Optimal. Leaf size=86 \[ \frac {d (2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3} \]
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Rubi [A] time = 0.11, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1639, 12, 785, 780, 217, 203} \[ \frac {d (2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 217
Rule 780
Rule 785
Rule 1639
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {\int \frac {3 d e^3 x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx}{3 e^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {d \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx}{e}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {\int \frac {x \left (d^2 e-d e^2 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}\\ &=\frac {d (2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {d^3 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=\frac {d (2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2}\\ &=\frac {d (2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.80 \[ \frac {\sqrt {d^2-e^2 x^2} \left (4 d^2-3 d e x+2 e^2 x^2\right )+3 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{6 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 73, normalized size = 0.85 \[ -\frac {6 \, d^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (2 \, e^{2} x^{2} - 3 \, d e x + 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 54, normalized size = 0.63 \[ \frac {1}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\relax (d) + \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (4 \, d^{2} e^{\left (-3\right )} + {\left (2 \, x e^{\left (-1\right )} - 3 \, d e^{\left (-2\right )}\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 160, normalized size = 1.86 \[ \frac {d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e^{2}}-\frac {d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{2}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d x}{2 e^{2}}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2}}{e^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 77, normalized size = 0.90 \[ \frac {d^{3} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d x}{2 \, e^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, e^{3}} \]
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 \[ \int \frac {x^2\,\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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