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 = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2014} \begin {gather*} \frac {x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (n-p)} \end {gather*}
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*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 1.03 \begin {gather*} -\frac {x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (p-n)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 76, normalized size = 1.95 \begin {gather*} \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 {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*} \int {\left (a x^{n} + b x^{p}\right )}^{q} x^{-p {\left (q + 1\right )} + n - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \begin {gather*} \int x^{n -\left (q +1\right ) p -1} \left (a \,x^{n}+b \,x^{p}\right )^{q}\, dx \end {gather*}
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*} \int {\left (a x^{n} + b x^{p}\right )}^{q} x^{-p {\left (q + 1\right )} + n - 1}\,{d x} \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 = -0.03 \begin {gather*} \int x^{n-p\,\left (q+1\right )-1}\,{\left (a\,x^n+b\,x^p\right )}^q \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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