Integrand size = 15, antiderivative size = 58 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {(c+d x)^8}{9 (b c-a d) (a+b x)^9}+\frac {d (c+d x)^8}{72 (b c-a d)^2 (a+b x)^8} \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=\frac {d (c+d x)^8}{72 (a+b x)^8 (b c-a d)^2}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^8}{9 (b c-a d) (a+b x)^9}-\frac {d \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{9 (b c-a d)} \\ & = -\frac {(c+d x)^8}{9 (b c-a d) (a+b x)^9}+\frac {d (c+d x)^8}{72 (b c-a d)^2 (a+b x)^8} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(367\) vs. \(2(58)=116\).
Time = 0.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 6.33 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (2 c+9 d x)+3 a^5 b^2 d^5 \left (c^2+6 c d x+12 d^2 x^2\right )+a^4 b^3 d^4 \left (4 c^3+27 c^2 d x+72 c d^2 x^2+84 d^3 x^3\right )+a^3 b^4 d^3 \left (5 c^4+36 c^3 d x+108 c^2 d^2 x^2+168 c d^3 x^3+126 d^4 x^4\right )+3 a^2 b^5 d^2 \left (2 c^5+15 c^4 d x+48 c^3 d^2 x^2+84 c^2 d^3 x^3+84 c d^4 x^4+42 d^5 x^5\right )+a b^6 d \left (7 c^6+54 c^5 d x+180 c^4 d^2 x^2+336 c^3 d^3 x^3+378 c^2 d^4 x^4+252 c d^5 x^5+84 d^6 x^6\right )+b^7 \left (8 c^7+63 c^6 d x+216 c^5 d^2 x^2+420 c^4 d^3 x^3+504 c^3 d^4 x^4+378 c^2 d^5 x^5+168 c d^6 x^6+36 d^7 x^7\right )}{72 b^8 (a+b x)^9} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(54)=108\).
Time = 0.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 7.55
method | result | size |
risch | \(\frac {-\frac {d^{7} x^{7}}{2 b}-\frac {7 d^{6} \left (a d +2 b c \right ) x^{6}}{6 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) x^{5}}{4 b^{3}}-\frac {7 d^{4} \left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}-\frac {7 d^{3} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +5 b^{4} c^{4}\right ) x^{3}}{6 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+2 a^{4} b c \,d^{4}+3 a^{3} b^{2} c^{2} d^{3}+4 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +6 b^{5} c^{5}\right ) x^{2}}{2 b^{6}}-\frac {d \left (a^{6} d^{6}+2 a^{5} b c \,d^{5}+3 a^{4} b^{2} c^{2} d^{4}+4 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+6 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x}{8 b^{7}}-\frac {a^{7} d^{7}+2 a^{6} b c \,d^{6}+3 a^{5} b^{2} c^{2} d^{5}+4 a^{4} b^{3} c^{3} d^{4}+5 a^{3} b^{4} c^{4} d^{3}+6 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +8 b^{7} c^{7}}{72 b^{8}}}{\left (b x +a \right )^{9}}\) | \(438\) |
default | \(\frac {7 d^{6} \left (a d -b c \right )}{3 b^{8} \left (b x +a \right )^{3}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{9 b^{8} \left (b x +a \right )^{9}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{8 b^{8} \left (b x +a \right )^{8}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 b^{8} \left (b x +a \right )^{4}}+\frac {3 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {d^{7}}{2 b^{8} \left (b x +a \right )^{2}}+\frac {7 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )^{5}}\) | \(464\) |
