Optimal. Leaf size=79 \[ \frac {(a-x)^m \left (1-\frac {x}{a}\right )^{-m} (f x)^{p+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac {x}{a},-\frac {d x}{c}\right )}{f (p+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {135, 133} \[ \frac {(a-x)^m \left (1-\frac {x}{a}\right )^{-m} (f x)^{p+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac {x}{a},-\frac {d x}{c}\right )}{f (p+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rubi steps
\begin {align*} \int (a-x)^m (f x)^p (c+d x)^n \, dx &=\left ((a-x)^m \left (1-\frac {x}{a}\right )^{-m}\right ) \int (f x)^p \left (1-\frac {x}{a}\right )^m (c+d x)^n \, dx\\ &=\left ((a-x)^m \left (1-\frac {x}{a}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int (f x)^p \left (1-\frac {x}{a}\right )^m \left (1+\frac {d x}{c}\right )^n \, dx\\ &=\frac {(a-x)^m (f x)^{1+p} \left (1-\frac {x}{a}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+p;-m,-n;2+p;\frac {x}{a},-\frac {d x}{c}\right )}{f (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 77, normalized size = 0.97 \[ \frac {x (a-x)^m \left (\frac {a-x}{a}\right )^{-m} (f x)^p (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac {x}{a},-\frac {d x}{c}\right )}{p+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{n} \left (f x\right )^{p} {\left (a - x\right )}^{m}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{n} \left (f x\right )^{p} {\left (a - x\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{p} \left (a -x \right )^{m} \left (d x +c \right )^{n}\, dx \]
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 \[ \int {\left (d x + c\right )}^{n} \left (f x\right )^{p} {\left (a - x\right )}^{m}\,{d x} \]
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 \[ \int {\left (f\,x\right )}^p\,{\left (a-x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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 \[ \int \left (f x\right )^{p} \left (a - x\right )^{m} \left (c + d x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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