Optimal. Leaf size=91 \[ \frac {x^2 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}+\frac {(4 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 e^4}+\frac {3 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4} \]
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Rubi [A] time = 0.06, antiderivative size = 91, 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 = {850, 819, 780, 217, 203} \[ \frac {(4 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 e^4}+\frac {x^2 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}+\frac {3 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 819
Rule 850
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\int \frac {x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {x^2 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (2 d^3-3 d^2 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac {x^2 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}+\frac {(4 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 e^4}+\frac {\left (3 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^3}\\ &=\frac {x^2 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}+\frac {(4 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 e^4}+\frac {\left (3 d^2\right ) \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^3}\\ &=\frac {x^2 (d-e x)}{e^2 \sqrt {d^2-e^2 x^2}}+\frac {(4 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 e^4}+\frac {3 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 80, normalized size = 0.88 \[ \frac {\sqrt {d^2-e^2 x^2} \left (4 d^2+d e x-e^2 x^2\right )+3 d^2 (d+e x) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4 (d+e x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 101, normalized size = 1.11 \[ \frac {4 \, d^{2} e x + 4 \, d^{3} - 6 \, {\left (d^{2} e x + d^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (e^{2} x^{2} - d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, {\left (e^{5} x + d e^{4}\right )}} \]
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 = 120, normalized size = 1.32 \[ \frac {3 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{3}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, x}{2 e^{3}}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2}}{\left (x +\frac {d}{e}\right ) e^{5}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 86, normalized size = 0.95 \[ \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{5} x + d e^{4}} + \frac {3 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} x}{2 \, e^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{e^{4}} \]
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 \[ \int \frac {x^3}{\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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