Optimal. Leaf size=89 \[ \frac {x^{m+1} \left (a+b x^{n-j}\right ) \left (a x^j+b x^n\right )^p \, _2F_1\left (1,p+\frac {m+j p+1}{n-j}+1;\frac {m+j p+1}{n-j}+1;-\frac {b x^{n-j}}{a}\right )}{a (j p+m+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2032, 365, 364} \[ \frac {x^{m+1} \left (\frac {a x^{j-n}}{b}+1\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac {m+n p+1}{j-n};\frac {m+n p+1}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{m+n p+1} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2032
Rubi steps
\begin {align*} \int x^m \left (a x^j+b x^n\right )^p \, dx &=\left (x^{-n p} \left (b+a x^{j-n}\right )^{-p} \left (a x^j+b x^n\right )^p\right ) \int x^{m+n p} \left (b+a x^{j-n}\right )^p \, dx\\ &=\left (x^{-n p} \left (1+\frac {a x^{j-n}}{b}\right )^{-p} \left (a x^j+b x^n\right )^p\right ) \int x^{m+n p} \left (1+\frac {a x^{j-n}}{b}\right )^p \, dx\\ &=\frac {x^{1+m} \left (1+\frac {a x^{j-n}}{b}\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac {1+m+n p}{j-n};1+\frac {1+m+n p}{j-n};-\frac {a x^{j-n}}{b}\right )}{1+m+n p}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 92, normalized size = 1.03 \[ \frac {x^{m+1} \left (\frac {a x^{j-n}}{b}+1\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac {m+n p+1}{j-n};\frac {m+n p+1}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{m+n p+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a x^{j} + b x^{n}\right )}^{p} x^{m}, x\right ) \]
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 \[ \int {\left (a x^{j} + b x^{n}\right )}^{p} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a \,x^{j}+b \,x^{n}\right )^{p}\, dx \]
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 \[ \int {\left (a x^{j} + b x^{n}\right )}^{p} x^{m}\,{d x} \]
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 \[ \int x^m\,{\left (a\,x^j+b\,x^n\right )}^p \,d x \]
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 \[ \int x^{m} \left (a x^{j} + b x^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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