Optimal. Leaf size=52 \[ \frac {a (d x)^{m+1}}{d (m+1)}+\frac {b (d x)^{m+3}}{d^3 (m+3)}+\frac {c (d x)^{m+5}}{d^5 (m+5)} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \[ \frac {a (d x)^{m+1}}{d (m+1)}+\frac {b (d x)^{m+3}}{d^3 (m+3)}+\frac {c (d x)^{m+5}}{d^5 (m+5)} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int (d x)^m \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a (d x)^m+\frac {b (d x)^{2+m}}{d^2}+\frac {c (d x)^{4+m}}{d^4}\right ) \, dx\\ &=\frac {a (d x)^{1+m}}{d (1+m)}+\frac {b (d x)^{3+m}}{d^3 (3+m)}+\frac {c (d x)^{5+m}}{d^5 (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.67 \[ x (d x)^m \left (\frac {a}{m+1}+\frac {b x^2}{m+3}+\frac {c x^4}{m+5}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 71, normalized size = 1.37 \[ \frac {{\left ({\left (c m^{2} + 4 \, c m + 3 \, c\right )} x^{5} + {\left (b m^{2} + 6 \, b m + 5 \, b\right )} x^{3} + {\left (a m^{2} + 8 \, a m + 15 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 119, normalized size = 2.29 \[ \frac {\left (d x\right )^{m} c m^{2} x^{5} + 4 \, \left (d x\right )^{m} c m x^{5} + \left (d x\right )^{m} b m^{2} x^{3} + 3 \, \left (d x\right )^{m} c x^{5} + 6 \, \left (d x\right )^{m} b m x^{3} + \left (d x\right )^{m} a m^{2} x + 5 \, \left (d x\right )^{m} b x^{3} + 8 \, \left (d x\right )^{m} a m x + 15 \, \left (d x\right )^{m} a x}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 78, normalized size = 1.50 \[ \frac {\left (c \,m^{2} x^{4}+4 c m \,x^{4}+b \,m^{2} x^{2}+3 c \,x^{4}+6 b m \,x^{2}+a \,m^{2}+5 b \,x^{2}+8 a m +15 a \right ) x \left (d x \right )^{m}}{\left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 50, normalized size = 0.96 \[ \frac {c d^{m} x^{5} x^{m}}{m + 5} + \frac {b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 89, normalized size = 1.71 \[ {\left (d\,x\right )}^m\,\left (\frac {b\,x^3\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}+\frac {c\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {a\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 314, normalized size = 6.04 \[ \begin {cases} \frac {- \frac {a}{4 x^{4}} - \frac {b}{2 x^{2}} + c \log {\relax (x )}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a}{2 x^{2}} + b \log {\relax (x )} + \frac {c x^{2}}{2}}{d^{3}} & \text {for}\: m = -3 \\\frac {a \log {\relax (x )} + \frac {b x^{2}}{2} + \frac {c x^{4}}{4}}{d} & \text {for}\: m = -1 \\\frac {a d^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 a d^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 a d^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {b d^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 b d^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 b d^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {c d^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 c d^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 c d^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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