Optimal. Leaf size=108 \[ \frac {f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}-\frac {(c f (1+m)-d e (2+m+n)) (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{b d (1+m) (2+m+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 99, normalized size of antiderivative = 0.92, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 68, 66}
\begin {gather*} \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (\frac {e}{m+1}-\frac {c f}{d (m+n+2)}\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{c}\right )}{b}+\frac {f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rule 81
Rubi steps
\begin {align*} \int (b x)^m (c+d x)^n (e+f x) \, dx &=\frac {f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\left (e-\frac {c f (1+m)}{d (2+m+n)}\right ) \int (b x)^m (c+d x)^n \, dx\\ &=\frac {f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\left (\left (e-\frac {c f (1+m)}{d (2+m+n)}\right ) (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 {f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\frac {\left (e-\frac {c f (1+m)}{d (2+m+n)}\right ) (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 = 0.06, size = 81, normalized size = 0.75 \begin {gather*} \frac {x (b x)^m (c+d x)^n \left (f (c+d x)+\frac {(-c f (1+m)+d e (2+m+n)) \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{1+m}\right )}{d (2+m+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )\, dx\]
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 [C] Result contains complex when optimal does not.
time = 6.02, size = 82, normalized size = 0.76 \begin {gather*} \frac {b^{m} c^{n} e 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 )} + \frac {b^{m} c^{n} f x^{2} x^{m} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} \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 (e+f\,x\right )\,{\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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