Integrand size = 15, antiderivative size = 28 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {(c+d x)^8}{8 (b c-a d) (a+b x)^8} \]
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Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {(c+d x)^8}{8 (a+b x)^8 (b c-a d)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^8}{8 (b c-a d) (a+b x)^8} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(353\) vs. \(2(28)=56\).
Time = 0.07 (sec) , antiderivative size = 353, normalized size of antiderivative = 12.61 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {a^7 d^7+a^6 b d^6 (c+8 d x)+a^5 b^2 d^5 \left (c^2+8 c d x+28 d^2 x^2\right )+a^4 b^3 d^4 \left (c^3+8 c^2 d x+28 c d^2 x^2+56 d^3 x^3\right )+a^3 b^4 d^3 \left (c^4+8 c^3 d x+28 c^2 d^2 x^2+56 c d^3 x^3+70 d^4 x^4\right )+a^2 b^5 d^2 \left (c^5+8 c^4 d x+28 c^3 d^2 x^2+56 c^2 d^3 x^3+70 c d^4 x^4+56 d^5 x^5\right )+a b^6 d \left (c^6+8 c^5 d x+28 c^4 d^2 x^2+56 c^3 d^3 x^3+70 c^2 d^4 x^4+56 c d^5 x^5+28 d^6 x^6\right )+b^7 \left (c^7+8 c^6 d x+28 c^5 d^2 x^2+56 c^4 d^3 x^3+70 c^3 d^4 x^4+56 c^2 d^5 x^5+28 c d^6 x^6+8 d^7 x^7\right )}{8 b^8 (a+b x)^8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(409\) vs. \(2(26)=52\).
Time = 0.22 (sec) , antiderivative size = 410, normalized size of antiderivative = 14.64
method | result | size |
risch | \(\frac {-\frac {d^{7} x^{7}}{b}-\frac {7 d^{6} \left (a d +b c \right ) x^{6}}{2 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{5}}{b^{3}}-\frac {35 d^{4} \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{4 b^{4}}-\frac {7 d^{3} \left (a^{4} d^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{3}}{b^{5}}-\frac {7 d^{2} \left (a^{5} d^{5}+a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{2}}{2 b^{6}}-\frac {d \left (a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x}{b^{7}}-\frac {a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}}{8 b^{8}}}{\left (b x +a \right )^{8}}\) | \(410\) |
norman | \(\frac {-\frac {d^{7} x^{7}}{b}+\frac {7 \left (-a \,d^{7}-b c \,d^{6}\right ) x^{6}}{2 b^{2}}+\frac {7 \left (-a^{2} d^{7}-a b c \,d^{6}-b^{2} c^{2} d^{5}\right ) x^{5}}{b^{3}}+\frac {35 \left (-a^{3} d^{7}-a^{2} b c \,d^{6}-a \,b^{2} c^{2} d^{5}-b^{3} c^{3} d^{4}\right ) x^{4}}{4 b^{4}}+\frac {7 \left (-a^{4} d^{7}-a^{3} b c \,d^{6}-a^{2} b^{2} c^{2} d^{5}-a \,b^{3} c^{3} d^{4}-b^{4} c^{4} d^{3}\right ) x^{3}}{b^{5}}+\frac {7 \left (-a^{5} d^{7}-a^{4} b c \,d^{6}-a^{3} b^{2} c^{2} d^{5}-a^{2} b^{3} c^{3} d^{4}-a \,b^{4} c^{4} d^{3}-b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}+\frac {\left (-a^{6} d^{7}-a^{5} b c \,d^{6}-a^{4} b^{2} c^{2} d^{5}-a^{3} b^{3} c^{3} d^{4}-a^{2} b^{4} c^{4} d^{3}-a \,b^{5} c^{5} d^{2}-b^{6} c^{6} d \right ) x}{b^{7}}+\frac {-a^{7} d^{7}-a^{6} b c \,d^{6}-a^{5} b^{2} c^{2} d^{5}-a^{4} b^{3} c^{3} d^{4}-a^{3} b^{4} c^{4} d^{3}-a^{2} b^{5} c^{5} d^{2}-a \,b^{6} c^{6} d -b^{7} c^{7}}{8 b^{8}}}{\left (b x +a \right )^{8}}\) | \(456\) |
default | \(-\frac {7 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )^{3}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{2 b^{8} \left (b x +a \right )^{6}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{8 b^{8} \left (b x +a \right )^{8}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{8} \left (b x +a \right )^{7}}+\frac {7 d^{6} \left (a d -b c \right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {7 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{8} \left (b x +a \right )^{5}}-\frac {d^{7}}{b^{8} \left (b x +a \right )}\) | \(464\) |
