Optimal. Leaf size=95 \[ -\frac {d \sqrt {d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac {8 \sqrt {d^2-e^2 x^2}}{15 e^3 (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{15 d e^3 (d+e x)} \]
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Rubi [A] time = 0.13, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1639, 793, 659, 651} \[ -\frac {d \sqrt {d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac {8 \sqrt {d^2-e^2 x^2}}{15 e^3 (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{15 d e^3 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 793
Rule 1639
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{e^3 (d+e x)^2}+\frac {\int \frac {2 d^2 e^2+d e^3 x}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{e^4}\\ &=-\frac {d \sqrt {d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac {\sqrt {d^2-e^2 x^2}}{e^3 (d+e x)^2}+\frac {(7 d) \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{5 e^2}\\ &=-\frac {d \sqrt {d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac {8 \sqrt {d^2-e^2 x^2}}{15 e^3 (d+e x)^2}+\frac {7 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{15 e^2}\\ &=-\frac {d \sqrt {d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac {8 \sqrt {d^2-e^2 x^2}}{15 e^3 (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{15 d e^3 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 52, normalized size = 0.55 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (2 d^2+6 d e x+7 e^2 x^2\right )}{15 d e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 104, normalized size = 1.09 \[ -\frac {2 \, e^{3} x^{3} + 6 \, d e^{2} x^{2} + 6 \, d^{2} e x + 2 \, d^{3} + {\left (7 \, e^{2} x^{2} + 6 \, d e x + 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{6} x^{3} + 3 \, d^{2} e^{5} x^{2} + 3 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
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 = 0.58 \[ -\frac {\left (-e x +d \right ) \left (7 e^{2} x^{2}+6 d e x +2 d^{2}\right )}{15 \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 125, normalized size = 1.32 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{5 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} + \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} - \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.76, size = 48, normalized size = 0.51 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^2+6\,d\,e\,x+7\,e^2\,x^2\right )}{15\,d\,e^3\,{\left (d+e\,x\right )}^3} \]
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 \[ \int \frac {x^{2}}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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