Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac{4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5} \]
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Rubi [A] time = 0.0586716, antiderivative size = 147, 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} \[ \frac{2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac{4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 \sqrt{d+e x}}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^{3/2}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{7/2}}{e^4}+\frac{c^2 (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac{2 d^2 (c d-b e)^2 (d+e x)^{3/2}}{3 e^5}-\frac{4 d (c d-b e) (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}\\ \end{align*}
Mathematica [A] time = 0.0716285, size = 124, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 141, normalized size = 1. \begin{align*}{\frac{630\,{c}^{2}{x}^{4}{e}^{4}+1540\,bc{e}^{4}{x}^{3}-560\,{c}^{2}d{e}^{3}{x}^{3}+990\,{b}^{2}{e}^{4}{x}^{2}-1320\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-792\,{b}^{2}d{e}^{3}x+1056\,bc{d}^{2}{e}^{2}x-384\,{c}^{2}{d}^{3}ex+528\,{b}^{2}{d}^{2}{e}^{2}-704\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{3465\,{e}^{5}} \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 = 1.11844, size = 188, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} - 770 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9181, size = 392, normalized size = 2.67 \begin{align*} \frac{2 \,{\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e + 264 \, b^{2} d^{3} e^{2} + 35 \,{\left (c^{2} d e^{4} + 22 \, b c e^{5}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{2} e^{3} - 22 \, b c d e^{4} - 99 \, b^{2} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 33 \, b^{2} d e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{4} e - 44 \, b c d^{3} e^{2} + 33 \, b^{2} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.65131, size = 173, normalized size = 1.18 \begin{align*} \frac{2 \left (\frac{c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28181, size = 227, normalized size = 1.54 \begin{align*} \frac{2}{3465} \,{\left (33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b^{2} e^{\left (-2\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b c e^{\left (-3\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} 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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