Optimal. Leaf size=207 \[ \frac{(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}+\frac{(c+d x)^{n-2} (e+f x)^{1-n} (2 a d f+b (c f (1-n)-d e (3-n)))}{d (2-n) (3-n) (d e-c f)^2}-\frac{f (c+d x)^{n-1} (e+f x)^{1-n} (2 a d f+b (c f (1-n)-d e (3-n)))}{d (1-n) (2-n) (3-n) (d e-c f)^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.359864, antiderivative size = 205, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}+\frac{(c+d x)^{n-2} (e+f x)^{1-n} (2 a d f+b c f (1-n)-b d e (3-n))}{d (2-n) (3-n) (d e-c f)^2}-\frac{f (c+d x)^{n-1} (e+f x)^{1-n} (2 a d f+b c f (1-n)-b d e (3-n))}{d (1-n) (2-n) (3-n) (d e-c f)^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 47.347, size = 151, normalized size = 0.73 \[ \frac{f \left (c + d x\right )^{n - 1} \left (e + f x\right )^{- n + 1} \left (2 a d f + b \left (c f \left (- n + 1\right ) - d e \left (- n + 3\right )\right )\right )}{d \left (- n + 1\right ) \left (- n + 2\right ) \left (- n + 3\right ) \left (c f - d e\right )^{3}} + \frac{\left (c + d x\right )^{n - 3} \left (e + f x\right )^{- n + 1} \left (a d - b c\right )}{d \left (- n + 3\right ) \left (c f - d e\right )} + \frac{\left (c + d x\right )^{n - 2} \left (e + f x\right )^{- n + 1} \left (2 a d f + b \left (c f \left (- n + 1\right ) - d e \left (- n + 3\right )\right )\right )}{d \left (- n + 2\right ) \left (- n + 3\right ) \left (c f - d e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(d*x+c)**(-4+n)/((f*x+e)**n),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.560298, size = 198, normalized size = 0.96 \[ \frac{(c+d x)^n (e+f x)^{-n} \left (\frac{f^2 (2 a d f-b c f (n-1)+b d e (n-3))}{(n-3) (n-2) (n-1) (d e-c f)^3}+\frac{f n (2 a d f-b c f (n-1)+b d e (n-3))}{(n-1) \left (n^2-5 n+6\right ) (c+d x) (d e-c f)^2}+\frac{a d f n+b c f (3-2 n)+b d e (n-3)}{(n-3) (n-2) (c+d x)^2 (d e-c f)}+\frac{a d-b c}{(n-3) (c+d x)^3}\right )}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.01, size = 506, normalized size = 2.4 \[ -{\frac{ \left ( dx+c \right ) ^{-3+n} \left ( fx+e \right ) \left ( b{c}^{2}{f}^{2}{n}^{2}x-2\,bcdef{n}^{2}x-bcd{f}^{2}n{x}^{2}+b{d}^{2}{e}^{2}{n}^{2}x+b{d}^{2}efn{x}^{2}+a{c}^{2}{f}^{2}{n}^{2}-2\,acdef{n}^{2}-2\,acd{f}^{2}nx+a{d}^{2}{e}^{2}{n}^{2}+2\,a{d}^{2}efnx+2\,a{d}^{2}{f}^{2}{x}^{2}-4\,b{c}^{2}{f}^{2}nx+8\,bcdefnx+bcd{f}^{2}{x}^{2}-4\,b{d}^{2}{e}^{2}nx-3\,b{d}^{2}ef{x}^{2}-5\,a{c}^{2}{f}^{2}n+8\,acdefn+6\,acd{f}^{2}x-3\,a{d}^{2}{e}^{2}n-2\,a{d}^{2}efx+b{c}^{2}efn+3\,b{c}^{2}{f}^{2}x-bcd{e}^{2}n-10\,bcdefx+3\,b{d}^{2}{e}^{2}x+6\,a{c}^{2}{f}^{2}-6\,acdef+2\,a{d}^{2}{e}^{2}-3\,b{c}^{2}ef+bcd{e}^{2} \right ) }{ \left ({c}^{3}{f}^{3}{n}^{3}-3\,{c}^{2}de{f}^{2}{n}^{3}+3\,c{d}^{2}{e}^{2}f{n}^{3}-{d}^{3}{e}^{3}{n}^{3}-6\,{c}^{3}{f}^{3}{n}^{2}+18\,{c}^{2}de{f}^{2}{n}^{2}-18\,c{d}^{2}{e}^{2}f{n}^{2}+6\,{d}^{3}{e}^{3}{n}^{2}+11\,{c}^{3}{f}^{3}n-33\,{c}^{2}de{f}^{2}n+33\,c{d}^{2}{e}^{2}fn-11\,{d}^{3}{e}^{3}n-6\,{c}^{3}{f}^{3}+18\,{c}^{2}de{f}^{2}-18\,c{d}^{2}{e}^{2}f+6\,{d}^{3}{e}^{3} \right ) \left ( fx+e \right ) ^{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (d x + c\right )}^{n - 4}{\left (f x + e\right )}^{-n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.268368, size = 1193, normalized size = 5.76 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)*(d*x+c)**(-4+n)/((f*x+e)**n),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (d x + c\right )}^{n - 4}}{{\left (f x + e\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n,x, algorithm="giac")
[Out]