Optimal. Leaf size=86 \[ \frac{d (a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (2-n) (b c-a d)^2}-\frac{(a+b x)^{n-2} (c+d x)^{1-n}}{(2-n) (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0600141, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{d (a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (2-n) (b c-a d)^2}-\frac{(a+b x)^{n-2} (c+d x)^{1-n}}{(2-n) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(-3 + n)/(c + d*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 12.6771, size = 60, normalized size = 0.7 \[ \frac{d \left (a + b x\right )^{n - 1} \left (c + d x\right )^{- n + 1}}{\left (- n + 1\right ) \left (- n + 2\right ) \left (a d - b c\right )^{2}} + \frac{\left (a + b x\right )^{n - 2} \left (c + d x\right )^{- n + 1}}{\left (- n + 2\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(-3+n)/((d*x+c)**n),x)
[Out]
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Mathematica [A] time = 0.114339, size = 59, normalized size = 0.69 \[ \frac{(a+b x)^{n-2} (c+d x)^{1-n} (-a d (n-2)+b c (n-1)+b d x)}{(n-2) (n-1) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(-3 + n)/(c + d*x)^n,x]
[Out]
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Maple [A] time = 0.007, size = 127, normalized size = 1.5 \[ -{\frac{ \left ( bx+a \right ) ^{-2+n} \left ( dx+c \right ) \left ( adn-bcn-bdx-2\,ad+bc \right ) }{ \left ({a}^{2}{d}^{2}{n}^{2}-2\,abcd{n}^{2}+{b}^{2}{c}^{2}{n}^{2}-3\,{a}^{2}{d}^{2}n+6\,abcdn-3\,{b}^{2}{c}^{2}n+2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2} \right ) \left ( dx+c \right ) ^{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(-3+n)/((d*x+c)^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n - 3}{\left (d x + c\right )}^{-n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(n - 3)/(d*x + c)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2272, size = 278, normalized size = 3.23 \[ \frac{{\left (b^{2} d^{2} x^{3} - a b c^{2} + 2 \, a^{2} c d +{\left (3 \, a b d^{2} +{\left (b^{2} c d - a b d^{2}\right )} n\right )} x^{2} +{\left (a b c^{2} - a^{2} c d\right )} n -{\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2} -{\left (b^{2} c^{2} - a^{2} d^{2}\right )} n\right )} x\right )}{\left (b x + a\right )}^{n - 3}}{{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n^{2} - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n\right )}{\left (d x + c\right )}^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(n - 3)/(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)**(-3+n)/((d*x+c)**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n - 3}}{{\left (d x + c\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(n - 3)/(d*x + c)^n,x, algorithm="giac")
[Out]