Optimal. Leaf size=224 \[ \frac{d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f) (d g-c h)}-\frac{f^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (d e-c f) (f g-e h)}+\frac{h^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (d g-c h) (f g-e h)} \]
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Rubi [A] time = 0.194952, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {180, 68} \[ \frac{d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f) (d g-c h)}-\frac{f^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (d e-c f) (f g-e h)}+\frac{h^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (d g-c h) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 180
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^m}{(c+d x) (e+f x) (g+h x)} \, dx &=\int \left (\frac{d^2 (a+b x)^m}{(d e-c f) (d g-c h) (c+d x)}+\frac{f^2 (a+b x)^m}{(d e-c f) (-f g+e h) (e+f x)}+\frac{h^2 (a+b x)^m}{(d g-c h) (f g-e h) (g+h x)}\right ) \, dx\\ &=\frac{d^2 \int \frac{(a+b x)^m}{c+d x} \, dx}{(d e-c f) (d g-c h)}-\frac{f^2 \int \frac{(a+b x)^m}{e+f x} \, dx}{(d e-c f) (f g-e h)}+\frac{h^2 \int \frac{(a+b x)^m}{g+h x} \, dx}{(d g-c h) (f g-e h)}\\ &=\frac{d^2 (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{(b c-a d) (d e-c f) (d g-c h) (1+m)}-\frac{f^2 (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{f (a+b x)}{b e-a f}\right )}{(b e-a f) (d e-c f) (f g-e h) (1+m)}+\frac{h^2 (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{h (a+b x)}{b g-a h}\right )}{(b g-a h) (d g-c h) (f g-e h) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.395389, size = 193, normalized size = 0.86 \[ \frac{(a+b x)^{m+1} \left (\frac{d^2 \, _2F_1\left (1,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )}{(b c-a d) (c f-d e) (c h-d g)}+\frac{f^2 \, _2F_1\left (1,m+1;m+2;\frac{f (a+b x)}{a f-b e}\right )}{(b e-a f) (d e-c f) (e h-f g)}+\frac{h^2 \, _2F_1\left (1,m+1;m+2;\frac{h (a+b x)}{a h-b g}\right )}{(b g-a h) (d g-c h) (f g-e h)}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( hx+g \right ) \left ( fx+e \right ) \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}{\left (f x + e\right )}{\left (h x + g\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}}{d f h x^{3} + c e g +{\left (d f g +{\left (d e + c f\right )} h\right )} x^{2} +{\left (c e h +{\left (d e + c f\right )} g\right )} x}, x\right ) \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 \frac{{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}{\left (f x + e\right )}{\left (h x + g\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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