Integrand size = 15, antiderivative size = 262 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {120 d^7 (b c-a d)^3 x}{b^{10}}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{3 b^{11}}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}} \]
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Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac {45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {120 d^7 x (b c-a d)^3}{b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {120 d^7 (b c-a d)^3}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^7}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^6}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^5}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^4}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^3}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^2}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^2}{b^{10}}+\frac {d^{10} (a+b x)^3}{b^{10}}\right ) \, dx \\ & = \frac {120 d^7 (b c-a d)^3 x}{b^{10}}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{3 b^{11}}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {12 b d^7 \left (120 b^3 c^3-315 a b^2 c^2 d+280 a^2 b c d^2-84 a^3 d^3\right ) x+6 b^2 d^8 \left (45 b^2 c^2-70 a b c d+28 a^2 d^2\right ) x^2+4 b^3 d^9 (10 b c-7 a d) x^3+3 b^4 d^{10} x^4-\frac {2 (b c-a d)^{10}}{(a+b x)^6}+\frac {24 d (-b c+a d)^9}{(a+b x)^5}-\frac {135 d^2 (b c-a d)^8}{(a+b x)^4}+\frac {480 d^3 (-b c+a d)^7}{(a+b x)^3}-\frac {1260 d^4 (b c-a d)^6}{(a+b x)^2}+\frac {3024 d^5 (-b c+a d)^5}{a+b x}+2520 d^6 (b c-a d)^4 \log (a+b x)}{12 b^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(849\) vs. \(2(252)=504\).
Time = 0.24 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.24
method | result | size |
norman | \(\frac {\frac {6174 a^{10} d^{10}-24696 a^{9} b c \,d^{9}+37044 a^{8} b^{2} c^{2} d^{8}-24696 a^{7} b^{3} c^{3} d^{7}+6174 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-84 a^{4} b^{6} c^{6} d^{4}-24 a^{3} b^{7} c^{7} d^{3}-9 a^{2} b^{8} c^{8} d^{2}-4 a \,b^{9} c^{9} d -2 b^{10} c^{10}}{12 b^{11}}+\frac {d^{10} x^{10}}{4 b}+\frac {6 \left (210 a^{5} d^{10}-840 a^{4} b c \,d^{9}+1260 a^{3} b^{2} c^{2} d^{8}-840 a^{2} b^{3} c^{3} d^{7}+210 a \,b^{4} c^{4} d^{6}-42 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}+\frac {15 \left (315 a^{6} d^{10}-1260 a^{5} b c \,d^{9}+1890 a^{4} b^{2} c^{2} d^{8}-1260 a^{3} b^{3} c^{3} d^{7}+315 a^{2} b^{4} c^{4} d^{6}-42 a \,b^{5} c^{5} d^{5}-7 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}+\frac {20 \left (385 a^{7} d^{10}-1540 a^{6} b c \,d^{9}+2310 a^{5} b^{2} c^{2} d^{8}-1540 a^{4} b^{3} c^{3} d^{7}+385 a^{3} b^{4} c^{4} d^{6}-42 a^{2} b^{5} c^{5} d^{5}-7 a \,b^{6} c^{6} d^{4}-2 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {15 \left (1750 a^{8} d^{10}-7000 a^{7} b c \,d^{9}+10500 a^{6} b^{2} c^{2} d^{8}-7000 a^{5} b^{3} c^{3} d^{7}+1750 a^{4} b^{4} c^{4} d^{6}-168 a^{3} b^{5} c^{5} d^{5}-28 a^{2} b^{6} c^{6} d^{4}-8 a \,b^{7} c^{7} d^{3}-3 b^{8} c^{8} d^{2}\right ) x^{2}}{4 b^{9}}+\frac {\left (5754 a^{9} d^{10}-23016 a^{8} b c \,d^{9}+34524 a^{7} b^{2} c^{2} d^{8}-23016 a^{6} b^{3} c^{3} d^{7}+5754 a^{5} b^{4} c^{4} d^{6}-504 a^{4} b^{5} c^{5} d^{5}-84 a^{3} b^{6} c^{6} d^{4}-24 a^{2} b^{7} c^{7} d^{3}-9 a \,b^{8} c^{8} d^{2}-4 b^{9} c^{9} d \right ) x}{2 b^{10}}-\frac {30 d^{7} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {15 d^{8} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {5 d^{9} \left (a d -4 b c \right ) x^{9}}{6 b^{2}}}{\left (b x +a \right )^{6}}+\frac {210 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(850\) |
