Optimal. Leaf size=252 \[ -\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6} \]
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Rubi [A]
time = 0.08, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {785}
\begin {gather*} \frac {8 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac {4 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {10 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^5}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{3/2}}{e^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{5/2}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{9/2}}{e^5}+\frac {2 c^3 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 291, normalized size = 1.15 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-10 c^3 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )+429 b e^3 \left (35 a^2 e^2+14 a b e (-2 d+3 e x)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-286 c e^2 \left (21 a^2 e^2 (2 d-3 e x)-9 a b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^2 \left (32 d^3-48 d^2 e x+60 d e^2 x^2-70 e^3 x^3\right )\right )+13 c^2 e \left (44 a e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.01, size = 265, normalized size = 1.05 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 323, normalized size = 1.28 \begin {gather*} \frac {2}{45045} \, {\left (6930 \, {\left (x e + d\right )}^{\frac {13}{2}} c^{3} - 20475 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 20020 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 6435 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 18018 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + a b^{2} e^{4} + a^{2} c e^{4} + 6 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{2} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 15015 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} - a^{2} b e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.69, size = 376, normalized size = 1.49 \begin {gather*} -\frac {2}{45045} \, {\left (2560 \, c^{3} d^{6} - {\left (6930 \, c^{3} x^{6} + 20475 \, b c^{2} x^{5} + 20020 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 15015 \, a^{2} b x + 6435 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 18018 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} - {\left (630 \, c^{3} d x^{5} + 2275 \, b c^{2} d x^{4} + 2860 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 15015 \, a^{2} b d + 1287 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 6006 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + 4 \, {\left (175 \, c^{3} d^{2} x^{4} + 650 \, b c^{2} d^{2} x^{3} + 858 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 429 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + 3003 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} - 8 \, {\left (100 \, c^{3} d^{3} x^{3} + 390 \, b c^{2} d^{3} x^{2} + 572 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + 429 \, {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 64 \, {\left (15 \, c^{3} d^{4} x^{2} + 65 \, b c^{2} d^{4} x + 143 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} - 640 \, {\left (2 \, c^{3} d^{5} x + 13 \, b c^{2} d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.54, size = 405, normalized size = 1.61 \begin {gather*} \frac {2 \cdot \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (5 b c^{2} e - 10 c^{3} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{3 e^{5}}\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. 966 vs.
\(2 (234) = 468\).
time = 2.20, size = 966, normalized size = 3.83 \begin {gather*} \frac {2}{45045} \, {\left (30030 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b^{2} d e^{\left (-1\right )} + 30030 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c d e^{\left (-1\right )} + 3003 \, {\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 )} b^{3} d e^{\left (-2\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 b c d e^{\left (-2\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 )} b^{2} c d e^{\left (-3\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 c^{2} d e^{\left (-3\right )} + 715 \, {\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 )} b c^{2} d e^{\left (-4\right )} + 130 \, {\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 e^{\left (-5\right )} + 6006 \, {\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 b^{2} e^{\left (-1\right )} + 6006 \, {\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 e^{\left (-1\right )} + 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 )} b^{3} e^{\left (-2\right )} + 7722 \, {\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 b c e^{\left (-2\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 )} b^{2} c e^{\left (-3\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 )} a c^{2} e^{\left (-3\right )} + 325 \, {\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 )} b c^{2} e^{\left (-4\right )} + 30 \, {\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} e^{\left (-5\right )} + 45045 \, \sqrt {x e + d} a^{2} b d + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.92, size = 267, normalized size = 1.06 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{5\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{9\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{7\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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