Optimal. Leaf size=145 \[ \frac{2 (d+e x)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac{2 d^2 \sqrt{d+e x} (c d-b e)^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac{4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]
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Rubi [A] time = 0.0589171, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {698} \[ \frac{2 (d+e x)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac{2 d^2 \sqrt{d+e x} (c d-b e)^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac{4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^2}{\sqrt{d+e x}} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 \sqrt{d+e x}}+\frac{2 d (c d-b e) (-2 c d+b e) \sqrt{d+e x}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{5/2}}{e^4}+\frac{c^2 (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac{2 d^2 (c d-b e)^2 \sqrt{d+e x}}{e^5}-\frac{4 d (c d-b e) (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}\\ \end{align*}
Mathematica [A] time = 0.0760272, size = 124, normalized size = 0.86 \[ \frac{2 \sqrt{d+e x} \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 141, normalized size = 1. \begin{align*}{\frac{70\,{c}^{2}{x}^{4}{e}^{4}+180\,bc{e}^{4}{x}^{3}-80\,{c}^{2}d{e}^{3}{x}^{3}+126\,{b}^{2}{e}^{4}{x}^{2}-216\,bcd{e}^{3}{x}^{2}+96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-168\,{b}^{2}d{e}^{3}x+288\,bc{d}^{2}{e}^{2}x-128\,{c}^{2}{d}^{3}ex+336\,{b}^{2}{d}^{2}{e}^{2}-576\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17526, size = 216, normalized size = 1.49 \begin{align*} \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} b c}{e^{3}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91698, size = 312, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 288 \, b c d^{3} e + 168 \, b^{2} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 9 \, b c e^{4}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{2} e^{2} - 36 \, b c d e^{3} + 21 \, b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{3} e - 36 \, b c d^{2} e^{2} + 21 \, b^{2} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 62.1308, size = 418, normalized size = 2.88 \begin{align*} \begin{cases} - \frac{\frac{2 b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 b c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{4 b c \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{2 c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{\frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27692, size = 227, normalized size = 1.57 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} b c e^{\left (-3\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} c^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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