Optimal. Leaf size=37 \[ \frac {x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2014} \[ \frac {x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int x^{-1-n (-1+p)} \left (b x^n+c x^{2 n}\right )^p \, dx &=\frac {x^{-n (1+p)} \left (b x^n+c x^{2 n}\right )^{1+p}}{c n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.03 \[ \frac {x^{-n p} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{c n (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 59, normalized size = 1.59 \[ \frac {{\left (c x x^{-n p + n - 1} x^{n} + b x x^{-n p + n - 1}\right )} {\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{{\left (c n p + c n\right )} x^{n}} \]
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 \[ \int {\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n {\left (p - 1\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int x^{-\left (p -1\right ) n -1} \left (b \,x^{n}+c \,x^{2 n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 43, normalized size = 1.16 \[ \frac {{\left (c x^{n} + b\right )} e^{\left (-n p \log \relax (x) + p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{c n {\left (p + 1\right )}} \]
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 \[ \int \frac {{\left (b\,x^n+c\,x^{2\,n}\right )}^p}{x^{n\,\left (p-1\right )+1}} \,d x \]
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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