gosper | \(-\frac {8 x^{7} d^{7} b^{7}+28 x^{6} a \,b^{6} d^{7}+28 x^{6} b^{7} c \,d^{6}+56 x^{5} a^{2} b^{5} d^{7}+56 x^{5} a \,b^{6} c \,d^{6}+56 x^{5} b^{7} c^{2} d^{5}+70 x^{4} a^{3} b^{4} d^{7}+70 x^{4} a^{2} b^{5} c \,d^{6}+70 x^{4} a \,b^{6} c^{2} d^{5}+70 x^{4} b^{7} c^{3} d^{4}+56 x^{3} a^{4} b^{3} d^{7}+56 x^{3} a^{3} b^{4} c \,d^{6}+56 x^{3} a^{2} b^{5} c^{2} d^{5}+56 x^{3} a \,b^{6} c^{3} d^{4}+56 x^{3} b^{7} c^{4} d^{3}+28 x^{2} a^{5} b^{2} d^{7}+28 x^{2} a^{4} b^{3} c \,d^{6}+28 x^{2} a^{3} b^{4} c^{2} d^{5}+28 x^{2} a^{2} b^{5} c^{3} d^{4}+28 x^{2} a \,b^{6} c^{4} d^{3}+28 x^{2} b^{7} c^{5} d^{2}+8 x \,a^{6} b \,d^{7}+8 x \,a^{5} b^{2} c \,d^{6}+8 x \,a^{4} b^{3} c^{2} d^{5}+8 x \,a^{3} b^{4} c^{3} d^{4}+8 x \,a^{2} b^{5} c^{4} d^{3}+8 x a \,b^{6} c^{5} d^{2}+8 x \,b^{7} c^{6} d +a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}}{8 \left (b x +a \right )^{8} b^{8}}\) | \(490\) |
parallelrisch | \(\frac {-8 x^{7} d^{7} b^{7}-28 x^{6} a \,b^{6} d^{7}-28 x^{6} b^{7} c \,d^{6}-56 x^{5} a^{2} b^{5} d^{7}-56 x^{5} a \,b^{6} c \,d^{6}-56 x^{5} b^{7} c^{2} d^{5}-70 x^{4} a^{3} b^{4} d^{7}-70 x^{4} a^{2} b^{5} c \,d^{6}-70 x^{4} a \,b^{6} c^{2} d^{5}-70 x^{4} b^{7} c^{3} d^{4}-56 x^{3} a^{4} b^{3} d^{7}-56 x^{3} a^{3} b^{4} c \,d^{6}-56 x^{3} a^{2} b^{5} c^{2} d^{5}-56 x^{3} a \,b^{6} c^{3} d^{4}-56 x^{3} b^{7} c^{4} d^{3}-28 x^{2} a^{5} b^{2} d^{7}-28 x^{2} a^{4} b^{3} c \,d^{6}-28 x^{2} a^{3} b^{4} c^{2} d^{5}-28 x^{2} a^{2} b^{5} c^{3} d^{4}-28 x^{2} a \,b^{6} c^{4} d^{3}-28 x^{2} b^{7} c^{5} d^{2}-8 x \,a^{6} b \,d^{7}-8 x \,a^{5} b^{2} c \,d^{6}-8 x \,a^{4} b^{3} c^{2} d^{5}-8 x \,a^{3} b^{4} c^{3} d^{4}-8 x \,a^{2} b^{5} c^{4} d^{3}-8 x a \,b^{6} c^{5} d^{2}-8 x \,b^{7} c^{6} d -a^{7} d^{7}-a^{6} b c \,d^{6}-a^{5} b^{2} c^{2} d^{5}-a^{4} b^{3} c^{3} d^{4}-a^{3} b^{4} c^{4} d^{3}-a^{2} b^{5} c^{5} d^{2}-a \,b^{6} c^{6} d -b^{7} c^{7}}{8 b^{8} \left (b x +a \right )^{8}}\) | \(498\) |
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Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (26) = 52\).
Time = 0.22 (sec) , antiderivative size = 509, normalized size of antiderivative = 18.18 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {8 \, b^{7} d^{7} x^{7} + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7} + 28 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 56 \, {\left (b^{7} c^{2} d^{5} + a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (b^{7} c^{3} d^{4} + a b^{6} c^{2} d^{5} + a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 56 \, {\left (b^{7} c^{4} d^{3} + a b^{6} c^{3} d^{4} + a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 28 \, {\left (b^{7} c^{5} d^{2} + a b^{6} c^{4} d^{3} + a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 8 \, {\left (b^{7} c^{6} d + a b^{6} c^{5} d^{2} + a^{2} b^{5} c^{4} d^{3} + a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{8 \, {\left (b^{16} x^{8} + 8 \, a b^{15} x^{7} + 28 \, a^{2} b^{14} x^{6} + 56 \, a^{3} b^{13} x^{5} + 70 \, a^{4} b^{12} x^{4} + 56 \, a^{5} b^{11} x^{3} + 28 \, a^{6} b^{10} x^{2} + 8 \, a^{7} b^{9} x + a^{8} b^{8}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (26) = 52\).
