Optimal. Leaf size=27 \[ \frac {x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1590} \[ \frac {x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int x^{1+p} \left (b+2 c x^2\right ) \left (b x+c x^3\right )^p \, dx &=\frac {x^{1+p} \left (b x+c x^3\right )^{1+p}}{2 (1+p)}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 97, normalized size = 3.59 \[ \frac {x^{p+2} \left (x \left (b+c x^2\right )\right )^p \left (\frac {c x^2}{b}+1\right )^{-p} \left (2 c (p+1) x^2 \, _2F_1\left (-p,p+2;p+3;-\frac {c x^2}{b}\right )+b (p+2) \, _2F_1\left (-p,p+1;p+2;-\frac {c x^2}{b}\right )\right )}{2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 32, normalized size = 1.19 \[ \frac {{\left (c x^{3} + b x\right )} {\left (c x^{3} + b x\right )}^{p} x^{p + 1}}{2 \, {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 54, normalized size = 2.00 \[ \frac {c x^{3} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \relax (x) + \log \relax (x)\right )} + b x e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \relax (x) + \log \relax (x)\right )}}{2 \, {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.15 \[ \frac {\left (c \,x^{2}+b \right ) x^{p +2} \left (c \,x^{3}+b x \right )^{p}}{2+2 p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 35, normalized size = 1.30 \[ \frac {{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \relax (x)\right )}}{2 \, {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 45, normalized size = 1.67 \[ {\left (c\,x^3+b\,x\right )}^p\,\left (\frac {b\,x\,x^{p+1}}{2\,p+2}+\frac {c\,x^{p+1}\,x^3}{2\,p+2}\right ) \]
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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