Optimal. Leaf size=134 \[ \frac {(a+b x)^{m+1} (d g-c h) (f g-e h) \, _2F_1\left (1,m+1;m+2;-\frac {h (a+b x)}{b g-a h}\right )}{h^2 (m+1) (b g-a h)}-\frac {(a+b x)^{m+1} (a d f h+b (m+2) (-c f h-d e h+d f g)-b d f h (m+1) x)}{b^2 h^2 (m+1) (m+2)} \]
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Rubi [A] time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {147, 68} \[ \frac {(a+b x)^{m+1} (d g-c h) (f g-e h) \, _2F_1\left (1,m+1;m+2;-\frac {h (a+b x)}{b g-a h}\right )}{h^2 (m+1) (b g-a h)}-\frac {(a+b x)^{m+1} (a d f h+b (m+2) (-c f h-d e h+d f g)-b d f h (m+1) x)}{b^2 h^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 147
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x) (e+f x)}{g+h x} \, dx &=-\frac {(a+b x)^{1+m} (a d f h+b (d f g-d e h-c f h) (2+m)-b d f h (1+m) x)}{b^2 h^2 (1+m) (2+m)}+\frac {((d g-c h) (f g-e h)) \int \frac {(a+b x)^m}{g+h x} \, dx}{h^2}\\ &=-\frac {(a+b x)^{1+m} (a d f h+b (d f g-d e h-c f h) (2+m)-b d f h (1+m) x)}{b^2 h^2 (1+m) (2+m)}+\frac {(d g-c h) (f g-e h) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {h (a+b x)}{b g-a h}\right )}{h^2 (b g-a h) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 120, normalized size = 0.90 \[ \frac {(a+b x)^{m+1} \left (\frac {b (c f h+d e h-d f g)-a d f h}{b^2 (m+1)}+\frac {d f h (a+b x)}{b^2 (m+2)}+\frac {(d g-c h) (f g-e h) \, _2F_1\left (1,m+1;m+2;\frac {h (a+b x)}{a h-b g}\right )}{(m+1) (b g-a h)}\right )}{h^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d f x^{2} + c e + {\left (d e + c f\right )} x\right )} {\left (b x + a\right )}^{m}}{h x + g}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )} {\left (b x + a\right )}^{m}}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right ) \left (f x +e \right ) \left (b x +a \right )^{m}}{h x +g}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )} {\left (b x + a\right )}^{m}}{h x + g}\,{d x} \]
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 \[ \int \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^m\,\left (c+d\,x\right )}{g+h\,x} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{m} \left (c + d x\right ) \left (e + f x\right )}{g + h x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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