Optimal. Leaf size=77 \[ \frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {653, 192, 191} \[ \frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 653
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {3}{5} \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d^2}\\ &=\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.82 \[ \frac {2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3}{5 d^4 e (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 116, normalized size = 1.51 \[ \frac {2 \, e^{4} x^{4} - 4 \, d e^{3} x^{3} + 4 \, d^{3} e x - 2 \, d^{4} - {\left (2 \, e^{3} x^{3} - 4 \, d e^{2} x^{2} + d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{4} e^{5} x^{4} - 2 \, d^{5} e^{4} x^{3} + 2 \, d^{7} e^{2} x - d^{8} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 61, normalized size = 0.79 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left (x^{2} {\left (\frac {2 \, x^{2} e^{4}}{d^{4}} - \frac {5 \, e^{2}}{d^{2}}\right )} + 5\right )} x + 2 \, d e^{\left (-1\right )}\right )}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.84 \[ \frac {\left (-e x +d \right ) \left (e x +d \right )^{3} \left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right )}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{4} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 78, normalized size = 1.01 \[ \frac {2 \, x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {2 \, x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.81, size = 66, normalized size = 0.86 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^3+d^2\,e\,x-4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \]
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 \[ \int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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