Optimal. Leaf size=123 \[ \frac{4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{8 c d \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac{2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A] time = 0.0472858, antiderivative size = 123, 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} \[ \frac{4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{8 c d \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac{2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^{3/2}}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 c \left (3 c d^2+a e^2\right ) \sqrt{d+e x}}{e^4}-\frac{4 c^2 d (d+e x)^{3/2}}{e^4}+\frac{c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{8 c d \left (c d^2+a e^2\right ) \sqrt{d+e x}}{e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}\\ \end{align*}
Mathematica [A] time = 0.0559045, size = 97, normalized size = 0.79 \[ -\frac{2 \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )}{105 e^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 106, normalized size = 0.9 \begin{align*} -{\frac{-30\,{c}^{2}{x}^{4}{e}^{4}+48\,{c}^{2}d{x}^{3}{e}^{3}-140\,ac{e}^{4}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+560\,acd{e}^{3}x+384\,{c}^{2}{d}^{3}ex+210\,{a}^{2}{e}^{4}+1120\,ac{d}^{2}{e}^{2}+768\,{c}^{2}{d}^{4}}{105\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10377, size = 163, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 84 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} d + 70 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 420 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{105 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{105 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69958, size = 261, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (15 \, c^{2} e^{4} x^{4} - 24 \, c^{2} d e^{3} x^{3} - 384 \, c^{2} d^{4} - 560 \, a c d^{2} e^{2} - 105 \, a^{2} e^{4} + 2 \,{\left (24 \, c^{2} d^{2} e^{2} + 35 \, a c e^{4}\right )} x^{2} - 8 \,{\left (24 \, c^{2} d^{3} e + 35 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.2785, size = 126, normalized size = 1.02 \begin{align*} - \frac{8 c^{2} d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 a c e^{2} + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (- 8 a c d e^{2} - 8 c^{2} d^{3}\right )}{e^{5}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2}}{e^{5} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30337, size = 185, normalized size = 1.5 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} e^{30} - 84 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d e^{30} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt{x e + d} c^{2} d^{3} e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c e^{32} - 420 \, \sqrt{x e + d} a c d e^{32}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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