Optimal. Leaf size=79 \[ \frac {a^4 x^{m+1}}{m+1}+\frac {4 a^3 b x^{m+3}}{m+3}+\frac {6 a^2 b^2 x^{m+5}}{m+5}+\frac {4 a b^3 x^{m+7}}{m+7}+\frac {b^4 x^{m+9}}{m+9} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {6 a^2 b^2 x^{m+5}}{m+5}+\frac {4 a^3 b x^{m+3}}{m+3}+\frac {a^4 x^{m+1}}{m+1}+\frac {4 a b^3 x^{m+7}}{m+7}+\frac {b^4 x^{m+9}}{m+9} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^m \left (a+b x^2\right )^4 \, dx &=\int \left (a^4 x^m+4 a^3 b x^{2+m}+6 a^2 b^2 x^{4+m}+4 a b^3 x^{6+m}+b^4 x^{8+m}\right ) \, dx\\ &=\frac {a^4 x^{1+m}}{1+m}+\frac {4 a^3 b x^{3+m}}{3+m}+\frac {6 a^2 b^2 x^{5+m}}{5+m}+\frac {4 a b^3 x^{7+m}}{7+m}+\frac {b^4 x^{9+m}}{9+m}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.91 \[ x^{m+1} \left (\frac {a^4}{m+1}+\frac {4 a^3 b x^2}{m+3}+\frac {6 a^2 b^2 x^4}{m+5}+\frac {4 a b^3 x^6}{m+7}+\frac {b^4 x^8}{m+9}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 251, normalized size = 3.18 \[ \frac {{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \, {\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \, {\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \, {\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} + {\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 365, normalized size = 4.62 \[ \frac {b^{4} m^{4} x^{9} x^{m} + 16 \, b^{4} m^{3} x^{9} x^{m} + 4 \, a b^{3} m^{4} x^{7} x^{m} + 86 \, b^{4} m^{2} x^{9} x^{m} + 72 \, a b^{3} m^{3} x^{7} x^{m} + 176 \, b^{4} m x^{9} x^{m} + 6 \, a^{2} b^{2} m^{4} x^{5} x^{m} + 416 \, a b^{3} m^{2} x^{7} x^{m} + 105 \, b^{4} x^{9} x^{m} + 120 \, a^{2} b^{2} m^{3} x^{5} x^{m} + 888 \, a b^{3} m x^{7} x^{m} + 4 \, a^{3} b m^{4} x^{3} x^{m} + 780 \, a^{2} b^{2} m^{2} x^{5} x^{m} + 540 \, a b^{3} x^{7} x^{m} + 88 \, a^{3} b m^{3} x^{3} x^{m} + 1800 \, a^{2} b^{2} m x^{5} x^{m} + a^{4} m^{4} x x^{m} + 656 \, a^{3} b m^{2} x^{3} x^{m} + 1134 \, a^{2} b^{2} x^{5} x^{m} + 24 \, a^{4} m^{3} x x^{m} + 1832 \, a^{3} b m x^{3} x^{m} + 206 \, a^{4} m^{2} x x^{m} + 1260 \, a^{3} b x^{3} x^{m} + 744 \, a^{4} m x x^{m} + 945 \, a^{4} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 291, normalized size = 3.68 \[ \frac {\left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 b^{4} m \,x^{8}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 a \,b^{3} m \,x^{6}+4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 a^{2} b^{2} m \,x^{4}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 a^{3} b m \,x^{2}+206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 a^{4} m +945 a^{4}\right ) x^{m +1}}{\left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 79, normalized size = 1.00 \[ \frac {b^{4} x^{m + 9}}{m + 9} + \frac {4 \, a b^{3} x^{m + 7}}{m + 7} + \frac {6 \, a^{2} b^{2} x^{m + 5}}{m + 5} + \frac {4 \, a^{3} b x^{m + 3}}{m + 3} + \frac {a^{4} x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.99, size = 272, normalized size = 3.44 \[ \frac {a^4\,x\,x^m\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b^4\,x^m\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {6\,a^2\,b^2\,x^m\,x^5\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {4\,a\,b^3\,x^m\,x^7\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {4\,a^3\,b\,x^m\,x^3\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.85, size = 1221, normalized size = 15.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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