}}} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {492, 411} \[ \frac {x \sqrt {c+d x^2}}{d \sqrt {x^2+4}}-\frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{d \sqrt {x^2+4} \sqrt {\frac {c+d x^2}{c \left (x^2+4\right )}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 492
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx &=\frac {x \sqrt {c+d x^2}}{d \sqrt {4+x^2}}-\frac {4 \int \frac {\sqrt {c+d x^2}}{\left (4+x^2\right )^{3/2}} \, dx}{d}\\ &=\frac {x \sqrt {c+d x^2}}{d \sqrt {4+x^2}}-\frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{d \sqrt {4+x^2} \sqrt {\frac {c+d x^2}{c \left (4+x^2\right )}}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 70, normalized size = 0.80 \[ -\frac {i c \sqrt {\frac {d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\frac {x}{2}\right )|\frac {4 d}{c}\right )-F\left (i \sinh ^{-1}\left (\frac {x}{2}\right )|\frac {4 d}{c}\right )\right )}{d \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, 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 = 76, normalized size = 0.86 \[ -\frac {2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \left (-\EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \frac {\sqrt {\frac {c}{d}}}{2}\right )+\EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \frac {\sqrt {\frac {c}{d}}}{2}\right )\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}} \]
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 \[ \int \frac {x^2}{\sqrt {x^2+4}\,\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {c + d x^{2}} \sqrt {x^{2} + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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