Optimal. Leaf size=69 \[ -\frac {x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \, _2F_1\left (1,1+p;2+p;1+\frac {b x^{n-q}}{a}\right )}{a (1+p) (n-q)} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {2057, 272, 67}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 2057
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 ) \text {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*}
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Mathematica [A]
time = 0.11, size = 73, normalized size = 1.06 \begin {gather*} \frac {x^{-p q} \left (b x^n+a x^q\right )^p \left (1+\frac {a x^{-n+q}}{b}\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac {a x^{-n+q}}{b}\right )}{p (n-q)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int x^{-p q -1} \left (b \,x^{n}+a \,x^{q}\right )^{p}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- p q - 1} \left (a x^{q} + b x^{n}\right )^{p}\, dx \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*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^n+a\,x^q\right )}^p}{x^{p\,q+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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