Optimal. Leaf size=76 \[ \frac {a^2 x^{3+m}}{3+m}+\frac {2 a b x^{5+m}}{5+m}+\frac {\left (b^2+2 a c\right ) x^{7+m}}{7+m}+\frac {2 b c x^{9+m}}{9+m}+\frac {c^2 x^{11+m}}{11+m} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1599, 1122}
\begin {gather*} \frac {a^2 x^{m+3}}{m+3}+\frac {x^{m+7} \left (2 a c+b^2\right )}{m+7}+\frac {2 a b x^{m+5}}{m+5}+\frac {2 b c x^{m+9}}{m+9}+\frac {c^2 x^{m+11}}{m+11} \end {gather*}
Antiderivative was successfully verified.
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Rule 1122
Rule 1599
Rubi steps
\begin {align*} \int x^m \left (a x+b x^3+c x^5\right )^2 \, dx &=\int x^{2+m} \left (a+b x^2+c x^4\right )^2 \, dx\\ &=\int \left (a^2 x^{2+m}+2 a b x^{4+m}+\left (b^2+2 a c\right ) x^{6+m}+2 b c x^{8+m}+c^2 x^{10+m}\right ) \, dx\\ &=\frac {a^2 x^{3+m}}{3+m}+\frac {2 a b x^{5+m}}{5+m}+\frac {\left (b^2+2 a c\right ) x^{7+m}}{7+m}+\frac {2 b c x^{9+m}}{9+m}+\frac {c^2 x^{11+m}}{11+m}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 70, normalized size = 0.92 \begin {gather*} x^m \left (\frac {a^2 x^3}{3+m}+\frac {2 a b x^5}{5+m}+\frac {\left (b^2+2 a c\right ) x^7}{7+m}+\frac {2 b c x^9}{9+m}+\frac {c^2 x^{11}}{11+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs.
\(2(76)=152\).
time = 0.01, size = 300, normalized size = 3.95
method | result | size |
gosper | \(\frac {x^{3+m} \left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 c^{2} x^{8}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 m \,x^{6} b c +2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 a c \,x^{4}+1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right )}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right )}\) | \(300\) |
risch | \(\frac {x^{m} \left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 c^{2} x^{8}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 m \,x^{6} b c +2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 a c \,x^{4}+1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right ) x^{3}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right )}\) | \(301\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 85, normalized size = 1.12 \begin {gather*} \frac {c^{2} x^{m + 11}}{m + 11} + \frac {2 \, b c x^{m + 9}}{m + 9} + \frac {b^{2} x^{m + 7}}{m + 7} + \frac {2 \, a c x^{m + 7}}{m + 7} + \frac {2 \, a b x^{m + 5}}{m + 5} + \frac {a^{2} x^{m + 3}}{m + 3} \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. 241 vs.
\(2 (76) = 152\).
time = 0.35, size = 241, normalized size = 3.17 \begin {gather*} \frac {{\left ({\left (c^{2} m^{4} + 24 \, c^{2} m^{3} + 206 \, c^{2} m^{2} + 744 \, c^{2} m + 945 \, c^{2}\right )} x^{11} + 2 \, {\left (b c m^{4} + 26 \, b c m^{3} + 236 \, b c m^{2} + 886 \, b c m + 1155 \, b c\right )} x^{9} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 28 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 274 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 1485 \, b^{2} + 2970 \, a c + 1092 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \, {\left (a b m^{4} + 30 \, a b m^{3} + 320 \, a b m^{2} + 1410 \, a b m + 2079 \, a b\right )} x^{5} + {\left (a^{2} m^{4} + 32 \, a^{2} m^{3} + 374 \, a^{2} m^{2} + 1888 \, a^{2} m + 3465 \, a^{2}\right )} x^{3}\right )} x^{m}}{m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1377 vs.
\(2 (66) = 132\).
