Optimal. Leaf size=61 \[ \frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3} \]
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Rubi [A] time = 0.0208208, antiderivative size = 61, 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} \[ \frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{\sqrt{d+e x}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 \sqrt{d+e x}}-\frac{2 c d \sqrt{d+e x}}{e^2}+\frac{c (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right ) \sqrt{d+e x}}{e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}\\ \end{align*}
Mathematica [A] time = 0.0333469, size = 44, normalized size = 0.72 \[ \frac{2 \sqrt{d+e x} \left (15 a e^2+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 41, normalized size = 0.7 \begin{align*}{\frac{6\,c{e}^{2}{x}^{2}-8\,cdex+30\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14683, size = 72, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{e x + d} a + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78838, size = 96, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (3 \, c e^{2} x^{2} - 4 \, c d e x + 8 \, c d^{2} + 15 \, a e^{2}\right )} \sqrt{e x + d}}{15 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.98915, size = 150, normalized size = 2.46 \begin{align*} \begin{cases} - \frac{\frac{2 a d}{\sqrt{d + e x}} + 2 a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38449, size = 74, normalized size = 1.21 \begin{align*} \frac{2}{15} \,{\left ({\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} c e^{\left (-2\right )} + 15 \, \sqrt{x e + d} a\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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