Optimal. Leaf size=23 \[ \frac {\left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \[ \frac {\left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^p \, dx &=\frac {\left (a+b x^2\right )^{1+p}}{2 b (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 0.96 \[ \frac {\left (a+b x^2\right )^{p+1}}{2 b p+2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 25, normalized size = 1.09 \[ \frac {{\left (b x^{2} + a\right )} {\left (b x^{2} + a\right )}^{p}}{2 \, {\left (b p + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 21, normalized size = 0.91 \[ \frac {{\left (b x^{2} + a\right )}^{p + 1}}{2 \, b {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.96 \[ \frac {\left (b \,x^{2}+a \right )^{p +1}}{2 \left (p +1\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 21, normalized size = 0.91 \[ \frac {{\left (b x^{2} + a\right )}^{p + 1}}{2 \, b {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.89, size = 21, normalized size = 0.91 \[ \frac {{\left (b\,x^2+a\right )}^{p+1}}{2\,b\,\left (p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.73, size = 97, normalized size = 4.22 \[ \begin {cases} \frac {x^{2}}{2 a} & \text {for}\: b = 0 \wedge p = -1 \\\frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{2 b} + \frac {\log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{2 b} & \text {for}\: p = -1 \\\frac {a \left (a + b x^{2}\right )^{p}}{2 b p + 2 b} + \frac {b x^{2} \left (a + b x^{2}\right )^{p}}{2 b p + 2 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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