Optimal. Leaf size=25 \[ \frac {a x^{m+1}}{m+1}+\frac {b x^{m+3}}{m+3} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \frac {a x^{m+1}}{m+1}+\frac {b x^{m+3}}{m+3} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int x^m \left (a+b x^2\right ) \, dx &=\int \left (a x^m+b x^{2+m}\right ) \, dx\\ &=\frac {a x^{1+m}}{1+m}+\frac {b x^{3+m}}{3+m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {a x^{m+1}}{m+1}+\frac {b x^{m+3}}{m+3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 33, normalized size = 1.32 \[ \frac {{\left ({\left (b m + b\right )} x^{3} + {\left (a m + 3 \, a\right )} x\right )} x^{m}}{m^{2} + 4 \, m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 43, normalized size = 1.72 \[ \frac {b m x^{3} x^{m} + b x^{3} x^{m} + a m x x^{m} + 3 \, a x x^{m}}{m^{2} + 4 \, m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 1.40 \[ \frac {\left (b m \,x^{2}+b \,x^{2}+a m +3 a \right ) x^{m +1}}{\left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 25, normalized size = 1.00 \[ \frac {b x^{m + 3}}{m + 3} + \frac {a x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.80, size = 34, normalized size = 1.36 \[ \frac {x^{m+1}\,\left (3\,a+a\,m+b\,x^2+b\,m\,x^2\right )}{m^2+4\,m+3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 94, normalized size = 3.76 \[ \begin {cases} - \frac {a}{2 x^{2}} + b \log {\relax (x )} & \text {for}\: m = -3 \\a \log {\relax (x )} + \frac {b x^{2}}{2} & \text {for}\: m = -1 \\\frac {a m x x^{m}}{m^{2} + 4 m + 3} + \frac {3 a x x^{m}}{m^{2} + 4 m + 3} + \frac {b m x^{3} x^{m}}{m^{2} + 4 m + 3} + \frac {b x^{3} x^{m}}{m^{2} + 4 m + 3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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