Optimal. Leaf size=207 \[ -\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-3}}{f (3-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-2} (b (2 c f+d e (1-n))-a d f (3-n))}{f (2-n) (3-n) (b e-a f)^2}+\frac{b (a+b x)^{1-n} (e+f x)^{n-1} (b (2 c f+d e (1-n))-a d f (3-n))}{f (1-n) (2-n) (3-n) (b e-a f)^3} \]
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Rubi [A] time = 0.117984, 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, Rules used = {79, 45, 37} \[ -\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-3}}{f (3-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-2} (-a d f (3-n)+2 b c f+b d e (1-n))}{f (2-n) (3-n) (b e-a f)^2}+\frac{b (a+b x)^{1-n} (e+f x)^{n-1} (-a d f (3-n)+2 b c f+b d e (1-n))}{f (1-n) (2-n) (3-n) (b e-a f)^3} \]
Antiderivative was successfully verified.
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Rule 79
Rule 45
Rule 37
Rubi steps
\begin{align*} \int (a+b x)^{-n} (c+d x) (e+f x)^{-4+n} \, dx &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f) (3-n)}-\frac{(-2 b c f-d (b e (1-n)+a f (-3+n))) \int (a+b x)^{-n} (e+f x)^{-3+n} \, dx}{f (-b e+a f) (-3+n)}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f) (3-n)}+\frac{(2 b c f+b d e (1-n)-a d f (3-n)) (a+b x)^{1-n} (e+f x)^{-2+n}}{f (b e-a f)^2 (2-n) (3-n)}-\frac{(b (-2 b c f-d (b e (1-n)+a f (-3+n)))) \int (a+b x)^{-n} (e+f x)^{-2+n} \, dx}{f (b e-a f) (-b e+a f) (2-n) (-3+n)}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f) (3-n)}+\frac{(2 b c f+b d e (1-n)-a d f (3-n)) (a+b x)^{1-n} (e+f x)^{-2+n}}{f (b e-a f)^2 (2-n) (3-n)}+\frac{b (2 b c f+b d e (1-n)-a d f (3-n)) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f)^3 (1-n) (2-n) (3-n)}\\ \end{align*}
Mathematica [A] time = 0.149136, size = 180, normalized size = 0.87 \[ \frac{(a+b x)^{1-n} (e+f x)^{n-3} \left (a^2 f (n-1) (c f (n-2)-d e+d f (n-3) x)+a b \left (2 c f (n-1) (f x-e (n-3))+d \left (e^2 (n-3)-2 e f \left (n^2-4 n+5\right ) x+f^2 (n-3) x^2\right )\right )+b^2 \left (c \left (e^2 \left (n^2-5 n+6\right )-2 e f (n-3) x+2 f^2 x^2\right )+d e (n-1) x (e (n-3)-f x)\right )\right )}{(n-3) (n-2) (n-1) (a f-b e)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 505, normalized size = 2.4 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( fx+e \right ) ^{-3+n} \left ({a}^{2}d{f}^{2}{n}^{2}x-2\,abdef{n}^{2}x+abd{f}^{2}n{x}^{2}+{b}^{2}d{e}^{2}{n}^{2}x-{b}^{2}defn{x}^{2}+{a}^{2}c{f}^{2}{n}^{2}-4\,{a}^{2}d{f}^{2}nx-2\,abcef{n}^{2}+2\,abc{f}^{2}nx+8\,abdefnx-3\,abd{f}^{2}{x}^{2}+{b}^{2}c{e}^{2}{n}^{2}-2\,{b}^{2}cefnx+2\,{b}^{2}c{f}^{2}{x}^{2}-4\,{b}^{2}d{e}^{2}nx+{b}^{2}def{x}^{2}-3\,{a}^{2}c{f}^{2}n-{a}^{2}defn+3\,{a}^{2}d{f}^{2}x+8\,abcefn-2\,abc{f}^{2}x+abd{e}^{2}n-10\,abdefx-5\,{b}^{2}c{e}^{2}n+6\,{b}^{2}cefx+3\,{b}^{2}d{e}^{2}x+2\,{a}^{2}c{f}^{2}+{a}^{2}def-6\,abcef-3\,abd{e}^{2}+6\,{b}^{2}c{e}^{2} \right ) }{ \left ({a}^{3}{f}^{3}{n}^{3}-3\,{a}^{2}be{f}^{2}{n}^{3}+3\,a{b}^{2}{e}^{2}f{n}^{3}-{b}^{3}{e}^{3}{n}^{3}-6\,{a}^{3}{f}^{3}{n}^{2}+18\,{a}^{2}be{f}^{2}{n}^{2}-18\,a{b}^{2}{e}^{2}f{n}^{2}+6\,{b}^{3}{e}^{3}{n}^{2}+11\,{a}^{3}{f}^{3}n-33\,{a}^{2}be{f}^{2}n+33\,a{b}^{2}{e}^{2}fn-11\,{b}^{3}{e}^{3}n-6\,{a}^{3}{f}^{3}+18\,{a}^{2}be{f}^{2}-18\,a{b}^{2}{e}^{2}f+6\,{b}^{3}{e}^{3} \right ) \left ( bx+a \right ) ^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 4}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76781, size = 1787, normalized size = 8.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}{\left (f x + e\right )}^{n - 4}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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