Optimal. Leaf size=204 \[ \frac{6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac{6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7} \]
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Rubi [A] time = 0.0826531, antiderivative size = 204, 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} \[ \frac{6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac{6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 \sqrt{d+e x}}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^6}-\frac{6 c^3 d (d+e x)^{11/2}}{e^6}+\frac{c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac{12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac{6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac{6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}\\ \end{align*}
Mathematica [A] time = 0.144417, size = 170, normalized size = 0.83 \[ \frac{2 (d+e x)^{3/2} \left (1287 a^2 c e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+15015 a^3 e^6+39 a c^2 e^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )+c^3 \left (1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-1536 d^5 e x+1024 d^6-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 205, normalized size = 1. \begin{align*}{\frac{6006\,{c}^{3}{x}^{6}{e}^{6}-5544\,{c}^{3}d{x}^{5}{e}^{5}+24570\,a{c}^{2}{e}^{6}{x}^{4}+5040\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-21840\,a{c}^{2}d{e}^{5}{x}^{3}-4480\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+38610\,{a}^{2}c{e}^{6}{x}^{2}+18720\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-30888\,{a}^{2}cd{e}^{5}x-14976\,a{c}^{2}{d}^{3}{e}^{3}x-3072\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}+20592\,{a}^{2}c{d}^{2}{e}^{4}+9984\,{d}^{4}{e}^{2}a{c}^{2}+2048\,{c}^{3}{d}^{6}}{45045\,{e}^{7}} \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 = 1.14468, size = 282, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} - 20790 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{3} d + 12285 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 19305 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 54054 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8093, size = 581, normalized size = 2.85 \begin{align*} \frac{2 \,{\left (3003 \, c^{3} e^{7} x^{7} + 231 \, c^{3} d e^{6} x^{6} + 1024 \, c^{3} d^{7} + 4992 \, a c^{2} d^{5} e^{2} + 10296 \, a^{2} c d^{3} e^{4} + 15015 \, a^{3} d e^{6} - 63 \,{\left (4 \, c^{3} d^{2} e^{5} - 195 \, a c^{2} e^{7}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{3} e^{4} + 39 \, a c^{2} d e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{4} e^{3} + 312 \, a c^{2} d^{2} e^{5} - 3861 \, a^{2} c e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{5} e^{2} + 624 \, a c^{2} d^{3} e^{4} + 1287 \, a^{2} c d e^{6}\right )} x^{2} -{\left (512 \, c^{3} d^{6} e + 2496 \, a c^{2} d^{4} e^{3} + 5148 \, a^{2} c d^{2} e^{5} - 15015 \, a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.06417, size = 265, normalized size = 1.3 \begin{align*} \frac{2 \left (- \frac{6 c^{3} d \left (d + e x\right )^{\frac{13}{2}}}{13 e^{6}} + \frac{c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36843, size = 301, normalized size = 1.48 \begin{align*} \frac{2}{45045} \,{\left (1287 \,{\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^{2} c e^{\left (-2\right )} + 39 \,{\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 )} a c^{2} e^{\left (-4\right )} +{\left (3003 \,{\left (x e + d\right )}^{\frac{15}{2}} - 20790 \,{\left (x e + d\right )}^{\frac{13}{2}} d + 61425 \,{\left (x e + d\right )}^{\frac{11}{2}} d^{2} - 100100 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{3} + 96525 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{4} - 54054 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{5} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{6}\right )} c^{3} e^{\left (-6\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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