Optimal. Leaf size=86 \[ \frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 86, 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 = {789, 653, 191} \[ \frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 653
Rule 789
Rubi steps
\begin {align*} \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e}\\ &=\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 55, normalized size = 0.64 \[ -\frac {(d+e x) \left (d^2-3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 104, normalized size = 1.21 \[ -\frac {e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} - {\left (e^{2} x^{2} - 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{2} e^{5} x^{3} - 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x - d^{5} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 60, normalized size = 0.70 \[ \frac {{\left (d^{3} e^{\left (-2\right )} + {\left (x {\left (\frac {x^{2} e^{3}}{d^{2}} - 5 \, e\right )} - 5 \, d\right )} x^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.60 \[ -\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (e^{2} x^{2}-3 d e x +d^{2}\right )}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 128, normalized size = 1.49 \[ \frac {e x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {d^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 46, normalized size = 0.53 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2-3\,d\,e\,x+e^2\,x^2\right )}{5\,d^2\,e^2\,{\left (d-e\,x\right )}^3} \]
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 \[ \int \frac {x \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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