Integrand size = 19, antiderivative size = 185 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {6 b^3 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \]
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Time = 0.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac {6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac {3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {(3 b) \int (a+b x)^n (c+d x)^{-4-n} \, dx}{(b c-a d) (4+n)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {\left (6 b^2\right ) \int (a+b x)^n (c+d x)^{-3-n} \, dx}{(b c-a d)^2 (3+n) (4+n)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {\left (6 b^3\right ) \int (a+b x)^n (c+d x)^{-2-n} \, dx}{(b c-a d)^3 (2+n) (3+n) (4+n)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {6 b^3 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.05 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-4-n} \left (-a^3 d^3 \left (6+11 n+6 n^2+n^3\right )+3 a^2 b d^2 \left (2+3 n+n^2\right ) (c (4+n)+d x)-3 a b^2 d (1+n) \left (c^2 \left (12+7 n+n^2\right )+2 c d (4+n) x+2 d^2 x^2\right )+b^3 \left (c^3 \left (24+26 n+9 n^2+n^3\right )+3 c^2 d \left (12+7 n+n^2\right ) x+6 c d^2 (4+n) x^2+6 d^3 x^3\right )\right )}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(661\) vs. \(2(185)=370\).
Time = 0.59 (sec) , antiderivative size = 662, normalized size of antiderivative = 3.58
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (d x +c \right )^{-4-n} \left (a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}-3 a^{2} b \,d^{3} n^{2} x +3 a \,b^{2} c^{2} d \,n^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +6 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-3 b^{3} c^{2} d \,n^{2} x -6 b^{3} c \,d^{2} n \,x^{2}-6 d^{3} x^{3} b^{3}+6 a^{3} d^{3} n^{2}-21 a^{2} b c \,d^{2} n^{2}-9 a^{2} b \,d^{3} n x +24 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} n^{2}-21 b^{3} c^{2} d n x -24 x^{2} b^{3} c \,d^{2}+11 a^{3} d^{3} n -42 a^{2} b c \,d^{2} n -6 x \,a^{2} b \,d^{3}+57 a \,b^{2} c^{2} d n +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} n -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right )}{a^{4} d^{4} n^{4}-4 a^{3} b c \,d^{3} n^{4}+6 a^{2} b^{2} c^{2} d^{2} n^{4}-4 a \,b^{3} c^{3} d \,n^{4}+b^{4} c^{4} n^{4}+10 a^{4} d^{4} n^{3}-40 a^{3} b c \,d^{3} n^{3}+60 a^{2} b^{2} c^{2} d^{2} n^{3}-40 a \,b^{3} c^{3} d \,n^{3}+10 b^{4} c^{4} n^{3}+35 a^{4} d^{4} n^{2}-140 a^{3} b c \,d^{3} n^{2}+210 a^{2} b^{2} c^{2} d^{2} n^{2}-140 a \,b^{3} c^{3} d \,n^{2}+35 b^{4} c^{4} n^{2}+50 a^{4} d^{4} n -200 a^{3} b c \,d^{3} n +300 a^{2} b^{2} c^{2} d^{2} n -200 a \,b^{3} c^{3} d n +50 b^{4} c^{4} n +24 a^{4} d^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 b^{4} c^{4}}\) | \(662\) |
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Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (185) = 370\).
Time = 0.28 (sec) , antiderivative size = 954, normalized size of antiderivative = 5.16 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {{\left (6 \, b^{4} d^{4} x^{5} + 24 \, a b^{3} c^{4} - 36 \, a^{2} b^{2} c^{3} d + 24 \, a^{3} b c^{2} d^{2} - 6 \, a^{4} c d^{3} + 6 \, {\left (5 \, b^{4} c d^{3} + {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} n\right )} x^{4} + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} n^{3} + 3 \, {\left (20 \, b^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} n^{2} + {\left (9 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} n\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{4} - 8 \, a^{2} b^{2} c^{3} d + 7 \, a^{3} b c^{2} d^{2} - 2 \, a^{4} c d^{3}\right )} n^{2} + {\left (60 \, b^{4} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} n^{3} + 3 \, {\left (4 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} n^{2} + {\left (47 \, b^{4} c^{3} d - 60 \, a b^{3} c^{2} d^{2} + 15 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} n\right )} x^{2} + {\left (26 \, a b^{3} c^{4} - 57 \, a^{2} b^{2} c^{3} d + 42 \, a^{3} b c^{2} d^{2} - 11 \, a^{4} c d^{3}\right )} n + {\left (24 \, b^{4} c^{4} + 24 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 24 \, a^{3} b c d^{3} - 6 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} n^{3} + 3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 6 \, a^{3} b c d^{3} - 2 \, a^{4} d^{4}\right )} n^{2} + {\left (26 \, b^{4} c^{4} - 10 \, a b^{3} c^{3} d - 45 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5}}{24 \, b^{4} c^{4} - 96 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 96 \, a^{3} b c d^{3} + 24 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n^{4} + 10 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n^{3} + 35 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n^{2} + 50 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n} \]
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Exception generated. \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5} \,d x } \]
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\[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5} \,d x } \]
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Time = 1.71 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.11 \[ \int (a+b x)^n (c+d x)^{-5-n} \, dx=\frac {6\,b^4\,d^4\,x^5\,{\left (a+b\,x\right )}^n}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^n\,\left (a^3\,d^3\,n^3+6\,a^3\,d^3\,n^2+11\,a^3\,d^3\,n+6\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-21\,a^2\,b\,c\,d^2\,n^2-42\,a^2\,b\,c\,d^2\,n-24\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,n^3+24\,a\,b^2\,c^2\,d\,n^2+57\,a\,b^2\,c^2\,d\,n+36\,a\,b^2\,c^2\,d-b^3\,c^3\,n^3-9\,b^3\,c^3\,n^2-26\,b^3\,c^3\,n-24\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x\,{\left (a+b\,x\right )}^n\,\left (a^4\,d^4\,n^3+6\,a^4\,d^4\,n^2+11\,a^4\,d^4\,n+6\,a^4\,d^4-2\,a^3\,b\,c\,d^3\,n^3-18\,a^3\,b\,c\,d^3\,n^2-40\,a^3\,b\,c\,d^3\,n-24\,a^3\,b\,c\,d^3+9\,a^2\,b^2\,c^2\,d^2\,n^2+45\,a^2\,b^2\,c^2\,d^2\,n+36\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d\,n^3+12\,a\,b^3\,c^3\,d\,n^2+10\,a\,b^3\,c^3\,d\,n-24\,a\,b^3\,c^3\,d-b^4\,c^4\,n^3-9\,b^4\,c^4\,n^2-26\,b^4\,c^4\,n-24\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (a^2\,d^2\,n^2+a^2\,d^2\,n-2\,a\,b\,c\,d\,n^2-10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^n\,\left (5\,b\,c-a\,d\,n+b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n\,\left (-a^3\,d^3\,n^3-3\,a^3\,d^3\,n^2-2\,a^3\,d^3\,n+3\,a^2\,b\,c\,d^2\,n^3+18\,a^2\,b\,c\,d^2\,n^2+15\,a^2\,b\,c\,d^2\,n-3\,a\,b^2\,c^2\,d\,n^3-27\,a\,b^2\,c^2\,d\,n^2-60\,a\,b^2\,c^2\,d\,n+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]
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