Optimal. Leaf size=185 \[ \frac{6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac{6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac{(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac{3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]
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Rubi [A] time = 0.177219, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac{6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac{(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac{3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n*(c + d*x)^(-5 - n),x]
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Rubi in Sympy [A] time = 51.6278, size = 153, normalized size = 0.83 \[ \frac{6 b^{3} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 1}}{\left (n + 1\right ) \left (n + 2\right ) \left (n + 3\right ) \left (n + 4\right ) \left (a d - b c\right )^{4}} - \frac{6 b^{2} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 2}}{\left (n + 2\right ) \left (n + 3\right ) \left (n + 4\right ) \left (a d - b c\right )^{3}} + \frac{3 b \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 3}}{\left (n + 3\right ) \left (n + 4\right ) \left (a d - b c\right )^{2}} - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 4}}{\left (n + 4\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x+c)**(-5-n),x)
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Mathematica [A] time = 0.533182, size = 181, normalized size = 0.98 \[ \frac{(a+b x)^n (c+d x)^{-n} \left (\frac{6 b^4}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac{6 b^3 n}{(n+1) \left (n^3+9 n^2+26 n+24\right ) (c+d x) (b c-a d)^3}+\frac{3 b^2 n}{(n+2) \left (n^2+7 n+12\right ) (c+d x)^2 (b c-a d)^2}+\frac{b n}{(n+3) (n+4) (c+d x)^3 (b c-a d)}-\frac{1}{(n+4) (c+d x)^4}\right )}{d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n*(c + d*x)^(-5 - n),x]
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Maple [B] time = 0.013, size = 662, normalized size = 3.6 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{-4-n} \left ({a}^{3}{d}^{3}{n}^{3}-3\,{a}^{2}bc{d}^{2}{n}^{3}-3\,{a}^{2}b{d}^{3}{n}^{2}x+3\,a{b}^{2}{c}^{2}d{n}^{3}+6\,a{b}^{2}c{d}^{2}{n}^{2}x+6\,a{b}^{2}{d}^{3}n{x}^{2}-{b}^{3}{c}^{3}{n}^{3}-3\,{b}^{3}{c}^{2}d{n}^{2}x-6\,{b}^{3}c{d}^{2}n{x}^{2}-6\,{x}^{3}{b}^{3}{d}^{3}+6\,{a}^{3}{d}^{3}{n}^{2}-21\,{a}^{2}bc{d}^{2}{n}^{2}-9\,{a}^{2}b{d}^{3}nx+24\,a{b}^{2}{c}^{2}d{n}^{2}+30\,a{b}^{2}c{d}^{2}nx+6\,a{b}^{2}{d}^{3}{x}^{2}-9\,{b}^{3}{c}^{3}{n}^{2}-21\,{b}^{3}{c}^{2}dnx-24\,{b}^{3}c{d}^{2}{x}^{2}+11\,{a}^{3}{d}^{3}n-42\,{a}^{2}bc{d}^{2}n-6\,{a}^{2}b{d}^{3}x+57\,a{b}^{2}{c}^{2}dn+24\,a{b}^{2}c{d}^{2}x-26\,{b}^{3}{c}^{3}n-36\,{b}^{3}{c}^{2}dx+6\,{a}^{3}{d}^{3}-24\,{a}^{2}cb{d}^{2}+36\,a{b}^{2}{c}^{2}d-24\,{b}^{3}{c}^{3} \right ) }{{a}^{4}{d}^{4}{n}^{4}-4\,{a}^{3}bc{d}^{3}{n}^{4}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{4}-4\,a{b}^{3}{c}^{3}d{n}^{4}+{b}^{4}{c}^{4}{n}^{4}+10\,{a}^{4}{d}^{4}{n}^{3}-40\,{a}^{3}bc{d}^{3}{n}^{3}+60\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{3}-40\,a{b}^{3}{c}^{3}d{n}^{3}+10\,{b}^{4}{c}^{4}{n}^{3}+35\,{a}^{4}{d}^{4}{n}^{2}-140\,{a}^{3}bc{d}^{3}{n}^{2}+210\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{2}-140\,a{b}^{3}{c}^{3}d{n}^{2}+35\,{b}^{4}{c}^{4}{n}^{2}+50\,{a}^{4}{d}^{4}n-200\,{a}^{3}bc{d}^{3}n+300\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}n-200\,a{b}^{3}{c}^{3}dn+50\,{b}^{4}{c}^{4}n+24\,{a}^{4}{d}^{4}-96\,{a}^{3}bc{d}^{3}+144\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-96\,a{b}^{3}{c}^{3}d+24\,{b}^{4}{c}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x+c)^(-5-n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^(-n - 5),x, algorithm="maxima")
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Fricas [A] time = 0.252753, size = 1288, normalized size = 6.96 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^(-n - 5),x, algorithm="fricas")
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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)**n*(d*x+c)**(-5-n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^(-n - 5),x, algorithm="giac")
[Out]