norman | \(\frac {-\frac {d^{7} x^{7}}{2 b}+\frac {7 \left (-a b \,d^{7}-2 b^{2} c \,d^{6}\right ) x^{6}}{6 b^{3}}+\frac {7 \left (-a^{2} b \,d^{7}-2 a \,b^{2} c \,d^{6}-3 b^{3} c^{2} d^{5}\right ) x^{5}}{4 b^{4}}+\frac {7 \left (-a^{3} b \,d^{7}-2 a^{2} b^{2} c \,d^{6}-3 a \,b^{3} c^{2} d^{5}-4 b^{4} c^{3} d^{4}\right ) x^{4}}{4 b^{5}}+\frac {7 \left (-a^{4} b \,d^{7}-2 a^{3} b^{2} c \,d^{6}-3 a^{2} b^{3} c^{2} d^{5}-4 a \,b^{4} c^{3} d^{4}-5 b^{5} c^{4} d^{3}\right ) x^{3}}{6 b^{6}}+\frac {\left (-a^{5} b \,d^{7}-2 a^{4} b^{2} c \,d^{6}-3 a^{3} b^{3} c^{2} d^{5}-4 a^{2} b^{4} c^{3} d^{4}-5 a \,b^{5} c^{4} d^{3}-6 b^{6} c^{5} d^{2}\right ) x^{2}}{2 b^{7}}+\frac {\left (-a^{6} b \,d^{7}-2 a^{5} b^{2} c \,d^{6}-3 a^{4} b^{3} c^{2} d^{5}-4 a^{3} b^{4} c^{3} d^{4}-5 a^{2} b^{5} c^{4} d^{3}-6 a \,b^{6} c^{5} d^{2}-7 b^{7} c^{6} d \right ) x}{8 b^{8}}+\frac {-a^{7} b \,d^{7}-2 a^{6} b^{2} c \,d^{6}-3 a^{5} b^{3} c^{2} d^{5}-4 a^{4} b^{4} c^{3} d^{4}-5 a^{3} b^{5} c^{4} d^{3}-6 a^{2} b^{6} c^{5} d^{2}-7 a \,b^{7} c^{6} d -8 b^{8} c^{7}}{72 b^{9}}}{\left (b x +a \right )^{9}}\) | \(478\) |
gosper | \(-\frac {36 x^{7} d^{7} b^{7}+84 x^{6} a \,b^{6} d^{7}+168 x^{6} b^{7} c \,d^{6}+126 x^{5} a^{2} b^{5} d^{7}+252 x^{5} a \,b^{6} c \,d^{6}+378 x^{5} b^{7} c^{2} d^{5}+126 x^{4} a^{3} b^{4} d^{7}+252 x^{4} a^{2} b^{5} c \,d^{6}+378 x^{4} a \,b^{6} c^{2} d^{5}+504 x^{4} b^{7} c^{3} d^{4}+84 x^{3} a^{4} b^{3} d^{7}+168 x^{3} a^{3} b^{4} c \,d^{6}+252 x^{3} a^{2} b^{5} c^{2} d^{5}+336 x^{3} a \,b^{6} c^{3} d^{4}+420 x^{3} b^{7} c^{4} d^{3}+36 x^{2} a^{5} b^{2} d^{7}+72 x^{2} a^{4} b^{3} c \,d^{6}+108 x^{2} a^{3} b^{4} c^{2} d^{5}+144 x^{2} a^{2} b^{5} c^{3} d^{4}+180 x^{2} a \,b^{6} c^{4} d^{3}+216 x^{2} b^{7} c^{5} d^{2}+9 x \,a^{6} b \,d^{7}+18 x \,a^{5} b^{2} c \,d^{6}+27 x \,a^{4} b^{3} c^{2} d^{5}+36 x \,a^{3} b^{4} c^{3} d^{4}+45 x \,a^{2} b^{5} c^{4} d^{3}+54 x a \,b^{6} c^{5} d^{2}+63 x \,b^{7} c^{6} d +a^{7} d^{7}+2 a^{6} b c \,d^{6}+3 a^{5} b^{2} c^{2} d^{5}+4 a^{4} b^{3} c^{3} d^{4}+5 a^{3} b^{4} c^{4} d^{3}+6 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +8 b^{7} c^{7}}{72 b^{8} \left (b x +a \right )^{9}}\) | \(497\) |
parallelrisch | \(\frac {-36 d^{7} x^{7} b^{8}-84 a \,b^{7} d^{7} x^{6}-168 b^{8} c \,d^{6} x^{6}-126 a^{2} b^{6} d^{7} x^{5}-252 a \,b^{7} c \,d^{6} x^{5}-378 b^{8} c^{2} d^{5} x^{5}-126 a^{3} b^{5} d^{7} x^{4}-252 a^{2} b^{6} c \,d^{6} x^{4}-378 a \,b^{7} c^{2} d^{5} x^{4}-504 b^{8} c^{3} d^{4} x^{4}-84 a^{4} b^{4} d^{7} x^{3}-168 a^{3} b^{5} c \,d^{6} x^{3}-252 a^{2} b^{6} c^{2} d^{5} x^{3}-336 a \,b^{7} c^{3} d^{4} x^{3}-420 b^{8} c^{4} d^{3} x^{3}-36 a^{5} b^{3} d^{7} x^{2}-72 a^{4} b^{4} c \,d^{6} x^{2}-108 a^{3} b^{5} c^{2} d^{5} x^{2}-144 a^{2} b^{6} c^{3} d^{4} x^{2}-180 a \,b^{7} c^{4} d^{3} x^{2}-216 b^{8} c^{5} d^{2} x^{2}-9 a^{6} b^{2} d^{7} x -18 a^{5} b^{3} c \,d^{6} x -27 a^{4} b^{4} c^{2} d^{5} x -36 a^{3} b^{5} c^{3} d^{4} x -45 a^{2} b^{6} c^{4} d^{3} x -54 a \,b^{7} c^{5} d^{2} x -63 b^{8} c^{6} d x -a^{7} b \,d^{7}-2 a^{6} b^{2} c \,d^{6}-3 a^{5} b^{3} c^{2} d^{5}-4 a^{4} b^{4} c^{3} d^{4}-5 a^{3} b^{5} c^{4} d^{3}-6 a^{2} b^{6} c^{5} d^{2}-7 a \,b^{7} c^{6} d -8 b^{8} c^{7}}{72 b^{9} \left (b x +a \right )^{9}}\) | \(503\) |
