Optimal. Leaf size=52 \[ \frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{b (1+m)} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {68, 66}
\begin {gather*} \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{c}\right )}{b (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rubi steps
\begin {align*} \int (b x)^m (c+d x)^n \, dx &=\left ((c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int (b x)^m \left (1+\frac {d x}{c}\right )^n \, dx\\ &=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{b (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.92 \begin {gather*} \frac {x (b x)^m (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 3.15, size = 39, normalized size = 0.75 \begin {gather*} \frac {b^m c^n x^{1+m} \text {hyper}\left [\left \{-n,1+m\right \},\left \{2+m\right \},\frac {d x \text {exp\_polar}\left [I \text {Pi}\right ]}{c}\right ]}{1+m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \left (d x +c \right )^{n}\, dx\]
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 \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 = 0.31, 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 [C] Result contains complex when optimal does not.
time = 1.55, size = 37, normalized size = 0.71 \begin {gather*} \frac {b^{m} c^{n} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 = -0.02 \begin {gather*} \int {\left (b\,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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