Optimal. Leaf size=76 \[ \frac {a^2 x^{m+3}}{m+3}+\frac {x^{m+7} \left (2 a c+b^2\right )}{m+7}+\frac {2 a b x^{m+5}}{m+5}+\frac {2 b c x^{m+9}}{m+9}+\frac {c^2 x^{m+11}}{m+11} \]
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Rubi [A] time = 0.05, 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 = {1585, 1108} \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 1108
Rule 1585
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 = 0.07, size = 69, normalized size = 0.91 \begin {gather*} x^{m+3} \left (\frac {a^2}{m+3}+\frac {x^4 \left (2 a c+b^2\right )}{m+7}+\frac {2 a b x^2}{m+5}+\frac {2 b c x^6}{m+9}+\frac {c^2 x^8}{m+11}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a x+b x^3+c x^5\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.45, 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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giac [B] time = 0.52, 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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maple [B] time = 0.00, size = 300, normalized size = 3.95 \begin {gather*} \frac {\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 c^{2} m \,x^{8}+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 b c m \,x^{6}+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^{m +3}}{\left (m +11\right ) \left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, 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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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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sympy [A] time = 4.29, size = 1377, normalized size = 18.12
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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