Optimal. Leaf size=56 \[ \frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {639, 191} \begin {gather*} \frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}+\frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 639
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {d+e x}{3 d e \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}{3 d}\\ &=\frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.91 \begin {gather*} \frac {d^2+2 d e x-2 e^2 x^2}{3 d^3 e (d-e x) \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 58, normalized size = 1.04 \begin {gather*} \frac {\left (d^2+2 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 d^3 e (d-e x)^2 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 102, normalized size = 1.82 \begin {gather*} \frac {e^{3} x^{3} - d e^{2} x^{2} - d^{2} e x + d^{3} - {\left (2 \, e^{2} x^{2} - 2 \, d e x - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{3} e^{4} x^{3} - d^{4} e^{3} x^{2} - d^{5} e^{2} x + d^{6} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 52, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (x {\left (\frac {2 \, x^{2} e^{2}}{d^{3}} - \frac {3}{d}\right )} - e^{\left (-1\right )}\right )}}{3 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 53, normalized size = 0.95 \begin {gather*} \frac {\left (e x +d \right )^{2} \left (-e x +d \right ) \left (-2 e^{2} x^{2}+2 d e x +d^{2}\right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 60, normalized size = 1.07 \begin {gather*} \frac {x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {2 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 54, normalized size = 0.96 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+2\,d\,e\,x-2\,e^2\,x^2\right )}{3\,d^3\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.18, size = 296, normalized size = 5.29 \begin {gather*} d \left (\begin {cases} \frac {3 i d^{2} x}{- 3 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {2 i e^{2} x^{3}}{- 3 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {3 d^{2} x}{- 3 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {2 e^{2} x^{3}}{- 3 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} - \frac {1}{- 3 d^{2} e^{2} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 \left (d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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