Optimal. Leaf size=61 \[ -\frac{2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.0225897, 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} \[ -\frac{2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^{7/2}}-\frac{2 c d}{e^2 (d+e x)^{5/2}}+\frac{c}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )}{5 e^3 (d+e x)^{5/2}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0385796, size = 44, normalized size = 0.72 \[ -\frac{2 \left (3 a e^2+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 41, normalized size = 0.7 \begin{align*} -{\frac{30\,c{e}^{2}{x}^{2}+40\,cdex+6\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}} \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 = 1.09602, size = 59, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c - 10 \,{\left (e x + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82718, size = 158, normalized size = 2.59 \begin{align*} -\frac{2 \,{\left (15 \, c e^{2} x^{2} + 20 \, c d e x + 8 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.8407, size = 252, normalized size = 4.13 \begin{align*} \begin{cases} - \frac{6 a e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{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 = 1.35545, size = 61, normalized size = 1. \begin{align*} -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} c - 10 \,{\left (x e + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )} e^{\left (-3\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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