Optimal. Leaf size=36 \[ \frac {x^{-((1-n) (p+1))} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]
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Rubi [A] time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2036} \begin {gather*} \frac {x^{-(1-n) (p+1)} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2036
Rubi steps
\begin {align*} \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx &=\frac {x^{-((1-n) (1+p))} \left (b x+c x^{1+n}\right )^{1+p}}{n (1+p)}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 108, normalized size = 3.00 \begin {gather*} \frac {x^{-p} \left (x \left (b+c x^n\right )\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)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.85, size = 42, normalized size = 1.17 \begin {gather*} \frac {{\left (b x + c x^{n + 1}\right )} {\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} p + n - 1}}{n p + n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (2 \, c x^{n} + b\right )} {\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} {\left (p + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2 c \,x^{n}+b \right ) x^{\left (n -1\right ) \left (p +1\right )} \left (b x +c \,x^{n +1}\right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 39, normalized size = 1.08 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^{\left (n-1\right )\,\left (p+1\right )}\,{\left (b\,x+c\,x^{n+1}\right )}^p\,\left (b+2\,c\,x^n\right ) \,d x \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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