Optimal. Leaf size=33 \[ \frac {(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {651} \begin {gather*} \frac {(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 41, normalized size = 1.24 \begin {gather*} \frac {(d+e x)^2}{3 d e (d-e x) \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 40, normalized size = 1.21 \begin {gather*} \frac {(d+e x) \sqrt {d^2-e^2 x^2}}{3 d e (e x-d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 63, normalized size = 1.91 \begin {gather*} \frac {e^{2} x^{2} - 2 \, d e x + d^{2} + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}}{3 \, {\left (d e^{3} x^{2} - 2 \, d^{2} e^{2} x + d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 56, normalized size = 1.70 \begin {gather*} \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (d^{2} e^{\left (-1\right )} + {\left (x {\left (\frac {x e^{2}}{d} + 3 \, e\right )} + 3 \, d\right )} x\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 = 36, normalized size = 1.09 \begin {gather*} \frac {\left (e x +d \right )^{4} \left (-e x +d \right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 80, normalized size = 2.42 \begin {gather*} \frac {e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {4 \, d x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {d^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 35, normalized size = 1.06 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )}{3\,d\,e\,{\left (d-e\,x\right )}^2} \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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