Optimal. Leaf size=166 \[ \frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712}
\begin {gather*} \frac {2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac {4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{9/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{11/2}}{e^4}+\frac {c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 173, normalized size = 1.04 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+65 e^2 \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-10 c e \left (-13 a e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 b \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )\right )}{45045 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 136, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (e^{2} a -b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(136\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (e^{2} a -b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(136\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} x^{4} e^{4}+6930 b c \,e^{4} x^{3}-1848 c^{2} d \,e^{3} x^{3}+8190 a c \,e^{4} x^{2}+4095 b^{2} e^{4} x^{2}-3780 b c d \,e^{3} x^{2}+1008 c^{2} d^{2} e^{2} x^{2}+10010 a b \,e^{4} x -3640 a c d \,e^{3} x -1820 b^{2} d \,e^{3} x +1680 b c \,d^{2} e^{2} x -448 c^{2} d^{3} e x +6435 a^{2} e^{4}-2860 d \,e^{3} a b +1040 a c \,d^{2} e^{2}+520 b^{2} d^{2} e^{2}-480 d^{3} e b c +128 c^{2} d^{4}\right )}{45045 e^{5}}\) | \(194\) |
trager | \(\frac {2 \left (3003 c^{2} e^{7} x^{7}+6930 b c \,e^{7} x^{6}+7161 c^{2} d \,e^{6} x^{6}+8190 a c \,e^{7} x^{5}+4095 b^{2} e^{7} x^{5}+17010 b c d \,e^{6} x^{5}+4473 c^{2} d^{2} e^{5} x^{5}+10010 a b \,e^{7} x^{4}+20930 a c d \,e^{6} x^{4}+10465 b^{2} d \,e^{6} x^{4}+11130 b c \,d^{2} e^{5} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+27170 a b d \,e^{6} x^{3}+14690 a c \,d^{2} e^{5} x^{3}+7345 b^{2} d^{2} e^{5} x^{3}+150 b c \,d^{3} e^{4} x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+21450 a b \,d^{2} e^{5} x^{2}+390 a c \,d^{3} e^{4} x^{2}+195 b^{2} d^{3} e^{4} x^{2}-180 b c \,d^{4} e^{3} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x +1430 a b \,d^{3} e^{4} x -520 a c \,d^{4} e^{3} x -260 b^{2} d^{4} e^{3} x +240 b c \,d^{5} e^{2} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}-2860 a b \,d^{4} e^{3}+1040 a c \,d^{5} e^{2}+520 b^{2} d^{5} e^{2}-480 b c \,d^{6} e +128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(433\) |
risch | \(\frac {2 \left (3003 c^{2} e^{7} x^{7}+6930 b c \,e^{7} x^{6}+7161 c^{2} d \,e^{6} x^{6}+8190 a c \,e^{7} x^{5}+4095 b^{2} e^{7} x^{5}+17010 b c d \,e^{6} x^{5}+4473 c^{2} d^{2} e^{5} x^{5}+10010 a b \,e^{7} x^{4}+20930 a c d \,e^{6} x^{4}+10465 b^{2} d \,e^{6} x^{4}+11130 b c \,d^{2} e^{5} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+27170 a b d \,e^{6} x^{3}+14690 a c \,d^{2} e^{5} x^{3}+7345 b^{2} d^{2} e^{5} x^{3}+150 b c \,d^{3} e^{4} x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+21450 a b \,d^{2} e^{5} x^{2}+390 a c \,d^{3} e^{4} x^{2}+195 b^{2} d^{3} e^{4} x^{2}-180 b c \,d^{4} e^{3} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x +1430 a b \,d^{3} e^{4} x -520 a c \,d^{4} e^{3} x -260 b^{2} d^{4} e^{3} x +240 b c \,d^{5} e^{2} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}-2860 a b \,d^{4} e^{3}+1040 a c \,d^{5} e^{2}+520 b^{2} d^{5} e^{2}-480 b c \,d^{6} e +128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(433\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 185, normalized size = 1.11 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{2} - 6930 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {13}{2}} + 4095 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 10010 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-5\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. 343 vs.
\(2 (149) = 298\).
time = 2.72, size = 343, normalized size = 2.07 \begin {gather*} \frac {2}{45045} \, {\left (128 \, c^{2} d^{7} + {\left (3003 \, c^{2} x^{7} + 6930 \, b c x^{6} + 10010 \, a b x^{4} + 4095 \, {\left (b^{2} + 2 \, a c\right )} x^{5} + 6435 \, a^{2} x^{3}\right )} e^{7} + {\left (7161 \, c^{2} d x^{6} + 17010 \, b c d x^{5} + 27170 \, a b d x^{3} + 10465 \, {\left (b^{2} + 2 \, a c\right )} d x^{4} + 19305 \, a^{2} d x^{2}\right )} e^{6} + {\left (4473 \, c^{2} d^{2} x^{5} + 11130 \, b c d^{2} x^{4} + 21450 \, a b d^{2} x^{2} + 7345 \, {\left (b^{2} + 2 \, a c\right )} d^{2} x^{3} + 19305 \, a^{2} d^{2} x\right )} e^{5} + 5 \, {\left (7 \, c^{2} d^{3} x^{4} + 30 \, b c d^{3} x^{3} + 286 \, a b d^{3} x + 39 \, {\left (b^{2} + 2 \, a c\right )} d^{3} x^{2} + 1287 \, a^{2} d^{3}\right )} e^{4} - 20 \, {\left (2 \, c^{2} d^{4} x^{3} + 9 \, b c d^{4} x^{2} + 143 \, a b d^{4} + 13 \, {\left (b^{2} + 2 \, a c\right )} d^{4} x\right )} e^{3} + 8 \, {\left (6 \, c^{2} d^{5} x^{2} + 30 \, b c d^{5} x + 65 \, {\left (b^{2} + 2 \, a c\right )} d^{5}\right )} e^{2} - 32 \, {\left (2 \, c^{2} d^{6} x + 15 \, b c d^{6}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 23.45, size = 1129, normalized size = 6.80 \begin {gather*} a^{2} d^{2} \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 {4 a^{2} d \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 {2 a^{2} \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} + \frac {4 a b d^{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^{2}} + \frac {8 a b 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^{2}} + \frac {4 a b \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^{2}} + \frac {4 a c d^{2} \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 {8 a c d \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 {4 a c \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^{3}} + \frac {2 b^{2} d^{2} \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 b^{2} d \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 b^{2} \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^{3}} + \frac {4 b c d^{2} \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^{4}} + \frac {8 b c 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^{4}} + \frac {4 b c \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^{4}} + \frac {2 c^{2} d^{2} \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 {4 c^{2} d \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}} + \frac {2 c^{2} \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{5}} \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. 1498 vs.
\(2 (149) = 298\).
time = 1.67, size = 1498, normalized size = 9.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.84, size = 148, normalized size = 0.89 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{11\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{9\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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