Optimal. Leaf size=119 \[ -\frac {2 c^2 d^2 (d+e x)^{9/2} \left (c d^2-a e^2\right )}{3 e^4}+\frac {6 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \]
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Rubi [A] time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {626, 43} \begin {gather*} -\frac {2 c^2 d^2 (d+e x)^{9/2} \left (c d^2-a e^2\right )}{3 e^4}+\frac {6 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int (a e+c d x)^3 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^{3/2}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{9/2}}{e^3}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}{5 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{7 e^4}-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 98, normalized size = 0.82 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-385 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right )+495 c d (d+e x) \left (c d^2-a e^2\right )^2-231 \left (c d^2-a e^2\right )^3+105 c^3 d^3 (d+e x)^3\right )}{1155 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 151, normalized size = 1.27 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (231 a^3 e^6-693 a^2 c d^2 e^4+495 a^2 c d e^4 (d+e x)+693 a c^2 d^4 e^2-990 a c^2 d^3 e^2 (d+e x)+385 a c^2 d^2 e^2 (d+e x)^2-231 c^3 d^6+495 c^3 d^5 (d+e x)-385 c^3 d^4 (d+e x)^2+105 c^3 d^3 (d+e x)^3\right )}{1155 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 231, normalized size = 1.94 \begin {gather*} \frac {2 \, {\left (105 \, c^{3} d^{3} e^{5} x^{5} - 16 \, c^{3} d^{8} + 88 \, a c^{2} d^{6} e^{2} - 198 \, a^{2} c d^{4} e^{4} + 231 \, a^{3} d^{2} e^{6} + 35 \, {\left (4 \, c^{3} d^{4} e^{4} + 11 \, a c^{2} d^{2} e^{6}\right )} x^{4} + 5 \, {\left (c^{3} d^{5} e^{3} + 110 \, a c^{2} d^{3} e^{5} + 99 \, a^{2} c d e^{7}\right )} x^{3} - 3 \, {\left (2 \, c^{3} d^{6} e^{2} - 11 \, a c^{2} d^{4} e^{4} - 264 \, a^{2} c d^{2} e^{6} - 77 \, a^{3} e^{8}\right )} x^{2} + {\left (8 \, c^{3} d^{7} e - 44 \, a c^{2} d^{5} e^{3} + 99 \, a^{2} c d^{3} e^{5} + 462 \, a^{3} d e^{7}\right )} x\right )} \sqrt {e x + d}}{1155 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 185, normalized size = 1.55 \begin {gather*} \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} d^{3} e^{40} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d^{4} e^{40} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{5} e^{40} - 231 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{6} e^{40} + 385 \, {\left (x e + d\right )}^{\frac {9}{2}} a c^{2} d^{2} e^{42} - 990 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d^{3} e^{42} + 693 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{4} e^{42} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} c d e^{44} - 693 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c d^{2} e^{44} + 231 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} e^{46}\right )} e^{\left (-44\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 131, normalized size = 1.10 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 c^{3} d^{3} e^{3} x^{3}+385 a \,c^{2} d^{2} e^{4} x^{2}-70 c^{3} d^{4} e^{2} x^{2}+495 a^{2} c d \,e^{5} x -220 a \,c^{2} d^{3} e^{3} x +40 c^{3} d^{5} e x +231 a^{3} e^{6}-198 a^{2} c \,d^{2} e^{4}+88 a \,c^{2} d^{4} e^{2}-16 c^{3} d^{6}\right )}{1155 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 137, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} d^{3} - 385 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\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 {7}{2}} - 231 \, {\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 {5}{2}}\right )}}{1155 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 106, normalized size = 0.89 \begin {gather*} \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 110.40, size = 971, normalized size = 8.16
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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