Integrand size = 19, antiderivative size = 80 \[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d) (2+n)}+\frac {d (a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d)^2 (1+n) (2+n)} \]
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Time = 0.01 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\frac {d (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (b c-a d)^2}-\frac {(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d) (2+n)}-\frac {d \int (a+b x)^{-2-n} (c+d x)^n \, dx}{(b c-a d) (2+n)} \\ & = -\frac {(a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d) (2+n)}+\frac {d (a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d)^2 (1+n) (2+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\frac {(a+b x)^{-2-n} (c+d x)^{1+n} (a d (2+n)-b (c+c n-d x))}{(b c-a d)^2 (1+n) (2+n)} \]
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Time = 0.53 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.54
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{-2-n} \left (d x +c \right )^{1+n} \left (a d n -b c n +b d x +2 a d -b c \right )}{a^{2} d^{2} n^{2}-2 a b c d \,n^{2}+b^{2} c^{2} n^{2}+3 a^{2} d^{2} n -6 a b c d n +3 b^{2} c^{2} n +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(123\) |
parallelrisch | \(\frac {x^{3} \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} b^{3} d^{3}+x^{2} \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a \,b^{2} d^{3} n -x^{2} \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} b^{3} c \,d^{2} n +3 x^{2} \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a \,b^{2} d^{3}+x \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a^{2} b \,d^{3} n -x \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} b^{3} c^{2} d n +2 x \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a^{2} b \,d^{3}+2 x \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a \,b^{2} c \,d^{2}-x \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} b^{3} c^{2} d +\left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a^{2} b c \,d^{2} n -\left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a \,b^{2} c^{2} d n +2 \left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a^{2} b c \,d^{2}-\left (d x +c \right )^{n} \left (b x +a \right )^{-3-n} a \,b^{2} c^{2} d}{b d \left (a^{2} d^{2} n^{2}-2 a b c d \,n^{2}+b^{2} c^{2} n^{2}+3 a^{2} d^{2} n -6 a b c d n +3 b^{2} c^{2} n +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right )}\) | \(462\) |
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Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (80) = 160\).
Time = 0.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.59 \[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\frac {{\left (b^{2} d^{2} x^{3} - a b c^{2} + 2 \, a^{2} c d + {\left (3 \, a b d^{2} - {\left (b^{2} c d - a b d^{2}\right )} n\right )} x^{2} - {\left (a b c^{2} - a^{2} c d\right )} n - {\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2} + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{-n - 3} {\left (d x + c\right )}^{n}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n^{2} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n} \]
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Exception generated. \[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 3} {\left (d x + c\right )}^{n} \,d x } \]
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\[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 3} {\left (d x + c\right )}^{n} \,d x } \]
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Time = 0.76 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.68 \[ \int (a+b x)^{-3-n} (c+d x)^n \, dx=\frac {\frac {x\,{\left (c+d\,x\right )}^n\,\left (2\,a^2\,d^2-b^2\,c^2+a^2\,d^2\,n-b^2\,c^2\,n+2\,a\,b\,c\,d\right )}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}+\frac {a\,c\,{\left (c+d\,x\right )}^n\,\left (2\,a\,d-b\,c+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}+\frac {b^2\,d^2\,x^3\,{\left (c+d\,x\right )}^n}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n\,\left (3\,a\,d+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}}{{\left (a+b\,x\right )}^{n+3}} \]
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