Optimal. Leaf size=125 \[ -\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-2}}{f (2-n) (b e-a f)}-\frac{(a+b x)^{1-n} (e+f x)^{n-1} (a d f (2-n)-b (c f+d (e-e n)))}{f (1-n) (2-n) (b e-a f)^2} \]
[Out]
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Rubi [A] time = 0.203014, antiderivative size = 123, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^{1-n} (e+f x)^{n-1} (-a d f (2-n)+b c f+b d (e-e n))}{f (1-n) (2-n) (b e-a f)^2}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-2}}{f (2-n) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x)*(e + f*x)^(-3 + n))/(a + b*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 25.1949, size = 90, normalized size = 0.72 \[ - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 2} \left (c f - d e\right )}{f \left (- n + 2\right ) \left (a f - b e\right )} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 1} \left (- b c f + d \left (a f \left (- n + 2\right ) - b e \left (- n + 1\right )\right )\right )}{f \left (- n + 1\right ) \left (- n + 2\right ) \left (a f - b e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)*(f*x+e)**(-3+n)/((b*x+a)**n),x)
[Out]
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Mathematica [A] time = 0.256532, size = 84, normalized size = 0.67 \[ \frac{(a+b x)^{1-n} (e+f x)^{n-2} (a c f (n-1)-a d e+a d f (n-2) x+b c (f x-e (n-2))-b d e (n-1) x)}{(n-2) (n-1) (b e-a f)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x)*(e + f*x)^(-3 + n))/(a + b*x)^n,x]
[Out]
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Maple [A] time = 0.009, size = 160, normalized size = 1.3 \[{\frac{ \left ( bx+a \right ) \left ( fx+e \right ) ^{-2+n} \left ( adfnx-bdenx+acfn-2\,adfx-bcen+bcfx+bdex-acf-ade+2\,bce \right ) }{ \left ({a}^{2}{f}^{2}{n}^{2}-2\,abef{n}^{2}+{b}^{2}{e}^{2}{n}^{2}-3\,{a}^{2}{f}^{2}n+6\,abefn-3\,{b}^{2}{e}^{2}n+2\,{a}^{2}{f}^{2}-4\,abef+2\,{b}^{2}{e}^{2} \right ) \left ( bx+a \right ) ^{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)*(f*x+e)^(-3+n)/((b*x+a)^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}{\left (b x + a\right )}^{-n}{\left (f x + e\right )}^{n - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 3)/(b*x + a)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258246, size = 440, normalized size = 3.52 \[ -\frac{{\left (a^{2} c e f -{\left (b^{2} d e f +{\left (b^{2} c - 2 \, a b d\right )} f^{2} -{\left (b^{2} d e f - a b d f^{2}\right )} n\right )} x^{3} -{\left (2 \, a b c - a^{2} d\right )} e^{2} -{\left (b^{2} d e^{2} - 2 \, a^{2} d f^{2} +{\left (3 \, b^{2} c - 2 \, a b d\right )} e f -{\left (b^{2} d e^{2} + b^{2} c e f -{\left (a b c + a^{2} d\right )} f^{2}\right )} n\right )} x^{2} +{\left (a b c e^{2} - a^{2} c e f\right )} n -{\left (2 \, b^{2} c e^{2} - a^{2} c f^{2} +{\left (2 \, a b c - 3 \, a^{2} d\right )} e f +{\left (a^{2} d e f + a^{2} c f^{2} -{\left (b^{2} c + a b d\right )} e^{2}\right )} n\right )} x\right )}{\left (f x + e\right )}^{n - 3}}{{\left (2 \, b^{2} e^{2} - 4 \, a b e f + 2 \, a^{2} f^{2} +{\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} n^{2} - 3 \,{\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} n\right )}{\left (b x + a\right )}^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 3)/(b*x + a)^n,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((d*x+c)*(f*x+e)**(-3+n)/((b*x+a)**n),x)
[Out]
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GIAC/XCAS [A] time = 0.228602, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(f*x + e)^(n - 3)/(b*x + a)^n,x, algorithm="giac")
[Out]