Optimal. Leaf size=353 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{b d^2 (m+2) (m+3) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (a d f (m+3)-b (c f (m+2)+d e))}{b d^2 (m+3) (b c-a d)}-\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d} \]
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Rubi [A] time = 0.336113, antiderivative size = 351, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {90, 79, 45, 37} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{b d^2 (m+2) (m+3) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{b d^2 (m+3) (b c-a d)}-\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d} \]
Antiderivative was successfully verified.
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Rule 90
Rule 79
Rule 45
Rule 37
Rubi steps
\begin{align*} \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^2 \, dx &=-\frac{f (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d}-\frac{\int (a+b x)^m (c+d x)^{-4-m} \left (-b e (d e+c f (1+m))-a f (c f-d e (3+m))-(b c-a d) f^2 (2+m) x\right ) \, dx}{b d}\\ &=\frac{(d e-c f) (b d e+b c f (2+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d) (3+m)}-\frac{f (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d}+\frac{\left ((b c-a d) f^2 (2+m) (a d (-3-m)+b c (1+m))-2 b d (-b e (d e+c f (1+m))-a f (c f-d e (3+m)))\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{b d^2 (-b c+a d) (-3-m)}\\ &=\frac{(d e-c f) (b d e+b c f (2+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d) (3+m)}+\frac{\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^2 (2+m) (3+m)}-\frac{f (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d}+\frac{\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^2 (b c-a d)^2 (2+m) (3+m)}\\ &=\frac{(d e-c f) (b d e+b c f (2+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{b d^2 (b c-a d) (3+m)}+\frac{\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^2 (2+m) (3+m)}+\frac{\left (a^2 d^2 f^2 \left (6+5 m+m^2\right )-2 a b d f (3+m) (d e+c f (1+m))+b^2 \left (2 d^2 e^2+2 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^3 (1+m) (2+m) (3+m)}-\frac{f (a+b x)^{1+m} (c+d x)^{-3-m} (e+f x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.237863, size = 286, normalized size = 0.81 \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 \left (2 c^2 f^2+2 c d f (e (m+1)+f (m+3) x)+d^2 \left (e^2 \left (m^2+3 m+2\right )+2 e f \left (m^2+4 m+3\right ) x+f^2 \left (m^2+5 m+6\right ) x^2\right )\right )-2 a b \left (c^2 f (e (m+3)+f (m+1) x)+c d \left (e^2 \left (m^2+4 m+3\right )+2 e f \left (m^2+4 m+5\right ) x+f^2 \left (m^2+4 m+3\right ) x^2\right )+d^2 e x (e (m+1)+f (m+3) x)\right )+b^2 \left (c^2 \left (e^2 \left (m^2+5 m+6\right )+2 e f \left (m^2+4 m+3\right ) x+f^2 \left (m^2+3 m+2\right ) x^2\right )+2 c d e x (e (m+3)+f (m+1) x)+2 d^2 e^2 x^2\right )\right )}{(m+1) (m+2) (m+3) (a d-b c)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 741, normalized size = 2.1 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-3-m} \left ({a}^{2}{d}^{2}{f}^{2}{m}^{2}{x}^{2}-2\,abcd{f}^{2}{m}^{2}{x}^{2}+{b}^{2}{c}^{2}{f}^{2}{m}^{2}{x}^{2}+2\,{a}^{2}{d}^{2}ef{m}^{2}x+5\,{a}^{2}{d}^{2}{f}^{2}m{x}^{2}-4\,abcdef{m}^{2}x-8\,abcd{f}^{2}m{x}^{2}-2\,ab{d}^{2}efm{x}^{2}+2\,{b}^{2}{c}^{2}ef{m}^{2}x+3\,{b}^{2}{c}^{2}{f}^{2}m{x}^{2}+2\,{b}^{2}cdefm{x}^{2}+2\,{a}^{2}cd{f}^{2}mx+{a}^{2}{d}^{2}{e}^{2}{m}^{2}+8\,{a}^{2}{d}^{2}efmx+6\,{a}^{2}{d}^{2}{f}^{2}{x}^{2}-2\,ab{c}^{2}{f}^{2}mx-2\,abcd{e}^{2}{m}^{2}-16\,abcdefmx-6\,abcd{f}^{2}{x}^{2}-2\,ab{d}^{2}{e}^{2}mx-6\,ab{d}^{2}ef{x}^{2}+{b}^{2}{c}^{2}{e}^{2}{m}^{2}+8\,{b}^{2}{c}^{2}efmx+2\,{b}^{2}{c}^{2}{f}^{2}{x}^{2}+2\,{b}^{2}cd{e}^{2}mx+2\,{b}^{2}cdef{x}^{2}+2\,{b}^{2}{d}^{2}{e}^{2}{x}^{2}+2\,{a}^{2}cdefm+6\,{a}^{2}cd{f}^{2}x+3\,{a}^{2}{d}^{2}{e}^{2}m+6\,{a}^{2}{d}^{2}efx-2\,ab{c}^{2}efm-2\,ab{c}^{2}{f}^{2}x-8\,abcd{e}^{2}m-20\,abcdefx-2\,ab{d}^{2}{e}^{2}x+5\,{b}^{2}{c}^{2}{e}^{2}m+6\,{b}^{2}{c}^{2}efx+6\,{b}^{2}cd{e}^{2}x+2\,{a}^{2}{c}^{2}{f}^{2}+2\,{a}^{2}cdef+2\,{a}^{2}{d}^{2}{e}^{2}-6\,ab{c}^{2}ef-6\,abcd{e}^{2}+6\,{b}^{2}{c}^{2}{e}^{2} \right ) }{{a}^{3}{d}^{3}{m}^{3}-3\,{a}^{2}bc{d}^{2}{m}^{3}+3\,a{b}^{2}{c}^{2}d{m}^{3}-{b}^{3}{c}^{3}{m}^{3}+6\,{a}^{3}{d}^{3}{m}^{2}-18\,{a}^{2}bc{d}^{2}{m}^{2}+18\,a{b}^{2}{c}^{2}d{m}^{2}-6\,{b}^{3}{c}^{3}{m}^{2}+11\,{a}^{3}{d}^{3}m-33\,{a}^{2}bc{d}^{2}m+33\,a{b}^{2}{c}^{2}dm-11\,{b}^{3}{c}^{3}m+6\,{a}^{3}{d}^{3}-18\,{a}^{2}cb{d}^{2}+18\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3}}} \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{\left (f x + e\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76801, size = 2583, normalized size = 7.32 \begin{align*} \text{result too large to display} \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 (f x + e\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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