time = 0.70, size = 1377, normalized size = 18.12 \begin {gather*} \begin {cases} - \frac {a^{2}}{8 x^{8}} - \frac {a b}{3 x^{6}} - \frac {a c}{2 x^{4}} - \frac {b^{2}}{4 x^{4}} - \frac {b c}{x^{2}} + c^{2} \log {\left (x \right )} & \text {for}\: m = -11 \\- \frac {a^{2}}{6 x^{6}} - \frac {a b}{2 x^{4}} - \frac {a c}{x^{2}} - \frac {b^{2}}{2 x^{2}} + 2 b c \log {\left (x \right )} + \frac {c^{2} x^{2}}{2} & \text {for}\: m = -9 \\- \frac {a^{2}}{4 x^{4}} - \frac {a b}{x^{2}} + 2 a c \log {\left (x \right )} + b^{2} \log {\left (x \right )} + b c x^{2} + \frac {c^{2} x^{4}}{4} & \text {for}\: m = -7 \\- \frac {a^{2}}{2 x^{2}} + 2 a b \log {\left (x \right )} + a c x^{2} + \frac {b^{2} x^{2}}{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{6}}{6} & \text {for}\: m = -5 \\a^{2} \log {\left (x \right )} + a b x^{2} + \frac {a c x^{4}}{2} + \frac {b^{2} x^{4}}{4} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{8}}{8} & \text {for}\: m = -3 \\\frac {a^{2} m^{4} x^{3} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {32 a^{2} m^{3} x^{3} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {374 a^{2} m^{2} x^{3} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {1888 a^{2} m x^{3} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {3465 a^{2} x^{3} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2 a b m^{4} x^{5} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {60 a b m^{3} x^{5} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {640 a b m^{2} x^{5} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2820 a b m x^{5} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {4158 a b x^{5} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2 a c m^{4} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {56 a c m^{3} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {548 a c m^{2} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2184 a c m x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2970 a c x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {b^{2} m^{4} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {28 b^{2} m^{3} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {274 b^{2} m^{2} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {1092 b^{2} m x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {1485 b^{2} x^{7} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2 b c m^{4} x^{9} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {52 b c m^{3} x^{9} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {472 b c m^{2} x^{9} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {1772 b c m x^{9} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {2310 b c x^{9} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {c^{2} m^{4} x^{11} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {24 c^{2} m^{3} x^{11} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {206 c^{2} m^{2} x^{11} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {744 c^{2} m x^{11} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} + \frac {945 c^{2} x^{11} x^{m}}{m^{5} + 35 m^{4} + 470 m^{3} + 3010 m^{2} + 9129 m + 10395} & \text {otherwise} \end {cases} \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. 399 vs.
\(2 (76) = 152\).
time = 3.85, size = 399, normalized size = 5.25 \begin {gather*} \frac {c^{2} m^{4} x^{11} x^{m} + 24 \, c^{2} m^{3} x^{11} x^{m} + 2 \, b c m^{4} x^{9} x^{m} + 206 \, c^{2} m^{2} x^{11} x^{m} + 52 \, b c m^{3} x^{9} x^{m} + 744 \, c^{2} m x^{11} x^{m} + b^{2} m^{4} x^{7} x^{m} + 2 \, a c m^{4} x^{7} x^{m} + 472 \, b c m^{2} x^{9} x^{m} + 945 \, c^{2} x^{11} x^{m} + 28 \, b^{2} m^{3} x^{7} x^{m} + 56 \, a c m^{3} x^{7} x^{m} + 1772 \, b c m x^{9} x^{m} + 2 \, a b m^{4} x^{5} x^{m} + 274 \, b^{2} m^{2} x^{7} x^{m} + 548 \, a c m^{2} x^{7} x^{m} + 2310 \, b c x^{9} x^{m} + 60 \, a b m^{3} x^{5} x^{m} + 1092 \, b^{2} m x^{7} x^{m} + 2184 \, a c m x^{7} x^{m} + a^{2} m^{4} x^{3} x^{m} + 640 \, a b m^{2} x^{5} x^{m} + 1485 \, b^{2} x^{7} x^{m} + 2970 \, a c x^{7} x^{m} + 32 \, a^{2} m^{3} x^{3} x^{m} + 2820 \, a b m x^{5} x^{m} + 374 \, a^{2} m^{2} x^{3} x^{m} + 4158 \, a b x^{5} x^{m} + 1888 \, a^{2} m x^{3} x^{m} + 3465 \, a^{2} x^{3} x^{m}}{m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.19, size = 271, normalized size = 3.57 \begin {gather*} \frac {a^2\,x^m\,x^3\,\left (m^4+32\,m^3+374\,m^2+1888\,m+3465\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {c^2\,x^m\,x^{11}\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {x^m\,x^7\,\left (b^2+2\,a\,c\right )\,\left (m^4+28\,m^3+274\,m^2+1092\,m+1485\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {2\,a\,b\,x^m\,x^5\,\left (m^4+30\,m^3+320\,m^2+1410\,m+2079\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {2\,b\,c\,x^m\,x^9\,\left (m^4+26\,m^3+236\,m^2+886\,m+1155\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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