Optimal. Leaf size=69 \[ \frac {b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac {b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \]
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Rubi [A] time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, 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, 266, 65} \[ \frac {b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac {b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 2032
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\\ &=\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^2} \, dx,x,x^{n-q}\right )}{n-q}\\ &=\frac {b x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \, _2F_1\left (2,1+p;2+p;1+\frac {b x^{n-q}}{a}\right )}{a^2 (1+p) (n-q)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 82, normalized size = 1.19 \[ \frac {x^{-n-p q+q} \left (a x^q+b x^n\right )^p \left (\frac {a x^{q-n}}{b}+1\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {a x^{q-n}}{b}\right )}{(p-1) (n-q)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{n} + a x^{q}\right )}^{p} x^{-{\left (p - 1\right )} q - n - 1}, x\right ) \]
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 \[ \int {\left (b x^{n} + a x^{q}\right )}^{p} x^{-{\left (p - 1\right )} q - n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.79, size = 0, normalized size = 0.00 \[ \int x^{-n -\left (p -1\right ) q -1} \left (a \,x^{q}+b \,x^{n}\right )^{p}\, dx \]
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 \[ \int {\left (b x^{n} + a x^{q}\right )}^{p} x^{-{\left (p - 1\right )} q - n - 1}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^n+a\,x^q\right )}^p}{x^{n+q\,\left (p-1\right )+1}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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