Time = 0.24 (sec) , antiderivative size = 509, normalized size of antiderivative = 18.18 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {8 \, b^{7} d^{7} x^{7} + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7} + 28 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 56 \, {\left (b^{7} c^{2} d^{5} + a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (b^{7} c^{3} d^{4} + a b^{6} c^{2} d^{5} + a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 56 \, {\left (b^{7} c^{4} d^{3} + a b^{6} c^{3} d^{4} + a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 28 \, {\left (b^{7} c^{5} d^{2} + a b^{6} c^{4} d^{3} + a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 8 \, {\left (b^{7} c^{6} d + a b^{6} c^{5} d^{2} + a^{2} b^{5} c^{4} d^{3} + a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{8 \, {\left (b^{16} x^{8} + 8 \, a b^{15} x^{7} + 28 \, a^{2} b^{14} x^{6} + 56 \, a^{3} b^{13} x^{5} + 70 \, a^{4} b^{12} x^{4} + 56 \, a^{5} b^{11} x^{3} + 28 \, a^{6} b^{10} x^{2} + 8 \, a^{7} b^{9} x + a^{8} b^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 489, normalized size of antiderivative = 17.46 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {8 \, b^{7} d^{7} x^{7} + 28 \, b^{7} c d^{6} x^{6} + 28 \, a b^{6} d^{7} x^{6} + 56 \, b^{7} c^{2} d^{5} x^{5} + 56 \, a b^{6} c d^{6} x^{5} + 56 \, a^{2} b^{5} d^{7} x^{5} + 70 \, b^{7} c^{3} d^{4} x^{4} + 70 \, a b^{6} c^{2} d^{5} x^{4} + 70 \, a^{2} b^{5} c d^{6} x^{4} + 70 \, a^{3} b^{4} d^{7} x^{4} + 56 \, b^{7} c^{4} d^{3} x^{3} + 56 \, a b^{6} c^{3} d^{4} x^{3} + 56 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 56 \, a^{3} b^{4} c d^{6} x^{3} + 56 \, a^{4} b^{3} d^{7} x^{3} + 28 \, b^{7} c^{5} d^{2} x^{2} + 28 \, a b^{6} c^{4} d^{3} x^{2} + 28 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 28 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 28 \, a^{4} b^{3} c d^{6} x^{2} + 28 \, a^{5} b^{2} d^{7} x^{2} + 8 \, b^{7} c^{6} d x + 8 \, a b^{6} c^{5} d^{2} x + 8 \, a^{2} b^{5} c^{4} d^{3} x + 8 \, a^{3} b^{4} c^{3} d^{4} x + 8 \, a^{4} b^{3} c^{2} d^{5} x + 8 \, a^{5} b^{2} c d^{6} x + 8 \, a^{6} b d^{7} x + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7}}{8 \, {\left (b x + a\right )}^{8} b^{8}} \]
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Time = 0.20 (sec) , antiderivative size = 571, normalized size of antiderivative = 20.39 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {a^7\,d^7+a^6\,b\,c\,d^6+8\,a^6\,b\,d^7\,x+a^5\,b^2\,c^2\,d^5+8\,a^5\,b^2\,c\,d^6\,x+28\,a^5\,b^2\,d^7\,x^2+a^4\,b^3\,c^3\,d^4+8\,a^4\,b^3\,c^2\,d^5\,x+28\,a^4\,b^3\,c\,d^6\,x^2+56\,a^4\,b^3\,d^7\,x^3+a^3\,b^4\,c^4\,d^3+8\,a^3\,b^4\,c^3\,d^4\,x+28\,a^3\,b^4\,c^2\,d^5\,x^2+56\,a^3\,b^4\,c\,d^6\,x^3+70\,a^3\,b^4\,d^7\,x^4+a^2\,b^5\,c^5\,d^2+8\,a^2\,b^5\,c^4\,d^3\,x+28\,a^2\,b^5\,c^3\,d^4\,x^2+56\,a^2\,b^5\,c^2\,d^5\,x^3+70\,a^2\,b^5\,c\,d^6\,x^4+56\,a^2\,b^5\,d^7\,x^5+a\,b^6\,c^6\,d+8\,a\,b^6\,c^5\,d^2\,x+28\,a\,b^6\,c^4\,d^3\,x^2+56\,a\,b^6\,c^3\,d^4\,x^3+70\,a\,b^6\,c^2\,d^5\,x^4+56\,a\,b^6\,c\,d^6\,x^5+28\,a\,b^6\,d^7\,x^6+b^7\,c^7+8\,b^7\,c^6\,d\,x+28\,b^7\,c^5\,d^2\,x^2+56\,b^7\,c^4\,d^3\,x^3+70\,b^7\,c^3\,d^4\,x^4+56\,b^7\,c^2\,d^5\,x^5+28\,b^7\,c\,d^6\,x^6+8\,b^7\,d^7\,x^7}{8\,a^8\,b^8+64\,a^7\,b^9\,x+224\,a^6\,b^{10}\,x^2+448\,a^5\,b^{11}\,x^3+560\,a^4\,b^{12}\,x^4+448\,a^3\,b^{13}\,x^5+224\,a^2\,b^{14}\,x^6+64\,a\,b^{15}\,x^7+8\,b^{16}\,x^8} \]
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