Optimal. Leaf size=103 \[ \frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d-e x)} \]
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Rubi [A] time = 0.05, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {655, 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 655
Rule 659
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\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 = 0.06, size = 58, normalized size = 0.56 \[ \frac {(d+e x) \left (7 d^2-6 d e x+2 e^2 x^2\right )}{15 d^3 e (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 106, normalized size = 1.03 \[ \frac {7 \, e^{3} x^{3} - 21 \, d e^{2} x^{2} + 21 \, d^{2} e x - 7 \, d^{3} - {\left (2 \, e^{2} x^{2} - 6 \, d e x + 7 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{4} x^{3} - 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x - d^{6} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 70, normalized size = 0.68 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (7 \, d^{2} e^{\left (-1\right )} + {\left ({\left (x {\left (\frac {2 \, x^{2} e^{4}}{d^{3}} - \frac {5 \, e^{2}}{d}\right )} + 5 \, e\right )} x + 15 \, d\right )} x\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.53 \[ \frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (2 e^{2} x^{2}-6 d e x +7 d^{2}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 101, normalized size = 0.98 \[ \frac {e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7 \, d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {2 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.70, size = 49, 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 \[ \int \frac {\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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