Optimal. Leaf size=123 \[ -\frac{4 c \left (a e^2+3 c d^2\right )}{e^5 \sqrt{d+e x}}+\frac{8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d \sqrt{d+e x}}{e^5} \]
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Rubi [A] time = 0.046406, 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} \[ -\frac{4 c \left (a e^2+3 c d^2\right )}{e^5 \sqrt{d+e x}}+\frac{8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d \sqrt{d+e x}}{e^5} \]
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)^{7/2}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^{7/2}}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^{5/2}}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^{3/2}}-\frac{4 c^2 d}{e^4 \sqrt{d+e x}}+\frac{c^2 \sqrt{d+e x}}{e^4}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{8 c d \left (c d^2+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{4 c \left (3 c d^2+a e^2\right )}{e^5 \sqrt{d+e x}}-\frac{8 c^2 d \sqrt{d+e x}}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}\\ \end{align*}
Mathematica [A] time = 0.0636942, size = 96, normalized size = 0.78 \[ -\frac{2 \left (3 a^2 e^4+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+c^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 106, normalized size = 0.9 \begin{align*} -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}+80\,{c}^{2}d{x}^{3}{e}^{3}+60\,ac{e}^{4}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+80\,acd{e}^{3}x+640\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}+32\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1731, size = 163, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 12 \, \sqrt{e x + d} c^{2} d\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 30 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{2} - 20 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86978, size = 289, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} - 128 \, c^{2} d^{4} - 16 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 30 \,{\left (8 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} - 40 \,{\left (8 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.14516, size = 592, normalized size = 4.81 \begin{align*} \begin{cases} - \frac{6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.10165, size = 176, normalized size = 1.43 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} + 30 \,{\left (x e + d\right )}^{2} a c e^{2} - 20 \,{\left (x e + d\right )} a c d e^{2} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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