default | \(-\frac {d^{7} \left (-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {7}{3} x^{3} a \,b^{2} d^{3}-\frac {10}{3} x^{3} b^{3} c \,d^{2}-14 x^{2} a^{2} b \,d^{3}+35 x^{2} a \,b^{2} c \,d^{2}-\frac {45}{2} x^{2} b^{3} c^{2} d +84 a^{3} d^{3} x -280 a^{2} b c \,d^{2} x +315 a \,b^{2} c^{2} d x -120 b^{3} c^{3} x \right )}{b^{10}}+\frac {40 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{3}}+\frac {210 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{6 b^{11} \left (b x +a \right )^{6}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{4 b^{11} \left (b x +a \right )^{4}}-\frac {105 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{11} \left (b x +a \right )^{2}}+\frac {2 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{b^{11} \left (b x +a \right )^{5}}+\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{11} \left (b x +a \right )}\) | \(862\) |
risch | \(\frac {d^{10} x^{4}}{4 b^{7}}-\frac {7 d^{10} x^{3} a}{3 b^{8}}+\frac {10 d^{9} x^{3} c}{3 b^{7}}+\frac {14 d^{10} x^{2} a^{2}}{b^{9}}-\frac {35 d^{9} x^{2} a c}{b^{8}}+\frac {45 d^{8} x^{2} c^{2}}{2 b^{7}}-\frac {84 d^{10} a^{3} x}{b^{10}}+\frac {280 d^{9} a^{2} c x}{b^{9}}-\frac {315 d^{8} a \,c^{2} x}{b^{8}}+\frac {120 d^{7} c^{3} x}{b^{7}}+\frac {\left (252 a^{5} b^{4} d^{10}-1260 a^{4} b^{5} c \,d^{9}+2520 a^{3} b^{6} c^{2} d^{8}-2520 a^{2} b^{7} c^{3} d^{7}+1260 a \,b^{8} c^{4} d^{6}-252 b^{9} c^{5} d^{5}\right ) x^{5}+105 b^{3} d^{4} \left (11 a^{6} d^{6}-54 a^{5} b c \,d^{5}+105 a^{4} b^{2} c^{2} d^{4}-100 a^{3} b^{3} c^{3} d^{3}+45 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d -b^{6} c^{6}\right ) x^{4}+20 b^{2} d^{3} \left (107 a^{7} d^{7}-518 a^{6} b c \,d^{6}+987 a^{5} b^{2} c^{2} d^{5}-910 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}\right ) x^{3}+\frac {15 b \,d^{2} \left (533 a^{8} d^{8}-2552 a^{7} b c \,d^{7}+4788 a^{6} b^{2} c^{2} d^{6}-4312 a^{5} b^{3} c^{3} d^{5}+1750 a^{4} b^{4} c^{4} d^{4}-168 a^{3} b^{5} c^{5} d^{3}-28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d -3 b^{8} c^{8}\right ) x^{2}}{4}+\frac {d \left (1879 a^{9} d^{9}-8916 a^{8} b c \,d^{8}+16524 a^{7} b^{2} c^{2} d^{7}-14616 a^{6} b^{3} c^{3} d^{6}+5754 a^{5} b^{4} c^{4} d^{5}-504 a^{4} b^{5} c^{5} d^{4}-84 a^{3} b^{6} c^{6} d^{3}-24 a^{2} b^{7} c^{7} d^{2}-9 a \,b^{8} c^{8} d -4 b^{9} c^{9}\right ) x}{2}+\frac {2131 a^{10} d^{10}-10036 a^{9} b c \,d^{9}+18414 a^{8} b^{2} c^{2} d^{8}-16056 a^{7} b^{3} c^{3} d^{7}+6174 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-84 a^{4} b^{6} c^{6} d^{4}-24 a^{3} b^{7} c^{7} d^{3}-9 a^{2} b^{8} c^{8} d^{2}-4 a \,b^{9} c^{9} d -2 b^{10} c^{10}}{12 b}}{b^{10} \left (b x +a \right )^{6}}+\frac {210 d^{10} \ln \left (b x +a \right ) a^{4}}{b^{11}}-\frac {840 d^{9} \ln \left (b x +a \right ) a^{3} c}{b^{10}}+\frac {1260 d^{8} \ln \left (b x +a \right ) a^{2} c^{2}}{b^{9}}-\frac {840 d^{7} \ln \left (b x +a \right ) a \,c^{3}}{b^{8}}+\frac {210 d^{6} \ln \left (b x +a \right ) c^{4}}{b^{7}}\) | \(884\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1619\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1386 vs. \(2 (252) = 504\).
Time = 0.24 (sec) , antiderivative size = 1386, normalized size of antiderivative = 5.29 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (252) = 504\).
