Optimal. Leaf size=39 \[ \frac{x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (n-p)} \]
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Rubi [A] time = 0.0498496, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2014} \[ \frac{x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (n-p)} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx &=\frac{x^{-p (1+q)} \left (a x^n+b x^p\right )^{1+q}}{a (n-p) (1+q)}\\ \end{align*}
Mathematica [A] time = 0.0240465, size = 40, normalized size = 1.03 \[ -\frac{x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (p-n)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.53, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1+n-p \left ( 1+q \right ) } \left ( a{x}^{n}+b{x}^{p} \right ) ^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a x^{n} + b x^{p}\right )}^{q} x^{-p{\left (q + 1\right )} + n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.87875, size = 155, normalized size = 3.97 \begin{align*} \frac{{\left (a x x^{-p q + n - p - 1} x^{n} + b x x^{-p q + n - p - 1} x^{p}\right )}{\left (a x^{n} + b x^{p}\right )}^{q}}{{\left (a n - a p +{\left (a n - a p\right )} q\right )} x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a x^{n} + b x^{p}\right )}^{q} x^{-p{\left (q + 1\right )} + n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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