Optimal. Leaf size=27 \[ \frac {x^{n (p+1)} \left (b+c x^n\right )^{p+1}}{n (p+1)} \]
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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} \[ \frac {x^{n (p+1)} \left (b+c x^n\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int x^{-1+n (1+p)} \left (b+c x^n\right )^p \left (b+2 c x^n\right ) \, dx &=\frac {x^{n (1+p)} \left (b+c x^n\right )^{1+p}}{n (1+p)}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 101, normalized size = 3.74 \[ \frac {\left (b+c x^n\right )^p \left (\frac {c x^n}{b}+1\right )^{-p} \left (b (p+2) x^{n (p+1)} \, _2F_1\left (-p,p+1;p+2;-\frac {c x^n}{b}\right )+2 c (p+1) x^{n (p+2)} \, _2F_1\left (-p,p+2;p+3;-\frac {c x^n}{b}\right )\right )}{n (p+1) (p+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 35, normalized size = 1.30 \[ \frac {{\left (c x x^{n} + b x\right )} {\left (c x^{n} + b\right )}^{p} x^{n p + n - 1}}{n p + n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 66, normalized size = 2.44 \[ \frac {{\left (c x^{n} + b\right )}^{p} c x x^{n} e^{\left (n p \log \relax (x) + n \log \relax (x) - \log \relax (x)\right )} + {\left (c x^{n} + b\right )}^{p} b x e^{\left (n p \log \relax (x) + n \log \relax (x) - \log \relax (x)\right )}}{n p + n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.69, size = 0, normalized size = 0.00 \[ \int \left (2 c \,x^{n}+b \right ) x^{\left (p +1\right ) n -1} \left (c \,x^{n}+b \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 39, normalized size = 1.44 \[ \frac {{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (n p \log \relax (x) + p \log \left (c x^{n} + b\right )\right )}}{n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.86, size = 54, normalized size = 2.00 \[ \left (\frac {b\,x\,x^{n\,\left (p+1\right )-1}}{n\,\left (p+1\right )}+\frac {c\,x\,x^n\,x^{n\,\left (p+1\right )-1}}{n\,\left (p+1\right )}\right )\,{\left (b+c\,x^n\right )}^p \]
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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