Optimal. Leaf size=29 \[ \frac {x^{2 (p+1)} \left (b x+c x^4\right )^{p+1}}{3 (p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 29, 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 = {1590} \[ \frac {x^{2 (p+1)} \left (b x+c x^4\right )^{p+1}}{3 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int x^{2 (1+p)} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^p \, dx &=\frac {x^{2 (1+p)} \left (b x+c x^4\right )^{1+p}}{3 (1+p)}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 99, normalized size = 3.41 \[ \frac {x^{2 p+3} \left (x \left (b+c x^3\right )\right )^p \left (\frac {c x^3}{b}+1\right )^{-p} \left (2 c (p+1) x^3 \, _2F_1\left (-p,p+2;p+3;-\frac {c x^3}{b}\right )+b (p+2) \, _2F_1\left (-p,p+1;p+2;-\frac {c x^3}{b}\right )\right )}{3 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 34, normalized size = 1.17 \[ \frac {{\left (c x^{4} + b x\right )} {\left (c x^{4} + b x\right )}^{p} x^{2 \, p + 2}}{3 \, {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 58, normalized size = 2.00 \[ \frac {c x^{4} e^{\left (p \log \left (c x^{3} + b\right ) + 3 \, p \log \relax (x) + 2 \, \log \relax (x)\right )} + b x e^{\left (p \log \left (c x^{3} + b\right ) + 3 \, p \log \relax (x) + 2 \, \log \relax (x)\right )}}{3 \, {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 1.14 \[ \frac {\left (c \,x^{3}+b \right ) x^{2 p +3} \left (c \,x^{4}+b x \right )^{p}}{3+3 p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 35, normalized size = 1.21 \[ \frac {{\left (c x^{6} + b x^{3}\right )} e^{\left (p \log \left (c x^{3} + b\right ) + 3 \, p \log \relax (x)\right )}}{3 \, {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 49, normalized size = 1.69 \[ {\left (c\,x^4+b\,x\right )}^p\,\left (\frac {c\,x^{2\,p+2}\,x^4}{3\,p+3}+\frac {b\,x\,x^{2\,p+2}}{3\,p+3}\right ) \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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