Optimal. Leaf size=69 \[ -\frac{x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (1,p+1;p+2;\frac{b x^{n-q}}{a}+1\right )}{a (p+1) (n-q)} \]
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Rubi [A] time = 0.0683911, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2032, 266, 65} \[ -\frac{x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (1,p+1;p+2;\frac{b x^{n-q}}{a}+1\right )}{a (p+1) (n-q)} \]
Antiderivative was successfully verified.
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Rule 2032
Rule 266
Rule 65
Rubi steps
\begin{align*} \int x^{-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 \frac{\left (a+b x^{n-q}\right )^p}{x} \, dx\\ &=\frac{\left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,x^{n-q}\right )}{n-q}\\ &=-\frac{x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \, _2F_1\left (1,1+p;2+p;1+\frac{b x^{n-q}}{a}\right )}{a (1+p) (n-q)}\\ \end{align*}
Mathematica [A] time = 0.116537, size = 73, normalized size = 1.06 \[ \frac{x^{-p q} \left (a x^q+b x^n\right )^p \left (\frac{a x^{q-n}}{b}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{a x^{q-n}}{b}\right )}{p (n-q)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.632, size = 0, normalized size = 0. \begin{align*} \int{x}^{-pq-1} \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^{-p 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^{-p 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^{-p q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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