Optimal. Leaf size=37 \[ -\frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0237575, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(-2 - n)*(c + d*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 4.25976, size = 27, normalized size = 0.73 \[ \frac{\left (a + b x\right )^{- n - 1} \left (c + d x\right )^{n + 1}}{\left (n + 1\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(-2-n)*(d*x+c)**n,x)
[Out]
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Mathematica [A] time = 0.060137, size = 36, normalized size = 0.97 \[ \frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(-2 - n)*(c + d*x)^n,x]
[Out]
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Maple [A] time = 0.006, size = 41, normalized size = 1.1 \[{\frac{ \left ( bx+a \right ) ^{-1-n} \left ( dx+c \right ) ^{1+n}}{adn-bcn+ad-bc}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(-2-n)*(d*x+c)^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{-n - 2}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(-n - 2)*(d*x + c)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234408, size = 80, normalized size = 2.16 \[ -\frac{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{-n - 2}{\left (d x + c\right )}^{n}}{b c - a d +{\left (b c - a d\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(-n - 2)*(d*x + c)^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((b*x+a)**(-2-n)*(d*x+c)**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{-n - 2}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(-n - 2)*(d*x + c)^n,x, algorithm="giac")
[Out]