Optimal. Leaf size=39 \[ \frac{2 x}{3 c^2 \sqrt{c+d x^2}}+\frac{x}{3 c \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0056034, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{2 x}{3 c^2 \sqrt{c+d x^2}}+\frac{x}{3 c \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{x}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac{x}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x}{3 c^2 \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0093256, size = 29, normalized size = 0.74 \[ \frac{x \left (3 c+2 d x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 26, normalized size = 0.7 \begin{align*}{\frac{x \left ( 2\,d{x}^{2}+3\,c \right ) }{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952149, size = 42, normalized size = 1.08 \begin{align*} \frac{2 \, x}{3 \, \sqrt{d x^{2} + c} c^{2}} + \frac{x}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49205, size = 99, normalized size = 2.54 \begin{align*} \frac{{\left (2 \, d x^{3} + 3 \, c x\right )} \sqrt{d x^{2} + c}}{3 \,{\left (c^{2} d^{2} x^{4} + 2 \, c^{3} d x^{2} + c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.792314, size = 95, normalized size = 2.44 \begin{align*} \frac{3 c x}{3 c^{\frac{7}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 3 c^{\frac{5}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{2 d x^{3}}{3 c^{\frac{7}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 3 c^{\frac{5}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16096, size = 36, normalized size = 0.92 \begin{align*} \frac{x{\left (\frac{2 \, d x^{2}}{c^{2}} + \frac{3}{c}\right )}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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