Optimal. Leaf size=91 \[ -\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 91, 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 = {793, 659, 191} \[ \frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}}-\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 659
Rule 793
Rubi steps
\begin {align*} \int \frac {x}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e}\\ &=\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}}+\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 69, normalized size = 0.76 \[ \frac {\sqrt {d^2-e^2 x^2} \left (d^3+2 d^2 e x+8 d e^2 x^2+4 e^3 x^3\right )}{15 d^3 e^2 (d-e x) (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 116, normalized size = 1.27 \[ \frac {e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4} - {\left (4 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} + 2 \, d^{2} e x + d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{6} x^{4} + 2 \, d^{4} e^{5} x^{3} - 2 \, d^{6} e^{3} x - d^{7} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 0.70 \[ \frac {\left (-e x +d \right ) \left (4 e^{3} x^{3}+8 d \,e^{2} x^{2}+2 d^{2} e x +d^{3}\right )}{15 \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 138, normalized size = 1.52 \[ \frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x^{2} + 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{3} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2}\right )}} - \frac {2}{15 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d e^{3} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2}\right )}} + \frac {4 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.88, size = 65, normalized size = 0.71 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (d^3+2\,d^2\,e\,x+8\,d\,e^2\,x^2+4\,e^3\,x^3\right )}{15\,d^3\,e^2\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \]
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 \[ \int \frac {x}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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