Optimal. Leaf size=59 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3} \]
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Rubi [A] time = 0.0208537, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ -\frac{2 \left (a e^2+c d^2\right )}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}+\frac{c \sqrt{d+e x}}{e^2}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )}{e^3 \sqrt{d+e x}}-\frac{4 c d \sqrt{d+e x}}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3}\\ \end{align*}
Mathematica [A] time = 0.034477, size = 43, normalized size = 0.73 \[ \frac{2 \left (c \left (-8 d^2-4 d e x+e^2 x^2\right )-3 a e^2\right )}{3 e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 41, normalized size = 0.7 \begin{align*} -{\frac{-2\,c{e}^{2}{x}^{2}+8\,cdex+6\,a{e}^{2}+16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12727, size = 73, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 6 \, \sqrt{e x + d} c d}{e^{2}} - \frac{3 \,{\left (c d^{2} + a e^{2}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78122, size = 107, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (c e^{2} x^{2} - 4 \, c d e x - 8 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.83845, size = 58, normalized size = 0.98 \begin{align*} - \frac{4 c d \sqrt{d + e x}}{e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} - \frac{2 \left (a e^{2} + c d^{2}\right )}{e^{3} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.72146, size = 73, normalized size = 1.24 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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