Optimal. Leaf size=186 \[ \frac{6 d^3 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}-\frac{6 d^2 (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (n+4) (b c-a d)^3}-\frac{(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}+\frac{3 d (a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (n+4) (b c-a d)^2} \]
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Rubi [A] time = 0.23032, antiderivative size = 186, 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 \[ \frac{6 d^3 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}-\frac{6 d^2 (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (n+4) (b c-a d)^3}-\frac{(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}+\frac{3 d (a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (n+4) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(-5 - n)*(c + d*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 50.6477, size = 153, normalized size = 0.82 \[ \frac{6 d^{3} \left (a + b x\right )^{- n - 1} \left (c + d x\right )^{n + 1}}{\left (n + 1\right ) \left (n + 2\right ) \left (n + 3\right ) \left (n + 4\right ) \left (a d - b c\right )^{4}} + \frac{6 d^{2} \left (a + b x\right )^{- n - 2} \left (c + d x\right )^{n + 1}}{\left (n + 2\right ) \left (n + 3\right ) \left (n + 4\right ) \left (a d - b c\right )^{3}} + \frac{3 d \left (a + b x\right )^{- n - 3} \left (c + d x\right )^{n + 1}}{\left (n + 3\right ) \left (n + 4\right ) \left (a d - b c\right )^{2}} + \frac{\left (a + b x\right )^{- n - 4} \left (c + d x\right )^{n + 1}}{\left (n + 4\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(-5-n)*(d*x+c)**n,x)
[Out]
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Mathematica [A] time = 0.467563, size = 181, normalized size = 0.97 \[ -\frac{(a+b x)^{-n} (c+d x)^n \left (-\frac{6 d^4}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac{6 d^3 n}{(n+1) \left (n^3+9 n^2+26 n+24\right ) (a+b x) (b c-a d)^3}-\frac{3 d^2 n}{(n+2) \left (n^2+7 n+12\right ) (a+b x)^2 (b c-a d)^2}+\frac{d n}{(n+3) (n+4) (a+b x)^3 (b c-a d)}+\frac{1}{(n+4) (a+b x)^4}\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(-5 - n)*(c + d*x)^n,x]
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Maple [B] time = 0.013, size = 661, normalized size = 3.6 \[{\frac{ \left ( bx+a \right ) ^{-4-n} \left ( dx+c \right ) ^{1+n} \left ({a}^{3}{d}^{3}{n}^{3}-3\,{a}^{2}bc{d}^{2}{n}^{3}+3\,{a}^{2}b{d}^{3}{n}^{2}x+3\,a{b}^{2}{c}^{2}d{n}^{3}-6\,a{b}^{2}c{d}^{2}{n}^{2}x+6\,a{b}^{2}{d}^{3}n{x}^{2}-{b}^{3}{c}^{3}{n}^{3}+3\,{b}^{3}{c}^{2}d{n}^{2}x-6\,{b}^{3}c{d}^{2}n{x}^{2}+6\,{x}^{3}{b}^{3}{d}^{3}+9\,{a}^{3}{d}^{3}{n}^{2}-24\,{a}^{2}bc{d}^{2}{n}^{2}+21\,{a}^{2}b{d}^{3}nx+21\,a{b}^{2}{c}^{2}d{n}^{2}-30\,a{b}^{2}c{d}^{2}nx+24\,a{b}^{2}{d}^{3}{x}^{2}-6\,{b}^{3}{c}^{3}{n}^{2}+9\,{b}^{3}{c}^{2}dnx-6\,{b}^{3}c{d}^{2}{x}^{2}+26\,{a}^{3}{d}^{3}n-57\,{a}^{2}bc{d}^{2}n+36\,{a}^{2}b{d}^{3}x+42\,a{b}^{2}{c}^{2}dn-24\,a{b}^{2}c{d}^{2}x-11\,{b}^{3}{c}^{3}n+6\,{b}^{3}{c}^{2}dx+24\,{a}^{3}{d}^{3}-36\,{a}^{2}cb{d}^{2}+24\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3} \right ) }{{a}^{4}{d}^{4}{n}^{4}-4\,{a}^{3}bc{d}^{3}{n}^{4}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{4}-4\,a{b}^{3}{c}^{3}d{n}^{4}+{b}^{4}{c}^{4}{n}^{4}+10\,{a}^{4}{d}^{4}{n}^{3}-40\,{a}^{3}bc{d}^{3}{n}^{3}+60\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{3}-40\,a{b}^{3}{c}^{3}d{n}^{3}+10\,{b}^{4}{c}^{4}{n}^{3}+35\,{a}^{4}{d}^{4}{n}^{2}-140\,{a}^{3}bc{d}^{3}{n}^{2}+210\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{2}-140\,a{b}^{3}{c}^{3}d{n}^{2}+35\,{b}^{4}{c}^{4}{n}^{2}+50\,{a}^{4}{d}^{4}n-200\,{a}^{3}bc{d}^{3}n+300\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}n-200\,a{b}^{3}{c}^{3}dn+50\,{b}^{4}{c}^{4}n+24\,{a}^{4}{d}^{4}-96\,{a}^{3}bc{d}^{3}+144\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-96\,a{b}^{3}{c}^{3}d+24\,{b}^{4}{c}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(-5-n)*(d*x+c)^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{-n - 5}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(-n - 5)*(d*x + c)^n,x, algorithm="maxima")
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Fricas [A] time = 0.236676, size = 1295, 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 - 5)*(d*x + c)^n,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)**(-5-n)*(d*x+c)**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{-n - 5}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(-n - 5)*(d*x + c)^n,x, algorithm="giac")
[Out]