Optimal. Leaf size=38 \[ -\frac {x^{-2 n (1+p)} \left (b x^n+c x^{2 n}\right )^{1+p}}{b n (1+p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2039}
\begin {gather*} -\frac {x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int x^{-1-n (1+2 p)} \left (b x^n+c x^{2 n}\right )^p \, dx &=-\frac {x^{-2 n (1+p)} \left (b x^n+c x^{2 n}\right )^{1+p}}{b n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 43, normalized size = 1.13 \begin {gather*} -\frac {x^{-n (1+2 p)} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{b n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{-1-n \left (1+2 p \right )} \left (b \,x^{n}+c \,x^{2 n}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 59, normalized size = 1.55 \begin {gather*} -\frac {{\left (c x x^{-2 \, n p - n - 1} x^{n} + b x x^{-2 \, n p - n - 1}\right )} {\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{b n p + b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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*} \text {could not integrate} \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 \frac {{\left (b\,x^n+c\,x^{2\,n}\right )}^p}{x^{n\,\left (2\,p+1\right )+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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