Optimal. Leaf size=101 \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]
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Rubi [A] time = 0.033537, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 77, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.013, Rules used = {95} \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 95
Rubi steps
\begin{align*} \int (a+b x)^m (c+d x)^{-1-\frac{d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac{(b c-a d) f (1+m)}{b (d e-c f)}} \, dx &=\frac{b (a+b x)^{1+m} (c+d x)^{-\frac{d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac{(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.125438, size = 101, normalized size = 1. \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 162, normalized size = 1.6 \begin{align*}{\frac{b \left ( bx+a \right ) ^{1+m}}{{a}^{2}dfm-abcfm-abdem+{b}^{2}cem+{a}^{2}df-abcf-abde+{b}^{2}ce} \left ( fx+e \right ) ^{1+{\frac{adfm-bcfm+adf-2\,bcf+bde}{b \left ( cf-de \right ) }}} \left ( dx+c \right ) ^{1-{\frac{adfm-bdem+adf+bcf-2\,bde}{b \left ( cf-de \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.30364, size = 311, normalized size = 3.08 \begin{align*} \frac{{\left (b^{2} x + a b\right )} e^{\left (\frac{a d f m \log \left (d x + c\right )}{b d e - b c f} - \frac{a d f m \log \left (f x + e\right )}{b d e - b c f} + \frac{a d f \log \left (d x + c\right )}{b d e - b c f} - \frac{d e m \log \left (d x + c\right )}{d e - c f} - \frac{a d f \log \left (f x + e\right )}{b d e - b c f} + \frac{c f m \log \left (f x + e\right )}{d e - c f} + m \log \left (b x + a\right ) - \frac{d e \log \left (d x + c\right )}{d e - c f} + \frac{c f \log \left (f x + e\right )}{d e - c f}\right )}}{b^{2} c e{\left (m + 1\right )} + a^{2} d f{\left (m + 1\right )} -{\left (d e{\left (m + 1\right )} + c f{\left (m + 1\right )}\right )} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.9023, size = 458, normalized size = 4.53 \begin{align*} \frac{{\left (b^{2} d f x^{3} + a b c e +{\left (b^{2} d e +{\left (b^{2} c + a b d\right )} f\right )} x^{2} +{\left (a b c f +{\left (b^{2} c + a b d\right )} e\right )} x\right )}{\left (b x + a\right )}^{m}}{{\left ({\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f +{\left ({\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f\right )} m\right )}{\left (d x + c\right )}^{\frac{2 \, b d e -{\left (b c + a d\right )} f +{\left (b d e - a d f\right )} m}{b d e - b c f}}{\left (f x + e\right )}^{\frac{b d e -{\left (b c - a d\right )} f m -{\left (2 \, b c - a d\right )} f}{b d e - b c f}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-\frac{{\left (b e - a f\right )} d{\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1}{\left (f x + e\right )}^{\frac{{\left (b c - a d\right )} f{\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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