Optimal. Leaf size=79 \[ \frac {(a-x)^m (f x)^{1+p} \left (1-\frac {x}{a}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+p;-m,-n;2+p;\frac {x}{a},-\frac {d x}{c}\right )}{f (1+p)} \]
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Rubi [A]
time = 0.03, 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 = {140, 138}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
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 \begin {gather*} \frac {(a-x)^m \left (\frac {a-x}{a}\right )^{-m} x (f x)^p (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} F_1\left (1+p;-m,-n;2+p;\frac {x}{a},-\frac {d x}{c}\right )}{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a -x \right )^{m} \left (f x \right )^{p} \left (d x +c \right )^{n}\, dx\]
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 \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 \left (f x\right )^{p} \left (a - x\right )^{m} \left (c + d x\right )^{n}\, 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 {\left (f\,x\right )}^p\,{\left (a-x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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