Integrand size = 19, antiderivative size = 130 \[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d) (3+n)}+\frac {2 b (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^2 (2+n) (3+n)}+\frac {2 b^2 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^3 (1+n) (2+n) (3+n)} \]
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Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\frac {2 b^2 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (b c-a d)^3}+\frac {(a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (b c-a d)}+\frac {2 b (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (b c-a d)^2} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d) (3+n)}+\frac {(2 b) \int (a+b x)^n (c+d x)^{-3-n} \, dx}{(b c-a d) (3+n)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d) (3+n)}+\frac {2 b (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^2 (2+n) (3+n)}+\frac {\left (2 b^2\right ) \int (a+b x)^n (c+d x)^{-2-n} \, dx}{(b c-a d)^2 (2+n) (3+n)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d) (3+n)}+\frac {2 b (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^2 (2+n) (3+n)}+\frac {2 b^2 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^3 (1+n) (2+n) (3+n)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-3-n} \left (a^2 d^2 \left (2+3 n+n^2\right )-2 a b d (1+n) (c (3+n)+d x)+b^2 \left (c^2 \left (6+5 n+n^2\right )+2 c d (3+n) x+2 d^2 x^2\right )\right )}{(b c-a d)^3 (1+n) (2+n) (3+n)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(130)=260\).
Time = 0.59 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.45
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (d x +c \right )^{-3-n} \left (a^{2} d^{2} n^{2}-2 a b c d \,n^{2}-2 a b \,d^{2} n x +b^{2} c^{2} n^{2}+2 b^{2} c d n x +2 d^{2} x^{2} b^{2}+3 a^{2} d^{2} n -8 a b c d n -2 x a b \,d^{2}+5 b^{2} c^{2} n +6 x \,b^{2} c d +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right )}{a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}+3 a \,b^{2} c^{2} d \,n^{3}-b^{3} c^{3} n^{3}+6 a^{3} d^{3} n^{2}-18 a^{2} b c \,d^{2} n^{2}+18 a \,b^{2} c^{2} d \,n^{2}-6 b^{3} c^{3} n^{2}+11 a^{3} d^{3} n -33 a^{2} b c \,d^{2} n +33 a \,b^{2} c^{2} d n -11 b^{3} c^{3} n +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\) | \(319\) |
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Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (130) = 260\).
Time = 0.25 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.90 \[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\frac {{\left (2 \, b^{3} d^{3} x^{4} + 6 \, a b^{2} c^{3} - 6 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 2 \, {\left (4 \, b^{3} c d^{2} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} n\right )} x^{3} + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} n^{2} + {\left (12 \, b^{3} c^{2} d + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} n^{2} + {\left (7 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} n\right )} x^{2} + {\left (5 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} n + {\left (6 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} n^{2} + {\left (5 \, b^{3} c^{3} - a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 4}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{3} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{2} + 11 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n} \]
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Exception generated. \[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 4} \,d x } \]
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\[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 4} \,d x } \]
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Time = 0.98 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.06 \[ \int (a+b x)^n (c+d x)^{-4-n} \, dx=-\frac {x\,{\left (a+b\,x\right )}^n\,\left (a^3\,d^3\,n^2+3\,a^3\,d^3\,n+2\,a^3\,d^3-a^2\,b\,c\,d^2\,n^2-7\,a^2\,b\,c\,d^2\,n-6\,a^2\,b\,c\,d^2-a\,b^2\,c^2\,d\,n^2-a\,b^2\,c^2\,d\,n+6\,a\,b^2\,c^2\,d+b^3\,c^3\,n^2+5\,b^3\,c^3\,n+6\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^n\,\left (a^2\,d^2\,n^2+3\,a^2\,d^2\,n+2\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-8\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+b^2\,c^2\,n^2+5\,b^2\,c^2\,n+6\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {2\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^n}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n\,\left (a^2\,d^2\,n^2+a^2\,d^2\,n-2\,a\,b\,c\,d\,n^2-8\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {2\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (4\,b\,c-a\,d\,n+b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )} \]
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