Optimal. Leaf size=125 \[ \frac{4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5} \]
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Rubi [A] time = 0.0453817, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {697} \[ \frac{4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^2}{\sqrt{d+e x}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 \sqrt{d+e x}}-\frac{4 c d \left (c d^2+a e^2\right ) \sqrt{d+e x}}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^4}-\frac{4 c^2 d (d+e x)^{5/2}}{e^4}+\frac{c^2 (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}{e^5}-\frac{8 c d \left (c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}\\ \end{align*}
Mathematica [A] time = 0.0610542, size = 96, normalized size = 0.77 \[ \frac{2 \sqrt{d+e x} \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 106, normalized size = 0.9 \begin{align*}{\frac{70\,{c}^{2}{x}^{4}{e}^{4}-80\,{c}^{2}d{x}^{3}{e}^{3}+252\,ac{e}^{4}{x}^{2}+96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-336\,acd{e}^{3}x-128\,{c}^{2}{d}^{3}ex+630\,{a}^{2}{e}^{4}+672\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10894, size = 162, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} + \frac{42 \,{\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 )} a c}{e^{2}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79864, size = 242, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} + 128 \, c^{2} d^{4} + 336 \, a c d^{2} e^{2} + 315 \, a^{2} e^{4} + 6 \,{\left (8 \, c^{2} d^{2} e^{2} + 21 \, a c e^{4}\right )} x^{2} - 8 \,{\left (8 \, c^{2} d^{3} e + 21 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.2914, size = 330, normalized size = 2.64 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d}{\sqrt{d + e x}} + 2 a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a 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{4 a 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}} + \frac{2 c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{2 c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5}}{\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 = 1.23855, size = 170, normalized size = 1.36 \begin{align*} \frac{2}{315} \,{\left (42 \,{\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 )} a c e^{\left (-2\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} c^{2} e^{\left (-4\right )} + 315 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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