Optimal. Leaf size=235 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h \left (m^2-3 m+2\right )-a b d (1-m) (3 d (e h+f g)-2 c f h (m+1))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )-3 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{6 b^3 d^2 (m+1)}+\frac {(a+b x)^{m+1} (c+d x)^{1-m} (-a d f h (2-m)-b c f h (m+2)+3 b d (e h+f g)+2 b d f h x)}{6 b^2 d^2} \]
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Rubi [A] time = 0.14, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {147, 70, 69} \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h \left (m^2-3 m+2\right )-a b d (1-m) (3 d (e h+f g)-2 c f h (m+1))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )-3 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{6 b^3 d^2 (m+1)}+\frac {(a+b x)^{m+1} (c+d x)^{1-m} (-a d f h (2-m)-b c f h (m+2)+3 b d (e h+f g)+2 b d f h x)}{6 b^2 d^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 147
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-m} (e+f x) (g+h x) \, dx &=\frac {(a+b x)^{1+m} (c+d x)^{1-m} (3 b d (f g+e h)-a d f h (2-m)-b c f h (2+m)+2 b d f h x)}{6 b^2 d^2}+\frac {\left (a^2 d^2 f h \left (2-3 m+m^2\right )-a b d (1-m) (3 d (f g+e h)-2 c f h (1+m))+b^2 \left (6 d^2 e g-3 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{-m} \, dx}{6 b^2 d^2}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{1-m} (3 b d (f g+e h)-a d f h (2-m)-b c f h (2+m)+2 b d f h x)}{6 b^2 d^2}+\frac {\left (\left (a^2 d^2 f h \left (2-3 m+m^2\right )-a b d (1-m) (3 d (f g+e h)-2 c f h (1+m))+b^2 \left (6 d^2 e g-3 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-m} \, dx}{6 b^2 d^2}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{1-m} (3 b d (f g+e h)-a d f h (2-m)-b c f h (2+m)+2 b d f h x)}{6 b^2 d^2}+\frac {\left (a^2 d^2 f h \left (2-3 m+m^2\right )-a b d (1-m) (3 d (f g+e h)-2 c f h (1+m))+b^2 \left (6 d^2 e g-3 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 189, normalized size = 0.80 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (b \left (b (d e-c f) (d g-c h) \, _2F_1\left (m,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )-(b c-a d) (2 c f h-d (e h+f g)) \, _2F_1\left (m-1,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )\right )+f h (b c-a d)^2 \, _2F_1\left (m-2,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )\right )}{b^3 d^2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f h x^{2} + e g + {\left (f g + e h\right )} x\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \left (h x +g \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{-m}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (e+f\,x\right )\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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