Optimal. Leaf size=127 \[ \frac{4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5} \]
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Rubi [A] time = 0.0476634, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {697} \[ \frac{4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^4}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}-\frac{4 c^2 d (d+e x)^{9/2}}{e^4}+\frac{c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^5}-\frac{8 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (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 = 0.0870545, size = 97, normalized size = 0.76 \[ \frac{2 (d+e x)^{5/2} \left (9009 a^2 e^4+286 a c e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+3 c^2 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 106, normalized size = 0.8 \begin{align*}{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}-5040\,{c}^{2}d{x}^{3}{e}^{3}+20020\,ac{e}^{4}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-11440\,acd{e}^{3}x-1920\,{c}^{2}{d}^{3}ex+18018\,{a}^{2}{e}^{4}+4576\,ac{d}^{2}{e}^{2}+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 = 1.14843, size = 153, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 16380 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} d + 10010 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 25740 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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 = 1.81298, size = 423, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 4410 \, c^{2} d e^{5} x^{5} + 384 \, c^{2} d^{6} + 2288 \, a c d^{4} e^{2} + 9009 \, a^{2} d^{2} e^{4} + 35 \,{\left (3 \, c^{2} d^{2} e^{4} + 286 \, a c e^{6}\right )} x^{4} - 20 \,{\left (6 \, c^{2} d^{3} e^{3} - 715 \, a c d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} + 286 \, a c d^{2} e^{4} + 3003 \, a^{2} e^{6}\right )} x^{2} - 2 \,{\left (96 \, c^{2} d^{5} e + 572 \, a c d^{3} e^{3} - 9009 \, a^{2} d e^{5}\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 [A] time = 10.6265, size = 328, normalized size = 2.58 \begin{align*} a^{2} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{2} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{4 a c 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{4 a c \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{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 = 1.41704, size = 394, normalized size = 3.1 \begin{align*} \frac{2}{45045} \,{\left (858 \,{\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 )} a c d e^{\left (-2\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 )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d + 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 )} a c e^{\left (-2\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 )} + 3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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