Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5} \]
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Rubi [A] time = 0.0606813, 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} \[ \frac{2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 (d+e x)^{3/2}}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^{5/2}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{9/2}}{e^4}+\frac{c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 d^2 (c d-b e)^2 (d+e x)^{5/2}}{5 e^5}-\frac{4 d (c d-b e) (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}\\ \end{align*}
Mathematica [A] time = 0.0775826, size = 125, normalized size = 0.85 \[ \frac{2 (d+e x)^{5/2} \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (40 d^2 e x-16 d^3-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 141, normalized size = 1. \begin{align*}{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}+16380\,bc{e}^{4}{x}^{3}-5040\,{c}^{2}d{e}^{3}{x}^{3}+10010\,{b}^{2}{e}^{4}{x}^{2}-10920\,bcd{e}^{3}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-5720\,{b}^{2}d{e}^{3}x+6240\,bc{d}^{2}{e}^{2}x-1920\,{c}^{2}{d}^{3}ex+2288\,{b}^{2}{d}^{2}{e}^{2}-2496\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.218, size = 188, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 8190 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89437, size = 493, normalized size = 3.35 \begin{align*} \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e + 1144 \, b^{2} d^{4} e^{2} + 630 \,{\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \,{\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, b^{2} e^{6}\right )} x^{4} - 10 \,{\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 715 \, b^{2} d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 143 \, b^{2} d^{2} e^{4}\right )} x^{2} - 4 \,{\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 143 \, b^{2} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.7187, size = 413, normalized size = 2.81 \begin{align*} \frac{2 b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 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 b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{4 b c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{4 b c \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} + \frac{2 c^{2} d \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{5}} + \frac{2 c^{2} \left (- \frac{d^{5} \left (d + e x\right )^{\frac{3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac{5}{2}} - \frac{10 d^{3} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{10 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{9} - \frac{5 d \left (d + e x\right )^{\frac{11}{2}}}{11} + \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.42503, size = 506, normalized size = 3.44 \begin{align*} \frac{2}{45045} \,{\left (429 \,{\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} d e^{\left (-2\right )} + 286 \,{\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 d e^{\left (-3\right )} + 13 \,{\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} d e^{\left (-4\right )} + 143 \,{\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^{2} e^{\left (-2\right )} + 26 \,{\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 )} b c e^{\left (-3\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\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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