Optimal. Leaf size=64 \[ -\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]
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Rubi [A] time = 0.0271376, antiderivative size = 64, 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 = {698} \[ -\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{b x+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{d (c d-b e)}{e^2 (d+e x)^{3/2}}+\frac{-2 c d+b e}{e^2 \sqrt{d+e x}}+\frac{c \sqrt{d+e x}}{e^2}\right ) \, dx\\ &=-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}-\frac{2 (2 c d-b e) \sqrt{d+e x}}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3}\\ \end{align*}
Mathematica [A] time = 0.0305917, size = 48, normalized size = 0.75 \[ \frac{2 \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 46, normalized size = 0.7 \begin{align*}{\frac{2\,c{e}^{2}{x}^{2}+6\,b{e}^{2}x-8\,cdex+12\,bde-16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08546, size = 82, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 3 \,{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c d^{2} - b d e\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88869, size = 123, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (c e^{2} x^{2} - 8 \, c d^{2} + 6 \, b d e -{\left (4 \, c d e - 3 \, b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5883, size = 60, normalized size = 0.94 \begin{align*} \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 d \left (b e - c d\right )}{e^{3} \sqrt{d + e x}} + \frac{\sqrt{d + e x} \left (2 b e - 4 c d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28597, size = 93, normalized size = 1.45 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6} + 3 \, \sqrt{x e + d} b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} - b d e\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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