Optimal. Leaf size=37 \[ -\frac {(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {264} \[ -\frac {(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx &=-\frac {(c x)^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 1.00 \[ -\frac {x (c x)^{-n (p+1)-1} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 75, normalized size = 2.03 \[ -\frac {{\left (b x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \relax (c) - {\left (n p + n + 1\right )} \log \relax (x)\right )} + a x e^{\left (-{\left (n p + n + 1\right )} \log \relax (c) - {\left (n p + n + 1\right )} \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p}}{a n p + a n} \]
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 \[ \int {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{-n p -n -1} \left (b \,x^{n}+a \right )^{p}\, dx \]
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 \[ \int {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c\,x\right )}^{n+n\,p+1}} \,d x \]
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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