Optimal. Leaf size=59 \[ -\frac{2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}+\frac{4 c d}{e^3 \sqrt{d+e x}} \]
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Rubi [A] time = 0.0222372, antiderivative size = 59, 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} \[ -\frac{2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}+\frac{4 c d}{e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^{5/2}}-\frac{2 c d}{e^2 (d+e x)^{3/2}}+\frac{c}{e^2 \sqrt{d+e x}}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 c d}{e^3 \sqrt{d+e x}}+\frac{2 c \sqrt{d+e x}}{e^3}\\ \end{align*}
Mathematica [A] time = 0.037179, size = 44, normalized size = 0.75 \[ \frac{2 \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-a e^2\right )}{3 e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 40, normalized size = 0.7 \begin{align*} -{\frac{-6\,c{e}^{2}{x}^{2}-24\,cdex+2\,a{e}^{2}-16\,c{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11347, size = 70, normalized size = 1.19 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} c}{e^{2}} + \frac{6 \,{\left (e x + d\right )} c d - c d^{2} - a e^{2}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84766, size = 130, normalized size = 2.2 \begin{align*} \frac{2 \,{\left (3 \, c e^{2} x^{2} + 12 \, c d e x + 8 \, c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19617, size = 168, normalized size = 2.85 \begin{align*} \begin{cases} - \frac{2 a e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 c d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 c d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 c e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31088, size = 65, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x e + d} c e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} c d - c d^{2} - a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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