Optimal. Leaf size=100 \[ -\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]
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Rubi [A] time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}+\frac {2 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{5 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}+\frac {2 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{15 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.52 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (7 d^2+6 d e x+2 e^2 x^2\right )}{15 d^3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 104, normalized size = 1.04 \[ -\frac {7 \, e^{3} x^{3} + 21 \, d e^{2} x^{2} + 21 \, d^{2} e x + 7 \, d^{3} + {\left (2 \, e^{2} x^{2} + 6 \, d e x + 7 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} e\right )}} \]
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 = 55, normalized size = 0.55 \[ -\frac {\left (-e x +d \right ) \left (2 e^{2} x^{2}+6 d e x +7 d^{2}\right )}{15 \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 128, normalized size = 1.28 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{2} x + d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.62, size = 48, normalized size = 0.48 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (7\,d^2+6\,d\,e\,x+2\,e^2\,x^2\right )}{15\,d^3\,e\,{\left (d+e\,x\right )}^3} \]
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 \[ \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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