Optimal. Leaf size=77 \[ -\frac {d \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1639, 12, 793, 217, 203} \[ -\frac {d \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 217
Rule 793
Rule 1639
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {\int \frac {d e^3 x}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{e^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \int \frac {x}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{e}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {d \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 59, normalized size = 0.77 \[ -\frac {\frac {\sqrt {d^2-e^2 x^2} (2 d+e x)}{d+e x}+d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 85, normalized size = 1.10 \[ -\frac {2 \, d e x + 2 \, d^{2} - 2 \, {\left (d e x + d^{2}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + 2 \, d\right )}}{e^{4} x + d e^{3}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 1.26 \[ -\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{2}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d}{\left (x +\frac {d}{e}\right ) e^{4}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 63, normalized size = 0.82 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{e^{4} x + d e^{3}} - \frac {d \arcsin \left (\frac {e x}{d}\right )}{e^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e^{3}} \]
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 \[ \int \frac {x^2}{\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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