Optimal. Leaf size=24 \[ \frac{\left (b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.0141711, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1588} \[ \frac{\left (b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int x \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^p \, dx &=\frac{\left (b x^2+c x^4\right )^{1+p}}{2 (1+p)}\\ \end{align*}
Mathematica [C] time = 0.0736556, size = 97, normalized size = 4.04 \[ \frac{x^2 \left (x^2 \left (b+c x^2\right )\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \left (2 c (p+1) x^2 \, _2F_1\left (-p,p+2;p+3;-\frac{c x^2}{b}\right )+b (p+2) \, _2F_1\left (-p,p+1;p+2;-\frac{c x^2}{b}\right )\right )}{2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 31, normalized size = 1.3 \begin{align*}{\frac{{x}^{2} \left ( c{x}^{2}+b \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{p}}{2+2\,p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1558, size = 47, normalized size = 1.96 \begin{align*} \frac{{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08402, size = 63, normalized size = 2.62 \begin{align*} \frac{{\left (c x^{4} + b x^{2}\right )}{\left (c x^{4} + b x^{2}\right )}^{p}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 20.1521, size = 85, normalized size = 3.54 \begin{align*} \begin{cases} \frac{b x^{2} \left (b x^{2} + c x^{4}\right )^{p}}{2 p + 2} + \frac{c x^{4} \left (b x^{2} + c x^{4}\right )^{p}}{2 p + 2} & \text{for}\: p \neq -1 \\\log{\left (x \right )} + \frac{\log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + x \right )}}{2} + \frac{\log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + x \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14638, size = 59, normalized size = 2.46 \begin{align*} \frac{{\left (c x^{4} + b x^{2}\right )}^{p} c x^{4} +{\left (c x^{4} + b x^{2}\right )}^{p} b x^{2}}{2 \,{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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