Optimal. Leaf size=41 \[ \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b (2 p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1107, 609} \[ \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 609
Rule 1107
Rubi steps
\begin {align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^p \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 29, normalized size = 0.71 \[ \frac {\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^p}{4 b p+2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 37, normalized size = 0.90 \[ \frac {{\left (b x^{2} + a\right )} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{2 \, {\left (2 \, b p + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 58, normalized size = 1.41 \[ \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b x^{2} + {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a}{2 \, {\left (2 \, b p + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.98 \[ \frac {\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{2 \left (2 p +1\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 30, normalized size = 0.73 \[ \frac {{\left (b x^{2} + a\right )} {\left (b x^{2} + a\right )}^{2 \, p}}{2 \, b {\left (2 \, p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 46, normalized size = 1.12 \[ \left (\frac {x^2}{2\,\left (2\,p+1\right )}+\frac {a}{2\,b\,\left (2\,p+1\right )}\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {x^{2}}{2 \sqrt {a^{2}}} & \text {for}\: b = 0 \wedge p = - \frac {1}{2} \\\frac {x^{2} \left (a^{2}\right )^{p}}{2} & \text {for}\: b = 0 \\\int \frac {x}{\sqrt {\left (a + b x^{2}\right )^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{4 b p + 2 b} + \frac {b x^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{4 b p + 2 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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