Optimal. Leaf size=119 \[ -\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{9/2}}{9 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{11/2}}{11 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{13/2}}{13 e^4}+\frac {2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \]
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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 = {640, 45}
\begin {gather*} -\frac {6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac {6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac {2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac {2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^{7/2}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{9/2}}{9 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{11/2}}{11 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{13/2}}{13 e^4}+\frac {2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 111, normalized size = 0.93 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (715 a^3 e^6-195 a^2 c d e^4 (2 d-9 e x)+15 a c^2 d^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 97, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(97\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(97\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 c^{3} d^{3} e^{3} x^{3}+1485 a \,c^{2} d^{2} e^{4} x^{2}-198 c^{3} d^{4} e^{2} x^{2}+1755 a^{2} c d \,e^{5} x -540 a \,c^{2} d^{3} e^{3} x +72 c^{3} d^{5} e x +715 e^{6} a^{3}-390 e^{4} d^{2} a^{2} c +120 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{6435 e^{4}}\) | \(131\) |
trager | \(\frac {2 \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 a \,c^{2} d^{4} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(359\) |
risch | \(\frac {2 \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 a \,c^{2} d^{4} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(359\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 134, normalized size = 1.13 \begin {gather*} \frac {2}{6435} \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{3} d^{3} - 1485 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {13}{2}} + 1755 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 715 \, {\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 (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 325 vs.
\(2 (100) = 200\).
time = 3.47, size = 325, normalized size = 2.73 \begin {gather*} \frac {2}{6435} \, {\left (8 \, c^{3} d^{9} x e - 16 \, c^{3} d^{10} + 715 \, a^{3} x^{4} e^{10} + 65 \, {\left (27 \, a^{2} c d x^{5} + 44 \, a^{3} d x^{3}\right )} e^{9} + 15 \, {\left (99 \, a c^{2} d^{2} x^{6} + 442 \, a^{2} c d^{2} x^{4} + 286 \, a^{3} d^{2} x^{2}\right )} e^{8} + {\left (429 \, c^{3} d^{3} x^{7} + 5400 \, a c^{2} d^{3} x^{5} + 8970 \, a^{2} c d^{3} x^{3} + 2860 \, a^{3} d^{3} x\right )} e^{7} + {\left (1518 \, c^{3} d^{4} x^{6} + 6870 \, a c^{2} d^{4} x^{4} + 4680 \, a^{2} c d^{4} x^{2} + 715 \, a^{3} d^{4}\right )} e^{6} + 3 \, {\left (618 \, c^{3} d^{5} x^{5} + 1060 \, a c^{2} d^{5} x^{3} + 65 \, a^{2} c d^{5} x\right )} e^{5} + 5 \, {\left (160 \, c^{3} d^{6} x^{4} + 9 \, a c^{2} d^{6} x^{2} - 78 \, a^{2} c d^{6}\right )} e^{4} + 5 \, {\left (c^{3} d^{7} x^{3} - 12 \, a c^{2} d^{7} x\right )} e^{3} - 6 \, {\left (c^{3} d^{8} x^{2} - 20 \, a c^{2} d^{8}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.01, size = 165, normalized size = 1.39 \begin {gather*} \frac {2 \left (\frac {c^{3} d^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{3}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{13 e^{3}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{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 )}{9 e^{3}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1293 vs.
\(2 (100) = 200\).
time = 1.76, size = 1293, normalized size = 10.87 \begin {gather*} \frac {2}{45045} \, {\left (1287 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c^{3} d^{7} e^{\left (-3\right )} + 9009 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a c^{2} d^{6} e^{\left (-1\right )} + 572 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{3} d^{6} e^{\left (-3\right )} + 45045 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c d^{5} e + 15444 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a c^{2} d^{5} e^{\left (-1\right )} + 390 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{3} d^{5} e^{\left (-3\right )} + 36036 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} c d^{4} e + 2574 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c^{2} d^{4} e^{\left (-1\right )} + 60 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} d^{4} e^{\left (-3\right )} + 45045 \, \sqrt {x e + d} a^{3} d^{4} e^{3} + 60060 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d^{3} e^{3} + 23166 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} c d^{3} e + 780 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a c^{2} d^{3} e^{\left (-1\right )} + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{3} d^{3} e^{\left (-3\right )} + 18018 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3} d^{2} e^{3} + 1716 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} c d^{2} e + 45 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} a c^{2} d^{2} e^{\left (-1\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{3} d e^{3} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a^{2} c d e + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{3} e^{3}\right )} e^{\left (-1\right )} \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 )}^{9/2}}{9\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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