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 = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (b x^2+c x^4\right )^{p+1}}{2 (p+1)} \end {gather*}
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*}
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Mathematica [C] time = 0.07, size = 97, normalized size = 4.04 \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.85, size = 31, normalized size = 1.29 \begin {gather*} \frac {{\left (c x^{4} + b x^{2}\right )} {\left (c x^{4} + b x^{2}\right )}^{p}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 22, normalized size = 0.92 \begin {gather*} \frac {{\left (c x^{4} + b x^{2}\right )}^{p + 1}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.29 \begin {gather*} \frac {\left (c \,x^{2}+b \right ) x^{2} \left (c \,x^{4}+b \,x^{2}\right )^{p}}{2 p +2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 35, normalized size = 1.46 \begin {gather*} \frac {{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \relax (x)\right )}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.07, size = 31, normalized size = 1.29 \begin {gather*} \frac {x^2\,\left (c\,x^2+b\right )\,{\left (c\,x^4+b\,x^2\right )}^p}{2\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.12, size = 85, normalized size = 3.54 \begin {gather*} \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 {\relax (x )} + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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