Optimal. Leaf size=27 \[ \frac {x^{2 (p+1)} \left (b+c x^2\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {449} \begin {gather*} \frac {x^{2 (p+1)} \left (b+c x^2\right )^{p+1}}{2 (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int x^{-1+2 (1+p)} \left (b+c x^2\right )^p \left (b+2 c x^2\right ) \, dx &=\frac {x^{2 (1+p)} \left (b+c x^2\right )^{1+p}}{2 (1+p)}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 97, normalized size = 3.59 \begin {gather*} \frac {x^{2 p+2} \left (b+c x^2\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 = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1+2 (1+p)} \left (b+c x^2\right )^p \left (b+2 c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 32, normalized size = 1.19 \begin {gather*} \frac {{\left (c x^{3} + b x\right )} {\left (c x^{2} + b\right )}^{p} x^{2 \, p + 1}}{2 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 52, normalized size = 1.93 \begin {gather*} \frac {{\left (c x^{2} + b\right )}^{p} c x^{3} e^{\left (2 \, p \log \relax (x) + \log \relax (x)\right )} + {\left (c x^{2} + b\right )}^{p} b x e^{\left (2 \, p \log \relax (x) + \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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maple [A] time = 0.04, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{2 p +2} \left (c \,x^{2}+b \right )^{p +1}}{2 p +2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 35, normalized size = 1.30 \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 = 4.88, size = 47, normalized size = 1.74 \begin {gather*} {\left (c\,x^2+b\right )}^p\,\left (\frac {c\,x^{2\,p+1}\,x^3}{2\,p+2}+\frac {b\,x\,x^{2\,p+1}}{2\,p+2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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