Optimal. Leaf size=87 \[ \frac{16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0364209, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {722, 636} \[ \frac{16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 722
Rule 636
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{(8 d (c d-b e)) \int \frac{d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac{2 (d+e x)^2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{16 d (c d-b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0482613, size = 105, normalized size = 1.21 \[ \frac{2 \left (6 b^2 c d x \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (-9 d^2 e x-d^3+9 d e^2 x^2+e^3 x^3\right )+24 b c^2 d^2 x^2 (d-e x)+16 c^3 d^3 x^3\right )}{3 b^4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 136, normalized size = 1.6 \begin{align*} -{\frac{2\,x \left ( cx+b \right ) \left ( -{b}^{3}{e}^{3}{x}^{3}-6\,{b}^{2}cd{e}^{2}{x}^{3}+24\,b{c}^{2}{d}^{2}e{x}^{3}-16\,{c}^{3}{d}^{3}{x}^{3}-9\,{b}^{3}d{e}^{2}{x}^{2}+36\,{b}^{2}c{d}^{2}e{x}^{2}-24\,b{c}^{2}{d}^{3}{x}^{2}+9\,{b}^{3}{d}^{2}ex-6\,{b}^{2}c{d}^{3}x+{d}^{3}{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 = 1.17347, size = 401, normalized size = 4.61 \begin{align*} -\frac{e^{3} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{4 \, c d^{3} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, c^{2} d^{3} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} + \frac{2 \, d^{2} e x}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{16 \, c d^{2} e x}{\sqrt{c x^{2} + b x} b^{3}} + \frac{4 \, d e^{2} x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, d e^{2} x}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{b e^{3} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c^{2}} + \frac{2 \, e^{3} x}{3 \, \sqrt{c x^{2} + b x} b c} - \frac{2 \, d^{3}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, c d^{3}}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \, d^{2} e}{\sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, d e^{2}}{\sqrt{c x^{2} + b x} b c} + \frac{e^{3}}{3 \, \sqrt{c x^{2} + b x} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94477, size = 298, normalized size = 3.43 \begin{align*} -\frac{2 \,{\left (b^{3} d^{3} -{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{3} - 3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )} x^{2} - 3 \,{\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} 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{\left (d + e x\right )^{3}}{\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 = 1.28533, size = 188, normalized size = 2.16 \begin{align*} \frac{{\left (x{\left (\frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )}}{b^{4} c^{2}}\right )} x - \frac{d^{3}}{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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