Time = 0.26 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.53 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=-\frac {2 \, b^{10} c^{10} + 4 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} - 6174 \, a^{6} b^{4} c^{4} d^{6} + 16056 \, a^{7} b^{3} c^{3} d^{7} - 18414 \, a^{8} b^{2} c^{2} d^{8} + 10036 \, a^{9} b c d^{9} - 2131 \, a^{10} d^{10} + 3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 1260 \, {\left (b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} - 45 \, a^{2} b^{8} c^{4} d^{6} + 100 \, a^{3} b^{7} c^{3} d^{7} - 105 \, a^{4} b^{6} c^{2} d^{8} + 54 \, a^{5} b^{5} c d^{9} - 11 \, a^{6} b^{4} d^{10}\right )} x^{4} + 240 \, {\left (2 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 42 \, a^{2} b^{8} c^{5} d^{5} - 385 \, a^{3} b^{7} c^{4} d^{6} + 910 \, a^{4} b^{6} c^{3} d^{7} - 987 \, a^{5} b^{5} c^{2} d^{8} + 518 \, a^{6} b^{4} c d^{9} - 107 \, a^{7} b^{3} d^{10}\right )} x^{3} + 45 \, {\left (3 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} - 1750 \, a^{4} b^{6} c^{4} d^{6} + 4312 \, a^{5} b^{5} c^{3} d^{7} - 4788 \, a^{6} b^{4} c^{2} d^{8} + 2552 \, a^{7} b^{3} c d^{9} - 533 \, a^{8} b^{2} d^{10}\right )} x^{2} + 6 \, {\left (4 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} - 5754 \, a^{5} b^{5} c^{4} d^{6} + 14616 \, a^{6} b^{4} c^{3} d^{7} - 16524 \, a^{7} b^{3} c^{2} d^{8} + 8916 \, a^{8} b^{2} c d^{9} - 1879 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b^{17} x^{6} + 6 \, a b^{16} x^{5} + 15 \, a^{2} b^{15} x^{4} + 20 \, a^{3} b^{14} x^{3} + 15 \, a^{4} b^{13} x^{2} + 6 \, a^{5} b^{12} x + a^{6} b^{11}\right )}} + \frac {3 \, b^{3} d^{10} x^{4} + 4 \, {\left (10 \, b^{3} c d^{9} - 7 \, a b^{2} d^{10}\right )} x^{3} + 6 \, {\left (45 \, b^{3} c^{2} d^{8} - 70 \, a b^{2} c d^{9} + 28 \, a^{2} b d^{10}\right )} x^{2} + 12 \, {\left (120 \, b^{3} c^{3} d^{7} - 315 \, a b^{2} c^{2} d^{8} + 280 \, a^{2} b c d^{9} - 84 \, a^{3} d^{10}\right )} x}{12 \, b^{10}} + \frac {210 \, {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (252) = 504\).
Time = 0.37 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.35 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {210 \, {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {2 \, b^{10} c^{10} + 4 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} - 6174 \, a^{6} b^{4} c^{4} d^{6} + 16056 \, a^{7} b^{3} c^{3} d^{7} - 18414 \, a^{8} b^{2} c^{2} d^{8} + 10036 \, a^{9} b c d^{9} - 2131 \, a^{10} d^{10} + 3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 1260 \, {\left (b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} - 45 \, a^{2} b^{8} c^{4} d^{6} + 100 \, a^{3} b^{7} c^{3} d^{7} - 105 \, a^{4} b^{6} c^{2} d^{8} + 54 \, a^{5} b^{5} c d^{9} - 11 \, a^{6} b^{4} d^{10}\right )} x^{4} + 240 \, {\left (2 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 42 \, a^{2} b^{8} c^{5} d^{5} - 385 \, a^{3} b^{7} c^{4} d^{6} + 910 \, a^{4} b^{6} c^{3} d^{7} - 987 \, a^{5} b^{5} c^{2} d^{8} + 518 \, a^{6} b^{4} c d^{9} - 107 \, a^{7} b^{3} d^{10}\right )} x^{3} + 45 \, {\left (3 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} - 1750 \, a^{4} b^{6} c^{4} d^{6} + 4312 \, a^{5} b^{5} c^{3} d^{7} - 4788 \, a^{6} b^{4} c^{2} d^{8} + 2552 \, a^{7} b^{3} c d^{9} - 533 \, a^{8} b^{2} d^{10}\right )} x^{2} + 6 \, {\left (4 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} - 5754 \, a^{5} b^{5} c^{4} d^{6} + 14616 \, a^{6} b^{4} c^{3} d^{7} - 16524 \, a^{7} b^{3} c^{2} d^{8} + 8916 \, a^{8} b^{2} c d^{9} - 1879 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b x + a\right )}^{6} b^{11}} + \frac {3 \, b^{21} d^{10} x^{4} + 40 \, b^{21} c d^{9} x^{3} - 28 \, a b^{20} d^{10} x^{3} + 270 \, b^{21} c^{2} d^{8} x^{2} - 420 \, a b^{20} c d^{9} x^{2} + 168 \, a^{2} b^{19} d^{10} x^{2} + 1440 \, b^{21} c^{3} d^{7} x - 3780 \, a b^{20} c^{2} d^{8} x + 3360 \, a^{2} b^{19} c d^{9} x - 1008 \, a^{3} b^{18} d^{10} x}{12 \, b^{28}} \]
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Time = 0.62 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=x^2\,\left (\frac {7\,a\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{2\,b}-\frac {21\,a^2\,d^{10}}{2\,b^9}+\frac {45\,c^2\,d^8}{2\,b^7}\right )-\frac {x^4\,\left (-1155\,a^6\,b^3\,d^{10}+5670\,a^5\,b^4\,c\,d^9-11025\,a^4\,b^5\,c^2\,d^8+10500\,a^3\,b^6\,c^3\,d^7-4725\,a^2\,b^7\,c^4\,d^6+630\,a\,b^8\,c^5\,d^5+105\,b^9\,c^6\,d^4\right )+\frac {-2131\,a^{10}\,d^{10}+10036\,a^9\,b\,c\,d^9-18414\,a^8\,b^2\,c^2\,d^8+16056\,a^7\,b^3\,c^3\,d^7-6174\,a^6\,b^4\,c^4\,d^6+504\,a^5\,b^5\,c^5\,d^5+84\,a^4\,b^6\,c^6\,d^4+24\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+4\,a\,b^9\,c^9\,d+2\,b^{10}\,c^{10}}{12\,b}+x\,\left (-\frac {1879\,a^9\,d^{10}}{2}+4458\,a^8\,b\,c\,d^9-8262\,a^7\,b^2\,c^2\,d^8+7308\,a^6\,b^3\,c^3\,d^7-2877\,a^5\,b^4\,c^4\,d^6+252\,a^4\,b^5\,c^5\,d^5+42\,a^3\,b^6\,c^6\,d^4+12\,a^2\,b^7\,c^7\,d^3+\frac {9\,a\,b^8\,c^8\,d^2}{2}+2\,b^9\,c^9\,d\right )+x^3\,\left (-2140\,a^7\,b^2\,d^{10}+10360\,a^6\,b^3\,c\,d^9-19740\,a^5\,b^4\,c^2\,d^8+18200\,a^4\,b^5\,c^3\,d^7-7700\,a^3\,b^6\,c^4\,d^6+840\,a^2\,b^7\,c^5\,d^5+140\,a\,b^8\,c^6\,d^4+40\,b^9\,c^7\,d^3\right )+x^2\,\left (-\frac {7995\,a^8\,b\,d^{10}}{4}+9570\,a^7\,b^2\,c\,d^9-17955\,a^6\,b^3\,c^2\,d^8+16170\,a^5\,b^4\,c^3\,d^7-\frac {13125\,a^4\,b^5\,c^4\,d^6}{2}+630\,a^3\,b^6\,c^5\,d^5+105\,a^2\,b^7\,c^6\,d^4+30\,a\,b^8\,c^7\,d^3+\frac {45\,b^9\,c^8\,d^2}{4}\right )-x^5\,\left (252\,a^5\,b^4\,d^{10}-1260\,a^4\,b^5\,c\,d^9+2520\,a^3\,b^6\,c^2\,d^8-2520\,a^2\,b^7\,c^3\,d^7+1260\,a\,b^8\,c^4\,d^6-252\,b^9\,c^5\,d^5\right )}{a^6\,b^{10}+6\,a^5\,b^{11}\,x+15\,a^4\,b^{12}\,x^2+20\,a^3\,b^{13}\,x^3+15\,a^2\,b^{14}\,x^4+6\,a\,b^{15}\,x^5+b^{16}\,x^6}-x^3\,\left (\frac {7\,a\,d^{10}}{3\,b^8}-\frac {10\,c\,d^9}{3\,b^7}\right )-x\,\left (\frac {7\,a\,\left (\frac {7\,a\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{b}-\frac {21\,a^2\,d^{10}}{b^9}+\frac {45\,c^2\,d^8}{b^7}\right )}{b}+\frac {35\,a^3\,d^{10}}{b^{10}}-\frac {120\,c^3\,d^7}{b^7}-\frac {21\,a^2\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (210\,a^4\,d^{10}-840\,a^3\,b\,c\,d^9+1260\,a^2\,b^2\,c^2\,d^8-840\,a\,b^3\,c^3\,d^7+210\,b^4\,c^4\,d^6\right )}{b^{11}}+\frac {d^{10}\,x^4}{4\,b^7} \]
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