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 = 0.0738037, antiderivative size = 40, normalized size of antiderivative = 1., 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} \[ -\frac{x^{(q+1) (-(n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \]
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*}
Mathematica [A] time = 0.0397407, size = 38, normalized size = 0.95 \[ -\frac{x^{(q+1) (-(n+p))} \left (x^n \left (a+b x^p\right )\right )^{q+1}}{a p (q+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.47, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-nq-p \left ( 1+q \right ) } \left ({x}^{n} \left ( a+b{x}^{p} \right ) \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 ({\left (b x^{p} + a\right )} x^{n}\right )^{q} x^{-p{\left (q + 1\right )} - n q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.858222, size = 135, normalized size = 3.38 \begin{align*} -\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{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 ({\left (b x^{p} + a\right )} x^{n}\right )^{q} x^{-p{\left (q + 1\right )} - n q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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