Optimal. Leaf size=40 \[ -\frac {x^{-((q+1) (n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1980, 2014} \begin {gather*} -\frac {x^{(q+1) (-(n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 1980
Rule 2014
Rubi steps
\begin {align*} \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx &=\int x^{-1-n q-p (1+q)} \left (a x^n+b x^{n+p}\right )^q \, dx\\ &=-\frac {x^{-((n+p) (1+q))} \left (a x^n+b x^{n+p}\right )^{1+q}}{a p (1+q)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 38, normalized size = 0.95 \begin {gather*} -\frac {x^{-((q+1) (n+p))} \left (x^n \left (a+b x^p\right )\right )^{q+1}}{a p (q+1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 64, normalized size = 1.60 \begin {gather*} -\frac {{\left (b x x^{-{\left (n + p\right )} q - p - 1} x^{p} + a x x^{-{\left (n + p\right )} q - p - 1}\right )} {\left (b x^{n} x^{p} + a x^{n}\right )}^{q}}{a p q + a p} \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 ({\left (b x^{p} + a\right )} x^{n}\right )^{q} x^{-p {\left (q + 1\right )} - n q - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.87, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-n q -\left (q +1\right ) p -1} \left (\left (b \,x^{p}+a \right ) x^{n}\right )^{q}\, dx \end {gather*}
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*} \int \left ({\left (b x^{p} + a\right )} x^{n}\right )^{q} x^{-p {\left (q + 1\right )} - n q - 1}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {{\left (x^n\,\left (a+b\,x^p\right )\right )}^q}{x^{n\,q+p\,\left (q+1\right )+1}} \,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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