Optimal. Leaf size=84 \[ \frac{x^{n (-p)+n-q} \left (\frac{b x^{n-q}}{a}+1\right )^{-p} \left (a x^q+b x^n\right )^p \, _2F_1\left (1-p,-p;2-p;-\frac{b x^{n-q}}{a}\right )}{(1-p) (n-q)} \]
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Rubi [A] time = 0.081219, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2032, 365, 364} \[ \frac{x^{n (-p)+n-q} \left (\frac{b x^{n-q}}{a}+1\right )^{-p} \left (a x^q+b x^n\right )^p \, _2F_1\left (1-p,-p;2-p;-\frac{b x^{n-q}}{a}\right )}{(1-p) (n-q)} \]
Antiderivative was successfully verified.
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Rule 2032
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^{-1-n (-1+p)-q} \left (b x^n+a x^q\right )^p \, dx &=\left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \int x^{-1-n (-1+p)-q+p q} \left (a+b x^{n-q}\right )^p \, dx\\ &=\left (x^{-p q} \left (1+\frac{b x^{n-q}}{a}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \int x^{-1-n (-1+p)-q+p q} \left (1+\frac{b x^{n-q}}{a}\right )^p \, dx\\ &=\frac{x^{n-n p-q} \left (1+\frac{b x^{n-q}}{a}\right )^{-p} \left (b x^n+a x^q\right )^p \, _2F_1\left (1-p,-p;2-p;-\frac{b x^{n-q}}{a}\right )}{(1-p) (n-q)}\\ \end{align*}
Mathematica [A] time = 0.131903, size = 83, normalized size = 0.99 \[ -\frac{x^{n (-p)+n-q} \left (\frac{b x^{n-q}}{a}+1\right )^{-p} \left (a x^q+b x^n\right )^p \, _2F_1\left (1-p,-p;2-p;-\frac{b x^{n-q}}{a}\right )}{(p-1) (n-q)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.528, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n \left ( p-1 \right ) -q} \left ( b{x}^{n}+a{x}^{q} \right ) ^{p}\, 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 (b x^{n} + a x^{q}\right )}^{p} x^{-n{\left (p - 1\right )} - q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{n} + a x^{q}\right )}^{p} x^{-n p + n - q - 1}, x\right ) \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 (b x^{n} + a x^{q}\right )}^{p} x^{-n{\left (p - 1\right )} - q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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