Optimal. Leaf size=188 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b (c f (m+1)+2 d e))}{d (m+2) (m+3) (b c-a d)^2}-\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+3)-b (c f (m+1)+2 d e))}{d (m+1) (m+2) (m+3) (b c-a d)^3} \]
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Rubi [A] time = 0.304577, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}+\frac{(a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+2) (m+3) (b c-a d)^2}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+1) (m+2) (m+3) (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x),x]
[Out]
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Rubi in Sympy [A] time = 46.252, size = 156, normalized size = 0.83 \[ \frac{b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- 2 b d e + f \left (a d \left (m + 3\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{3}} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 3} \left (c f - d e\right )}{d \left (m + 3\right ) \left (a d - b c\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (- 2 b d e + f \left (a d \left (m + 3\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e),x)
[Out]
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Mathematica [A] time = 0.615199, size = 199, normalized size = 1.06 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{b^2 (-a d f (m+3)+b c f (m+1)+2 b d e)}{(m+1) (m+2) (m+3) (b c-a d)^3}+\frac{b m (-a d f (m+3)+b c f (m+1)+2 b d e)}{(m+1) \left (m^2+5 m+6\right ) (c+d x) (b c-a d)^2}+\frac{a d f (m+3)-b c f (2 m+3)+b d e m}{(m+2) (m+3) (c+d x)^2 (b c-a d)}+\frac{c f-d e}{(m+3) (c+d x)^3}\right )}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x),x]
[Out]
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Maple [B] time = 0.017, size = 503, normalized size = 2.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-3-m} \left ({a}^{2}{d}^{2}f{m}^{2}x-2\,abcdf{m}^{2}x-ab{d}^{2}fm{x}^{2}+{b}^{2}{c}^{2}f{m}^{2}x+{b}^{2}cdfm{x}^{2}+{a}^{2}{d}^{2}e{m}^{2}+4\,{a}^{2}{d}^{2}fmx-2\,abcde{m}^{2}-8\,abcdfmx-2\,ab{d}^{2}emx-3\,ab{d}^{2}f{x}^{2}+{b}^{2}{c}^{2}e{m}^{2}+4\,{b}^{2}{c}^{2}fmx+2\,{b}^{2}cdemx+{b}^{2}cdf{x}^{2}+2\,{b}^{2}{d}^{2}e{x}^{2}+{a}^{2}cdfm+3\,{a}^{2}{d}^{2}em+3\,{a}^{2}{d}^{2}fx-ab{c}^{2}fm-8\,abcdem-10\,abcdfx-2\,ab{d}^{2}ex+5\,{b}^{2}{c}^{2}em+3\,{b}^{2}{c}^{2}fx+6\,{b}^{2}cdex+{a}^{2}cdf+2\,{a}^{2}{d}^{2}e-3\,ab{c}^{2}f-6\,abcde+6\,{b}^{2}{c}^{2}e \right ) }{{a}^{3}{d}^{3}{m}^{3}-3\,{a}^{2}bc{d}^{2}{m}^{3}+3\,a{b}^{2}{c}^{2}d{m}^{3}-{b}^{3}{c}^{3}{m}^{3}+6\,{a}^{3}{d}^{3}{m}^{2}-18\,{a}^{2}bc{d}^{2}{m}^{2}+18\,a{b}^{2}{c}^{2}d{m}^{2}-6\,{b}^{3}{c}^{3}{m}^{2}+11\,{a}^{3}{d}^{3}m-33\,{a}^{2}bc{d}^{2}m+33\,a{b}^{2}{c}^{2}dm-11\,{b}^{3}{c}^{3}m+6\,{a}^{3}{d}^{3}-18\,{a}^{2}bc{d}^{2}+18\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238807, size = 1218, normalized size = 6.48 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="giac")
[Out]