Optimal. Leaf size=71 \[ \frac{8 (b+2 c x) (2 c d-b e)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0181793, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {638, 613} \[ \frac{8 (b+2 c x) (2 c d-b e)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 613
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{(4 (2 c d-b e)) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac{2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{8 (2 c d-b e) (b+2 c x)}{3 b^4 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0244812, size = 67, normalized size = 0.94 \[ -\frac{2 \left (-6 b^2 c x (d-2 e x)+b^3 (d+3 e x)+8 b c^2 x^2 (e x-3 d)-16 c^3 d x^3\right )}{3 b^4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 83, normalized size = 1.2 \begin{align*} -{\frac{2\,x \left ( cx+b \right ) \left ( 8\,b{c}^{2}e{x}^{3}-16\,{c}^{3}d{x}^{3}+12\,{b}^{2}ce{x}^{2}-24\,b{c}^{2}d{x}^{2}+3\,{b}^{3}ex-6\,{b}^{2}cdx+d{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14867, size = 176, normalized size = 2.48 \begin{align*} -\frac{4 \, c d x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, c^{2} d x}{3 \, \sqrt{c x^{2} + b x} b^{4}} + \frac{2 \, e x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{16 \, c e x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{2 \, d}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, c d}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \, e}{3 \, \sqrt{c x^{2} + b x} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96854, size = 209, normalized size = 2.94 \begin{align*} -\frac{2 \,{\left (b^{3} d - 8 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} - 12 \,{\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - 3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} x\right )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39542, size = 136, normalized size = 1.92 \begin{align*} \frac{{\left (4 \, x{\left (\frac{2 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (2 \, b^{2} c d - b^{3} e\right )}}{b^{4} c^{2}}\right )} x - \frac{d}{b c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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