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Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 548, normalized size of antiderivative = 9.45 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} d^{7} x^{7} + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7} + 84 \, {\left (2 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 126 \, {\left (3 \, b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 126 \, {\left (4 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 84 \, {\left (5 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 3 \, a^{2} b^{5} c^{2} d^{5} + 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 36 \, {\left (6 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 4 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} + 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 9 \, {\left (7 \, b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 4 \, a^{3} b^{4} c^{3} d^{4} + 3 \, a^{4} b^{3} c^{2} d^{5} + 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{72 \, {\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 548, normalized size of antiderivative = 9.45 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} d^{7} x^{7} + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7} + 84 \, {\left (2 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 126 \, {\left (3 \, b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 126 \, {\left (4 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 84 \, {\left (5 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 3 \, a^{2} b^{5} c^{2} d^{5} + 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 36 \, {\left (6 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 4 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} + 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 9 \, {\left (7 \, b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 4 \, a^{3} b^{4} c^{3} d^{4} + 3 \, a^{4} b^{3} c^{2} d^{5} + 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{72 \, {\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 496, normalized size of antiderivative = 8.55 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} d^{7} x^{7} + 168 \, b^{7} c d^{6} x^{6} + 84 \, a b^{6} d^{7} x^{6} + 378 \, b^{7} c^{2} d^{5} x^{5} + 252 \, a b^{6} c d^{6} x^{5} + 126 \, a^{2} b^{5} d^{7} x^{5} + 504 \, b^{7} c^{3} d^{4} x^{4} + 378 \, a b^{6} c^{2} d^{5} x^{4} + 252 \, a^{2} b^{5} c d^{6} x^{4} + 126 \, a^{3} b^{4} d^{7} x^{4} + 420 \, b^{7} c^{4} d^{3} x^{3} + 336 \, a b^{6} c^{3} d^{4} x^{3} + 252 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 168 \, a^{3} b^{4} c d^{6} x^{3} + 84 \, a^{4} b^{3} d^{7} x^{3} + 216 \, b^{7} c^{5} d^{2} x^{2} + 180 \, a b^{6} c^{4} d^{3} x^{2} + 144 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 108 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 72 \, a^{4} b^{3} c d^{6} x^{2} + 36 \, a^{5} b^{2} d^{7} x^{2} + 63 \, b^{7} c^{6} d x + 54 \, a b^{6} c^{5} d^{2} x + 45 \, a^{2} b^{5} c^{4} d^{3} x + 36 \, a^{3} b^{4} c^{3} d^{4} x + 27 \, a^{4} b^{3} c^{2} d^{5} x + 18 \, a^{5} b^{2} c d^{6} x + 9 \, a^{6} b d^{7} x + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7}}{72 \, {\left (b x + a\right )}^{9} b^{8}} \]
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Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=\frac {{\left (c+d\,x\right )}^8\,\left (9\,a\,d-8\,b\,c+b\,d\,x\right )}{72\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^9} \]
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