Optimal. Leaf size=97 \[ -\frac {\sqrt {d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac {\sqrt {d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]
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Rubi [A] time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {793, 659, 651} \[ -\frac {\sqrt {d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac {\sqrt {d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 793
Rubi steps
\begin {align*} \int \frac {x}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{5 e^2 (d+e x)^3}+\frac {3 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{5 e}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{5 e^2 (d+e x)^3}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac {\int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{5 d e}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{5 e^2 (d+e x)^3}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.51 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (d^2+3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 100, normalized size = 1.03 \[ -\frac {e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3} + {\left (e^{2} x^{2} + 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{2} e^{5} x^{3} + 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \]
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 = 52, normalized size = 0.54 \[ -\frac {\left (-e x +d \right ) \left (e^{2} x^{2}+3 d e x +d^{2}\right )}{5 \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 129, normalized size = 1.33 \[ \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d e^{4} x^{2} + 2 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{2} e^{3} x + d^{3} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 45, normalized size = 0.46 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+3\,d\,e\,x+e^2\,x^2\right )}{5\,d^2\,e^2\,{\left (d+e\,x\right )}^3} \]
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 \[ \int \frac {x}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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