Optimal. Leaf size=246 \[ \frac {f h (a+b x)^{m+3} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m+3,m+3;m+4;-\frac {d (a+b x)}{b c-a d}\right )}{(m+3) (b c-a d)^3}-\frac {(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^3 (-d) f h (m+1)-b x \left (a^2 d f h (2 m+3)-a b (2 c f h (m+1)+d (m+2) (e h+f g))+b^2 (c (m+1) (e h+f g)+d e g)\right )+a^2 b c f h m+a b^2 (c (e h+f g)+d e g (m+1))-b^3 c e g (m+2)\right )}{b^2 (m+1) (m+2) (b c-a d)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {145, 70, 69} \[ \frac {f h (a+b x)^{m+3} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m+3,m+3;m+4;-\frac {d (a+b x)}{b c-a d}\right )}{(m+3) (b c-a d)^3}-\frac {(a+b x)^{m+1} (c+d x)^{-m-2} \left (-b x \left (a^2 d f h (2 m+3)-a b (2 c f h (m+1)+d (m+2) (e h+f g))+b^2 (c (m+1) (e h+f g)+d e g)\right )+a^2 b c f h m+a^3 (-d) f h (m+1)+a b^2 (c (e h+f g)+d e g (m+1))-b^3 c e g (m+2)\right )}{b^2 (m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 145
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-3-m} (e+f x) (g+h x) \, dx &=-\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {(f h) \int (a+b x)^{2+m} (c+d x)^{-3-m} \, dx}{b^2}\\ &=-\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {\left (b f h (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-3-m} \, dx}{(b c-a d)^3}\\ &=-\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {f h (a+b x)^{3+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (3+m,3+m;4+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 237, normalized size = 0.96 \[ -\frac {(a+b x)^m (c+d x)^{-m-2} \left (d^3 (a+b x) \left (a^3 (-d) f h (m+1)+a^2 b f h (c m-d (2 m+3) x)+a b^2 (c e h+c f (g+2 h (m+1) x)+d e g (m+1)+d e h (m+2) x+d f g (m+2) x)-b^3 (c (e g (m+2)+e h (m+1) x+f g (m+1) x)+d e g x)\right )+f h (m+1) (b c-a d)^4 \left (\frac {d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m-2,-m-2;-m-1;\frac {b (c+d x)}{b c-a d}\right )\right )}{b^2 d^3 (m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f h x^{2} + e g + {\left (f g + e h\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}, x\right ) \]
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 \[ \int {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \left (h x +g \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{-m -3}\, dx \]
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 \[ \int {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}\,{d x} \]
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 \[ \int \frac {\left (e+f\,x\right )\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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